{"id":116868,"date":"2026-10-22T07:17:52","date_gmt":"2026-10-21T23:17:52","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/116868.html"},"modified":"2026-10-22T07:17:52","modified_gmt":"2026-10-21T23:17:52","slug":"%e7%ac%ac8%e8%ae%b2%ef%bc%9a%e5%9b%be","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/116868.html","title":{"rendered":"\u7b2c8\u8bb2\uff1a\u56fe"},"content":{"rendered":"<h3>\u76ee\u5f55<\/h3>\n<ul>\n<li>1. \u56fe\u7684\u57fa\u672c\u6982\u5ff5<\/li>\n<li>2. \u56fe\u7684\u5b58\u50a8\u7ed3\u6784<\/li>\n<li>3. \u56fe\u7684\u904d\u5386<\/li>\n<li>4. \u6700\u5c0f\u751f\u6210\u6811<\/li>\n<li>5. \u6700\u77ed\u8def\u5f84\u95ee\u9898<\/li>\n<\/ul>\n<h3>1. \u56fe\u7684\u57fa\u672c\u6982\u5ff5<\/h3>\n<p>\u56fe\u662f\u7531\u9876\u70b9\u96c6\u5408\u53ca\u9876\u70b9\u95f4\u7684\u5173\u7cfb\u7ec4\u6210\u7684\u4e00\u79cd\u6570\u636e\u7ed3\u6784&#xff1a;G &#061; (V, E)&#xff0c;\u5176\u4e2d&#xff1a;<\/p>\n<ul>\n<li>\u9876\u70b9\u96c6\u5408 V &#061; {x | x \u5c5e\u4e8e\u67d0\u4e2a\u6570\u636e\u5bf9\u8c61\u96c6} \u662f\u6709\u7a77\u975e\u7a7a\u96c6\u5408\u3002<\/li>\n<li>E &#061; {(x, y) | x, y \u5c5e\u4e8e V} \u6216\u8005 E &#061; {&lt;x, y&gt; | x, y \u5c5e\u4e8e V &amp;&amp; Path(x, y)} \u662f\u9876\u70b9\u95f4\u5173\u7cfb\u7684\u6709\u7a77\u96c6\u5408&#xff0c;\u4e5f\u53eb\u505a\u8fb9\u7684\u96c6\u5408\u3002<\/li>\n<\/ul>\n<p>(x, y) \u8868\u793a x \u5230 y \u7684\u4e00\u6761\u53cc\u5411\u901a\u8def&#xff0c;\u5373 (x, y) \u662f\u65e0\u65b9\u5411\u7684&#xff1b;Path(x, y) \u8868\u793a\u4ece x \u5230 y \u7684\u4e00\u6761\u5355\u5411\u901a\u8def&#xff0c;\u5373 Path(x, y) \u662f\u6709\u65b9\u5411\u7684\u3002<\/p>\n<p>\u9876\u70b9\u548c\u8fb9&#xff1a; \u56fe\u4e2d\u7ed3\u70b9\u79f0\u4e3a\u9876\u70b9&#xff0c;\u7b2c i \u4e2a\u9876\u70b9\u8bb0\u4f5c v_i\u3002\u4e24\u4e2a\u9876\u70b9 v_i \u548c v_j \u76f8\u5173\u8054&#xff0c;\u79f0\u4f5c\u9876\u70b9 v_i \u548c\u9876\u70b9 v_j \u4e4b\u95f4\u6709\u4e00\u6761\u8fb9&#xff0c;\u56fe\u4e2d\u7684\u7b2c k \u6761\u8fb9\u8bb0\u4f5c e_k&#xff0c;e_k &#061; (v_i, v_j) \u6216 &lt;v_i, v_j&gt;\u3002<\/p>\n<p>\u6709\u5411\u56fe\u548c\u65e0\u5411\u56fe&#xff1a; \u5728\u6709\u5411\u56fe\u4e2d&#xff0c;\u9876\u70b9\u5bf9 &lt;x, y&gt; \u662f\u6709\u5e8f\u7684&#xff0c;\u9876\u70b9\u5bf9 &lt;x, y&gt; \u79f0\u4e3a\u9876\u70b9 x \u5230\u9876\u70b9 y \u7684\u4e00\u6761\u8fb9&#xff08;\u5f27&#xff09;&#xff0c;&lt;x, y&gt; \u548c &lt;y, x&gt; \u662f\u4e24\u6761\u4e0d\u540c\u7684\u8fb9&#xff0c;\u6bd4\u5982\u4e0b\u56fe G3 \u548c G4 \u4e3a\u6709\u5411\u56fe\u3002\u5728\u65e0\u5411\u56fe\u4e2d&#xff0c;\u9876\u70b9\u5bf9 (x, y) \u662f\u65e0\u5e8f\u7684&#xff0c;\u9876\u70b9\u5bf9 (x, y) \u79f0\u4e3a\u9876\u70b9 x \u548c\u9876\u70b9 y \u76f8\u5173\u8054\u7684\u4e00\u6761\u8fb9&#xff0c;\u8fd9\u6761\u8fb9\u6ca1\u6709\u7279\u5b9a\u65b9\u5411&#xff0c;(x, y) \u548c (y, x) \u662f\u540c\u4e00\u6761\u8fb9&#xff0c;\u6bd4\u5982\u4e0b\u56fe G1 \u548c G2 \u4e3a\u65e0\u5411\u56fe\u3002\u6ce8\u610f&#xff1a;\u65e0\u5411\u8fb9 (x, y) \u7b49\u4e8e\u6709\u5411\u8fb9 &lt;x, y&gt; \u548c &lt;y, x&gt;\u3002<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/10\/20261021231742-6ad9481642123.png\" alt=\"\" \/><\/p>\n<p>\u5b8c\u5168\u56fe&#xff1a; \u5728\u6709 n \u4e2a\u9876\u70b9\u7684\u65e0\u5411\u56fe\u4e2d&#xff0c;\u82e5\u6709 n * (n &#8211; 1) \/ 2 \u6761\u8fb9&#xff0c;\u5373\u4efb\u610f\u4e24\u4e2a\u9876\u70b9\u4e4b\u95f4\u6709\u4e14\u4ec5\u6709\u4e00\u6761\u8fb9&#xff0c;\u5219\u79f0\u6b64\u56fe\u4e3a\u65e0\u5411\u5b8c\u5168\u56fe&#xff0c;\u6bd4\u5982\u4e0a\u56fe G1&#xff1b;\u5728 n \u4e2a\u9876\u70b9\u7684\u6709\u5411\u56fe\u4e2d&#xff0c;\u82e5\u6709 n * (n &#8211; 1) \u6761\u8fb9&#xff0c;\u5373\u4efb\u610f\u4e24\u4e2a\u9876\u70b9\u4e4b\u95f4\u6709\u4e14\u4ec5\u6709\u65b9\u5411\u76f8\u53cd\u7684\u8fb9&#xff0c;\u5219\u79f0\u6b64\u56fe\u4e3a\u6709\u5411\u5b8c\u5168\u56fe&#xff0c;\u6bd4\u5982\u4e0a\u56fe G4\u3002<\/p>\n<p>\u90bb\u63a5\u9876\u70b9&#xff1a; \u5728\u65e0\u5411\u56fe G \u4e2d&#xff0c;\u82e5 (u, v) \u662f E(G) \u4e2d\u7684\u4e00\u6761\u8fb9&#xff0c;\u5219\u79f0 u \u548c v \u4e92\u4e3a\u90bb\u63a5\u9876\u70b9&#xff0c;\u5e76\u79f0\u8fb9 (u, v) \u4f9d\u9644\u4e8e\u9876\u70b9 u \u548c v&#xff1b;\u5728\u6709\u5411\u56fe G \u4e2d&#xff0c;\u82e5 &lt;u, v&gt; \u662f E(G) \u4e2d\u7684\u4e00\u6761\u8fb9&#xff0c;\u5219\u79f0\u9876\u70b9 u \u90bb\u63a5\u5230 v&#xff0c;\u9876\u70b9 v \u90bb\u63a5\u81ea\u9876\u70b9 u&#xff0c;\u5e76\u79f0\u8fb9 &lt;u, v&gt; \u4e0e\u9876\u70b9 u \u548c\u9876\u70b9 v \u76f8\u5173\u8054\u3002<\/p>\n<p>\u9876\u70b9\u7684\u5ea6&#xff1a; \u9876\u70b9 v \u7684\u5ea6\u662f\u6307\u4e0e\u5b83\u76f8\u5173\u8054\u7684\u8fb9\u7684\u6761\u6570&#xff0c;\u8bb0\u4f5c deg(v)\u3002\u5728\u6709\u5411\u56fe\u4e2d&#xff0c;\u9876\u70b9\u7684\u5ea6\u7b49\u4e8e\u8be5\u9876\u70b9\u7684\u5165\u5ea6\u4e0e\u51fa\u5ea6\u4e4b\u548c&#xff0c;\u5176\u4e2d\u9876\u70b9 v \u7684\u5165\u5ea6\u662f\u4ee5 v \u4e3a\u7ec8\u70b9\u7684\u6709\u5411\u8fb9\u7684\u6761\u6570&#xff0c;\u8bb0\u4f5c indev(v)&#xff1b;\u9876\u70b9 v \u7684\u51fa\u5ea6\u662f\u4ee5 v \u4e3a\u8d77\u59cb\u70b9\u7684\u6709\u5411\u8fb9\u7684\u6761\u6570&#xff0c;\u8bb0\u4f5c outdev(v)\u3002\u56e0\u6b64&#xff1a;deg(v) &#061; indev(v) &#043; outdev(v)\u3002\u6ce8\u610f&#xff1a;\u5bf9\u4e8e\u65e0\u5411\u56fe&#xff0c;\u9876\u70b9\u7684\u5ea6\u7b49\u4e8e\u8be5\u9876\u70b9\u7684\u5165\u5ea6\u548c\u51fa\u5ea6&#xff0c;\u5373 deg(v) &#061; indev(v) &#061; outdev(v)\u3002<\/p>\n<p>\u8def\u5f84&#xff1a; \u5728\u56fe G &#061; (V, E) \u4e2d&#xff0c;\u82e5\u4ece\u9876\u70b9 v_i \u51fa\u53d1\u6709\u4e00\u7ec4\u8fb9\u4f7f\u5176\u53ef\u5230\u8fbe\u9876\u70b9 v_j&#xff0c;\u5219\u79f0\u9876\u70b9 v_i \u5230\u9876\u70b9 v_j \u7684\u9876\u70b9\u5e8f\u5217\u4e3a\u4ece\u9876\u70b9 v_i \u5230\u9876\u70b9 v_j \u7684\u8def\u5f84\u3002<\/p>\n<p>\u8def\u5f84\u957f\u5ea6&#xff1a; \u5bf9\u4e8e\u4e0d\u5e26\u6743\u7684\u56fe&#xff0c;\u4e00\u6761\u8def\u5f84\u7684\u8def\u5f84\u957f\u5ea6\u662f\u6307\u8be5\u8def\u5f84\u4e0a\u7684\u8fb9\u7684\u6761\u6570&#xff1b;\u5bf9\u4e8e\u5e26\u6743\u7684\u56fe&#xff0c;\u4e00\u6761\u8def\u5f84\u7684\u8def\u5f84\u957f\u5ea6\u662f\u6307\u8be5\u8def\u5f84\u4e0a\u5404\u4e2a\u8fb9\u6743\u503c\u7684\u603b\u548c\u3002<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/10\/20261021231743-6ad948171a6b5.png\" alt=\"\" \/><\/p>\n<p>\u7b80\u5355\u8def\u5f84\u4e0e\u56de\u8def&#xff1a; \u82e5\u8def\u5f84\u4e0a\u5404\u9876\u70b9 v_1, v_2, v_3, &#8230;, v_m \u5747\u4e0d\u91cd\u590d&#xff0c;\u5219\u79f0\u8fd9\u6837\u7684\u8def\u5f84\u4e3a\u7b80\u5355\u8def\u5f84\u3002\u82e5\u8def\u5f84\u4e0a\u7b2c\u4e00\u4e2a\u9876\u70b9 v_1 \u548c\u6700\u540e\u4e00\u4e2a\u9876\u70b9 v_m \u91cd\u5408&#xff0c;\u5219\u79f0\u8fd9\u6837\u7684\u8def\u5f84\u4e3a\u56de\u8def\u6216\u73af\u3002<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/10\/20261021231743-6ad948176fc8d.png\" alt=\"\" \/><\/p>\n<p>\u5b50\u56fe&#xff1a; \u8bbe\u56fe G &#061; {V, E} \u548c\u56fe G_1 &#061; {V_1, E_1}&#xff0c;\u82e5 V_1 \u5c5e\u4e8e V \u4e14 E_1 \u5c5e\u4e8e E&#xff0c;\u5219\u79f0 G_1 \u662f G \u7684\u5b50\u56fe\u3002<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/10\/20261021231743-6ad94817b97ca.png\" alt=\"\" \/><\/p>\n<p>\u8fde\u901a\u56fe&#xff1a; \u5728\u65e0\u5411\u56fe\u4e2d&#xff0c;\u82e5\u4ece\u9876\u70b9 v_1 \u5230\u9876\u70b9 v_2 \u6709\u8def\u5f84&#xff0c;\u5219\u79f0\u9876\u70b9 v_1 \u4e0e\u9876\u70b9 v_2 \u662f\u8fde\u901a\u7684\u3002\u5982\u679c\u56fe\u4e2d\u4efb\u610f\u4e00\u5bf9\u9876\u70b9\u90fd\u662f\u8fde\u901a\u7684&#xff0c;\u5219\u79f0\u6b64\u56fe\u4e3a\u8fde\u901a\u56fe\u3002<\/p>\n<p>\u5f3a\u8fde\u901a\u56fe&#xff1a; \u5728\u6709\u5411\u56fe\u4e2d&#xff0c;\u82e5\u5728\u6bcf\u4e00\u5bf9\u9876\u70b9 v_i \u548c v_j \u4e4b\u95f4\u90fd\u5b58\u5728\u4e00\u6761\u4ece v_i \u5230 v_j \u7684\u8def\u5f84&#xff0c;\u4e5f\u5b58\u5728\u4e00\u6761\u4ece v_j \u5230 v_i \u7684\u8def\u5f84&#xff0c;\u5219\u79f0\u6b64\u56fe\u662f\u5f3a\u8fde\u901a\u56fe\u3002<\/p>\n<p>\u751f\u6210\u6811&#xff1a; \u4e00\u4e2a\u8fde\u901a\u56fe\u7684\u6700\u5c0f\u8fde\u901a\u5b50\u56fe\u79f0\u4f5c\u8be5\u56fe\u7684\u751f\u6210\u6811\u3002\u6709 n \u4e2a\u9876\u70b9\u7684\u8fde\u901a\u56fe\u7684\u751f\u6210\u6811\u6709 n \u4e2a\u9876\u70b9\u548c n &#8211; 1 \u6761\u8fb9\u3002<\/p>\n<h3>2. \u56fe\u7684\u5b58\u50a8\u7ed3\u6784<\/h3>\n<p>\u56e0\u4e3a\u56fe\u4e2d\u65e2\u6709\u8282\u70b9&#xff0c;\u53c8\u6709\u8fb9&#xff08;\u8282\u70b9\u4e0e\u8282\u70b9\u4e4b\u95f4\u7684\u5173\u7cfb&#xff09;&#xff0c;\u56e0\u6b64&#xff0c;\u5728\u56fe\u7684\u5b58\u50a8\u4e2d&#xff0c;\u53ea\u9700\u8981\u4fdd\u5b58\u8282\u70b9\u548c\u8fb9\u5173\u7cfb\u5373\u53ef\u3002\u8282\u70b9\u4fdd\u5b58\u6bd4\u8f83\u7b80\u5355&#xff0c;\u53ea\u9700\u8981\u4e00\u6bb5\u8fde\u7eed\u7a7a\u95f4\u5373\u53ef&#xff0c;\u90a3\u8fb9\u5173\u7cfb\u8be5\u600e\u4e48\u4fdd\u5b58\u5462&#xff1f;<\/p>\n<h4>2.1 \u90bb\u63a5\u77e9\u9635<\/h4>\n<p>\u56e0\u4e3a\u8282\u70b9\u4e0e\u8282\u70b9\u4e4b\u95f4\u7684\u5173\u7cfb\u5c31\u662f\u8fde\u901a\u4e0e\u5426&#xff0c;\u5373\u4e3a 0 \u6216\u8005 1&#xff0c;\u56e0\u6b64\u90bb\u63a5\u77e9\u9635&#xff08;\u4e8c\u7ef4\u6570\u7ec4&#xff09;\u5373\u662f&#xff1a;\u5148\u7528\u4e00\u4e2a\u6570\u7ec4\u5c06\u9876\u70b9\u4fdd\u5b58&#xff0c;\u7136\u540e\u91c7\u7528\u77e9\u9635\u6765\u8868\u793a\u8282\u70b9\u4e0e\u8282\u70b9\u4e4b\u95f4\u7684\u5173\u7cfb\u3002<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/10\/20261021231744-6ad9481838a90.png\" alt=\"\" \/><\/p>\n<p>\u6ce8\u610f&#xff1a;<\/p>\n<li>\u65e0\u5411\u56fe\u7684\u90bb\u63a5\u77e9\u9635\u662f\u5bf9\u79f0\u7684&#xff0c;\u7b2c i \u884c&#xff08;\u5217&#xff09;\u5143\u7d20\u4e4b\u548c&#xff0c;\u5c31\u662f\u9876\u70b9 i \u7684\u5ea6\u3002\u6709\u5411\u56fe\u7684\u90bb\u63a5\u77e9\u9635\u5219\u4e0d\u4e00\u5b9a\u662f\u5bf9\u79f0\u7684&#xff0c;\u7b2c i \u884c&#xff08;\u5217&#xff09;\u5143\u7d20\u4e4b\u548c\u5c31\u662f\u9876\u70b9 i \u7684\u51fa&#xff08;\u5165&#xff09;\u5ea6\u3002<\/li>\n<li>\u5982\u679c\u8fb9\u5e26\u6709\u6743\u503c&#xff0c;\u5e76\u4e14\u4e24\u4e2a\u8282\u70b9\u4e4b\u95f4\u662f\u8fde\u901a\u7684&#xff0c;\u4e0a\u56fe\u4e2d\u7684\u8fb9\u7684\u5173\u7cfb\u5c31\u7528\u6743\u503c\u4ee3\u66ff&#xff1b;\u5982\u679c\u4e24\u4e2a\u9876\u70b9\u4e0d\u901a&#xff0c;\u5219\u4f7f\u7528\u65e0\u7a77\u5927\u4ee3\u66ff\u3002<\/li>\n<p><img decoding=\"async\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/10\/20261021231744-6ad94818b67da.png\" alt=\"\" \/><\/p>\n<li>\u7528\u90bb\u63a5\u77e9\u9635\u5b58\u50a8\u56fe\u7684\u4f18\u70b9\u662f\u80fd\u591f\u5feb\u901f\u77e5\u9053\u4e24\u4e2a\u9876\u70b9\u662f\u5426\u8fde\u901a&#xff0c;\u7f3a\u70b9\u662f\u5982\u679c\u9876\u70b9\u6bd4\u8f83\u591a\u3001\u8fb9\u6bd4\u8f83\u5c11\u65f6&#xff0c;\u77e9\u9635\u4e2d\u5b58\u50a8\u4e86\u5927\u91cf\u7684 0&#xff0c;\u6210\u4e3a\u7a00\u758f\u77e9\u9635&#xff0c;\u6bd4\u8f83\u6d6a\u8d39\u7a7a\u95f4&#xff0c;\u5e76\u4e14\u4e24\u4e2a\u8282\u70b9\u4e4b\u95f4\u7684\u8def\u5f84\u4e5f\u4e0d\u662f\u5f88\u597d\u6c42\u3002<\/li>\n<p><span class=\"token keyword\">template<\/span><span class=\"token operator\">&lt;<\/span><span class=\"token keyword\">class<\/span> <span class=\"token class-name\">V<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">class<\/span> <span class=\"token class-name\">W<\/span><span class=\"token punctuation\">,<\/span> W MAX_W <span class=\"token operator\">&#061;<\/span> INT_MAX<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">bool<\/span> Direction <span class=\"token operator\">&#061;<\/span> <span class=\"token boolean\">false<\/span><span class=\"token operator\">&gt;<\/span><br \/>\n<span class=\"token keyword\">class<\/span> <span class=\"token class-name\">Graph<\/span><br \/>\n<span class=\"token punctuation\">{<\/span><br \/>\n<span class=\"token keyword\">public<\/span><span class=\"token operator\">:<\/span><br \/>\n    <span class=\"token keyword\">typedef<\/span> Graph<span class=\"token operator\">&lt;<\/span>V<span class=\"token punctuation\">,<\/span> W<span class=\"token punctuation\">,<\/span> MAX_W<span class=\"token punctuation\">,<\/span> Direction<span class=\"token operator\">&gt;<\/span> Self<span class=\"token punctuation\">;<\/span><\/p>\n<p>    <span class=\"token function\">Graph<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token operator\">&#061;<\/span> <span class=\"token keyword\">default<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>    <span class=\"token function\">Graph<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">const<\/span> V<span class=\"token operator\">*<\/span> vertexs<span class=\"token punctuation\">,<\/span> size_t n<span class=\"token punctuation\">)<\/span><br \/>\n    <span class=\"token punctuation\">{<\/span><br \/>\n        _vertexs<span class=\"token punctuation\">.<\/span><span class=\"token function\">reserve<\/span><span class=\"token punctuation\">(<\/span>n<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>size_t i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> i <span class=\"token operator\">&lt;<\/span> n<span class=\"token punctuation\">;<\/span> <span class=\"token operator\">&#043;&#043;<\/span>i<span class=\"token punctuation\">)<\/span><br \/>\n        <span class=\"token punctuation\">{<\/span><br \/>\n            _vertexs<span class=\"token punctuation\">.<\/span><span class=\"token function\">push_back<\/span><span class=\"token punctuation\">(<\/span>vertexs<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            _vIndexMap<span class=\"token punctuation\">[<\/span>vertexs<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> i<span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><\/p>\n<p>        <span class=\"token comment\">\/\/ MAX_W \u4f5c\u4e3a\u4e0d\u5b58\u5728\u8fb9\u7684\u6807\u8bc6\u503c<\/span><br \/>\n        _matrix<span class=\"token punctuation\">.<\/span><span class=\"token function\">resize<\/span><span class=\"token punctuation\">(<\/span>n<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">auto<\/span><span class=\"token operator\">&amp;<\/span> e <span class=\"token operator\">:<\/span> _matrix<span class=\"token punctuation\">)<\/span><br \/>\n        <span class=\"token punctuation\">{<\/span><br \/>\n            e<span class=\"token punctuation\">.<\/span><span class=\"token function\">resize<\/span><span class=\"token punctuation\">(<\/span>n<span class=\"token punctuation\">,<\/span> MAX_W<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><\/p>\n<p>    size_t <span class=\"token function\">GetVertexIndex<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">const<\/span> V<span class=\"token operator\">&amp;<\/span> v<span class=\"token punctuation\">)<\/span><br \/>\n    <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token keyword\">auto<\/span> ret <span class=\"token operator\">&#061;<\/span> _vIndexMap<span class=\"token punctuation\">.<\/span><span class=\"token function\">find<\/span><span class=\"token punctuation\">(<\/span>v<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>ret <span class=\"token operator\">!&#061;<\/span> _vIndexMap<span class=\"token punctuation\">.<\/span><span class=\"token function\">end<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><br \/>\n        <span class=\"token punctuation\">{<\/span><br \/>\n            <span class=\"token keyword\">return<\/span> ret<span class=\"token operator\">-&gt;<\/span>second<span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><br \/>\n        <span class=\"token keyword\">else<\/span><br \/>\n        <span class=\"token punctuation\">{<\/span><br \/>\n            <span class=\"token keyword\">throw<\/span> <span class=\"token function\">invalid_argument<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string\">&#034;\u4e0d\u5b58\u5728\u7684\u9876\u70b9&#034;<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token keyword\">return<\/span> <span class=\"token operator\">&#8211;<\/span><span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><\/p>\n<p>    <span class=\"token keyword\">void<\/span> <span class=\"token function\">_AddEdge<\/span><span class=\"token punctuation\">(<\/span>size_t srci<span class=\"token punctuation\">,<\/span> size_t dsti<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">const<\/span> W<span class=\"token operator\">&amp;<\/span> w<span class=\"token punctuation\">)<\/span><br \/>\n    <span class=\"token punctuation\">{<\/span><br \/>\n        _matrix<span class=\"token punctuation\">[<\/span>srci<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>dsti<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> w<span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>Direction <span class=\"token operator\">&#061;&#061;<\/span> <span class=\"token boolean\">false<\/span><span class=\"token punctuation\">)<\/span><br \/>\n        <span class=\"token punctuation\">{<\/span><br \/>\n            _matrix<span class=\"token punctuation\">[<\/span>dsti<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>srci<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> w<span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><\/p>\n<p>    <span class=\"token keyword\">void<\/span> <span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">const<\/span> V<span class=\"token operator\">&amp;<\/span> src<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">const<\/span> V<span class=\"token operator\">&amp;<\/span> dst<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">const<\/span> W<span class=\"token operator\">&amp;<\/span> w<span class=\"token punctuation\">)<\/span><br \/>\n    <span class=\"token punctuation\">{<\/span><br \/>\n        size_t srci <span class=\"token operator\">&#061;<\/span> <span class=\"token function\">GetVertexIndex<\/span><span class=\"token punctuation\">(<\/span>src<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        size_t dsti <span class=\"token operator\">&#061;<\/span> <span class=\"token function\">GetVertexIndex<\/span><span class=\"token punctuation\">(<\/span>dst<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>        <span class=\"token function\">_AddEdge<\/span><span class=\"token punctuation\">(<\/span>srci<span class=\"token punctuation\">,<\/span> dsti<span class=\"token punctuation\">,<\/span> w<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><\/p>\n<p>    <span class=\"token keyword\">void<\/span> <span class=\"token function\">Print<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><br \/>\n    <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token comment\">\/\/ \u6253\u5370\u9876\u70b9\u548c\u4e0b\u6807\u6620\u5c04\u5173\u7cfb<\/span><br \/>\n        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>size_t i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> i <span class=\"token operator\">&lt;<\/span> _vertexs<span class=\"token punctuation\">.<\/span><span class=\"token function\">size<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span> <span class=\"token operator\">&#043;&#043;<\/span>i<span class=\"token punctuation\">)<\/span><br \/>\n        <span class=\"token punctuation\">{<\/span><br \/>\n            cout <span class=\"token operator\">&lt;&lt;<\/span> _vertexs<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&lt;&lt;<\/span> <span class=\"token string\">&#034;-&#034;<\/span> <span class=\"token operator\">&lt;&lt;<\/span> i <span class=\"token operator\">&lt;&lt;<\/span> <span class=\"token string\">&#034; &#034;<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><br \/>\n        cout <span class=\"token operator\">&lt;&lt;<\/span> endl <span class=\"token operator\">&lt;&lt;<\/span> endl<span class=\"token punctuation\">;<\/span><\/p>\n<p>        cout <span class=\"token operator\">&lt;&lt;<\/span> <span class=\"token string\">&#034; &#034;<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>size_t i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> i <span class=\"token operator\">&lt;<\/span> _vertexs<span class=\"token punctuation\">.<\/span><span class=\"token function\">size<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span> <span class=\"token operator\">&#043;&#043;<\/span>i<span class=\"token punctuation\">)<\/span><br \/>\n        <span class=\"token punctuation\">{<\/span><br \/>\n            cout <span class=\"token operator\">&lt;&lt;<\/span> i <span class=\"token operator\">&lt;&lt;<\/span> <span class=\"token string\">&#034; &#034;<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><br \/>\n        cout <span class=\"token operator\">&lt;&lt;<\/span> endl<span class=\"token punctuation\">;<\/span><\/p>\n<p>        <span class=\"token comment\">\/\/ \u6253\u5370\u77e9\u9635<\/span><br \/>\n        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>size_t i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> i <span class=\"token operator\">&lt;<\/span> _matrix<span class=\"token punctuation\">.<\/span><span class=\"token function\">size<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span> <span class=\"token operator\">&#043;&#043;<\/span>i<span class=\"token punctuation\">)<\/span><br \/>\n        <span class=\"token punctuation\">{<\/span><br \/>\n            cout <span class=\"token operator\">&lt;&lt;<\/span> i <span class=\"token operator\">&lt;&lt;<\/span> <span class=\"token string\">&#034; &#034;<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>size_t j <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> j <span class=\"token operator\">&lt;<\/span> _matrix<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">.<\/span><span class=\"token function\">size<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span> <span class=\"token operator\">&#043;&#043;<\/span>j<span class=\"token punctuation\">)<\/span><br \/>\n            <span class=\"token punctuation\">{<\/span><br \/>\n                <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>_matrix<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">!&#061;<\/span> MAX_W<span class=\"token punctuation\">)<\/span><br \/>\n                    cout <span class=\"token operator\">&lt;&lt;<\/span> _matrix<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&lt;&lt;<\/span> <span class=\"token string\">&#034; &#034;<\/span><span class=\"token punctuation\">;<\/span><br \/>\n                <span class=\"token keyword\">else<\/span><br \/>\n                    cout <span class=\"token operator\">&lt;&lt;<\/span> <span class=\"token string\">&#034;#&#034;<\/span> <span class=\"token operator\">&lt;&lt;<\/span> <span class=\"token string\">&#034; &#034;<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token punctuation\">}<\/span><br \/>\n            cout <span class=\"token operator\">&lt;&lt;<\/span> endl<span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><br \/>\n        cout <span class=\"token operator\">&lt;&lt;<\/span> endl <span class=\"token operator\">&lt;&lt;<\/span> endl<span class=\"token punctuation\">;<\/span><\/p>\n<p>        <span class=\"token comment\">\/\/ \u6253\u5370\u6240\u6709\u7684\u8fb9<\/span><br \/>\n        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>size_t i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> i <span class=\"token operator\">&lt;<\/span> _matrix<span class=\"token punctuation\">.<\/span><span class=\"token function\">size<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span> <span class=\"token operator\">&#043;&#043;<\/span>i<span class=\"token punctuation\">)<\/span><br \/>\n        <span class=\"token punctuation\">{<\/span><br \/>\n            <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>size_t j <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> j <span class=\"token operator\">&lt;<\/span> _matrix<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">.<\/span><span class=\"token function\">size<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span> <span class=\"token operator\">&#043;&#043;<\/span>j<span class=\"token punctuation\">)<\/span><br \/>\n            <span class=\"token punctuation\">{<\/span><br \/>\n                <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>i <span class=\"token operator\">&lt;<\/span> j <span class=\"token operator\">&amp;&amp;<\/span> _matrix<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">!&#061;<\/span> MAX_W<span class=\"token punctuation\">)<\/span><br \/>\n                <span class=\"token punctuation\">{<\/span><br \/>\n                    cout <span class=\"token operator\">&lt;&lt;<\/span> _vertexs<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&lt;&lt;<\/span> <span class=\"token string\">&#034;-&#034;<\/span> <span class=\"token operator\">&lt;&lt;<\/span> _vertexs<span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&lt;&lt;<\/span> <span class=\"token string\">&#034;:&#034;<\/span><br \/>\n                         <span class=\"token operator\">&lt;&lt;<\/span> _matrix<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&lt;&lt;<\/span> endl<span class=\"token punctuation\">;<\/span><br \/>\n                <span class=\"token punctuation\">}<\/span><br \/>\n            <span class=\"token punctuation\">}<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><\/p>\n<p><span class=\"token keyword\">private<\/span><span class=\"token operator\">:<\/span><br \/>\n    map<span class=\"token operator\">&lt;<\/span>V<span class=\"token punctuation\">,<\/span> size_t<span class=\"token operator\">&gt;<\/span> _vIndexMap<span class=\"token punctuation\">;<\/span><br \/>\n    vector<span class=\"token operator\">&lt;<\/span>V<span class=\"token operator\">&gt;<\/span> _vertexs<span class=\"token punctuation\">;<\/span> <span class=\"token comment\">\/\/ \u9876\u70b9\u96c6\u5408<\/span><br \/>\n    vector<span class=\"token operator\">&lt;<\/span>vector<span class=\"token operator\">&lt;<\/span>W<span class=\"token operator\">&gt;&gt;<\/span> _matrix<span class=\"token punctuation\">;<\/span> <span class=\"token comment\">\/\/ \u5b58\u50a8\u8fb9\u96c6\u5408\u7684\u77e9\u9635<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p><span class=\"token keyword\">void<\/span> <span class=\"token function\">TestGraph<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token punctuation\">{<\/span><br \/>\n    Graph<span class=\"token operator\">&lt;<\/span><span class=\"token keyword\">char<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span><span class=\"token punctuation\">,<\/span> INT_MAX<span class=\"token punctuation\">,<\/span> <span class=\"token boolean\">true<\/span><span class=\"token operator\">&gt;<\/span> <span class=\"token function\">g<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string\">&#034;0123&#034;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">4<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;0&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;1&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;0&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;3&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">4<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;1&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;3&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">2<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;1&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;2&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">9<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;2&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;3&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">8<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;2&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;1&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">5<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;2&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;0&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">3<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;3&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;2&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">6<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">Print<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<h4>2.2 \u90bb\u63a5\u8868<\/h4>\n<p>\u90bb\u63a5\u8868&#xff1a;\u4f7f\u7528\u6570\u7ec4\u8868\u793a\u9876\u70b9\u7684\u96c6\u5408&#xff0c;\u4f7f\u7528\u94fe\u8868\u8868\u793a\u8fb9\u7684\u5173\u7cfb\u3002<\/p>\n<li>\u65e0\u5411\u56fe\u90bb\u63a5\u8868\u5b58\u50a8<\/li>\n<p><img decoding=\"async\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/10\/20261021231745-6ad9481907388.png\" alt=\"\" \/><\/p>\n<p>\u6ce8\u610f&#xff1a; \u65e0\u5411\u56fe\u4e2d\u540c\u4e00\u6761\u8fb9\u5728\u90bb\u63a5\u8868\u4e2d\u51fa\u73b0\u4e86\u4e24\u6b21\u3002\u5982\u679c\u60f3\u77e5\u9053\u9876\u70b9 v_i \u7684\u5ea6&#xff0c;\u53ea\u9700\u8981\u77e5\u9053\u9876\u70b9 v_i \u8fb9\u94fe\u8868\u96c6\u5408\u4e2d\u7ed3\u70b9\u7684\u6570\u76ee\u5373\u53ef\u3002<\/p>\n<li>\u6709\u5411\u56fe\u90bb\u63a5\u8868\u5b58\u50a8<\/li>\n<p><img decoding=\"async\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/10\/20261021231745-6ad948196ad94.png\" alt=\"\" \/><\/p>\n<p>\u6ce8\u610f&#xff1a; \u6709\u5411\u56fe\u4e2d\u6bcf\u6761\u8fb9\u5728\u90bb\u63a5\u8868\u4e2d\u53ea\u51fa\u73b0\u4e00\u6b21&#xff0c;\u4e0e\u9876\u70b9 v_i \u5bf9\u5e94\u7684\u90bb\u63a5\u8868\u6240\u542b\u7ed3\u70b9\u7684\u4e2a\u6570&#xff0c;\u5c31\u662f\u8be5\u9876\u70b9\u7684\u51fa\u5ea6&#xff0c;\u4e5f\u79f0\u51fa\u5ea6\u8868\u3002\u8981\u5f97\u5230 v_i \u9876\u70b9\u7684\u5165\u5ea6&#xff0c;\u5fc5\u987b\u68c0\u6d4b\u5176\u4ed6\u6240\u6709\u9876\u70b9\u5bf9\u5e94\u7684\u8fb9\u94fe\u8868&#xff0c;\u770b\u6709\u591a\u5c11\u8fb9\u9876\u70b9\u7684 dst \u53d6\u503c\u662f i\u3002<\/p>\n<p><span class=\"token comment\">\/\/ \u90bb\u63a5\u8868<\/span><br \/>\n<span class=\"token keyword\">namespace<\/span> LinkTable<br \/>\n<span class=\"token punctuation\">{<\/span><br \/>\n<span class=\"token keyword\">template<\/span><span class=\"token operator\">&lt;<\/span><span class=\"token keyword\">class<\/span> <span class=\"token class-name\">W<\/span><span class=\"token operator\">&gt;<\/span><br \/>\n<span class=\"token keyword\">struct<\/span> <span class=\"token class-name\">LinkEdge<\/span><br \/>\n<span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token keyword\">int<\/span> _srcIndex<span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token keyword\">int<\/span> _dstIndex<span class=\"token punctuation\">;<\/span><br \/>\n    W _w<span class=\"token punctuation\">;<\/span><br \/>\n    LinkEdge<span class=\"token operator\">&lt;<\/span>W<span class=\"token operator\">&gt;<\/span><span class=\"token operator\">*<\/span> _next<span class=\"token punctuation\">;<\/span><\/p>\n<p>    <span class=\"token function\">LinkEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">const<\/span> W<span class=\"token operator\">&amp;<\/span> w<span class=\"token punctuation\">)<\/span><br \/>\n        <span class=\"token operator\">:<\/span> <span class=\"token function\">_srcIndex<\/span><span class=\"token punctuation\">(<\/span><span class=\"token operator\">&#8211;<\/span><span class=\"token number\">1<\/span><span class=\"token punctuation\">)<\/span><br \/>\n        <span class=\"token punctuation\">,<\/span> <span class=\"token function\">_dstIndex<\/span><span class=\"token punctuation\">(<\/span><span class=\"token operator\">&#8211;<\/span><span class=\"token number\">1<\/span><span class=\"token punctuation\">)<\/span><br \/>\n        <span class=\"token punctuation\">,<\/span> <span class=\"token function\">_w<\/span><span class=\"token punctuation\">(<\/span>w<span class=\"token punctuation\">)<\/span><br \/>\n        <span class=\"token punctuation\">,<\/span> <span class=\"token function\">_next<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">nullptr<\/span><span class=\"token punctuation\">)<\/span><br \/>\n    <span class=\"token punctuation\">{<\/span><span class=\"token punctuation\">}<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p><span class=\"token keyword\">template<\/span><span class=\"token operator\">&lt;<\/span><span class=\"token keyword\">class<\/span> <span class=\"token class-name\">V<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">class<\/span> <span class=\"token class-name\">W<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">bool<\/span> Direction <span class=\"token operator\">&#061;<\/span> <span class=\"token boolean\">false<\/span><span class=\"token operator\">&gt;<\/span><br \/>\n<span class=\"token keyword\">class<\/span> <span class=\"token class-name\">Graph<\/span><br \/>\n<span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token keyword\">typedef<\/span> LinkEdge<span class=\"token operator\">&lt;<\/span>W<span class=\"token operator\">&gt;<\/span> Edge<span class=\"token punctuation\">;<\/span><\/p>\n<p><span class=\"token keyword\">public<\/span><span class=\"token operator\">:<\/span><br \/>\n    <span class=\"token function\">Graph<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">const<\/span> V<span class=\"token operator\">*<\/span> vertexs<span class=\"token punctuation\">,<\/span> size_t n<span class=\"token punctuation\">)<\/span><br \/>\n    <span class=\"token punctuation\">{<\/span><br \/>\n        _vertexs<span class=\"token punctuation\">.<\/span><span class=\"token function\">reserve<\/span><span class=\"token punctuation\">(<\/span>n<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>size_t i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> i <span class=\"token operator\">&lt;<\/span> n<span class=\"token punctuation\">;<\/span> <span class=\"token operator\">&#043;&#043;<\/span>i<span class=\"token punctuation\">)<\/span><br \/>\n        <span class=\"token punctuation\">{<\/span><br \/>\n            _vertexs<span class=\"token punctuation\">.<\/span><span class=\"token function\">push_back<\/span><span class=\"token punctuation\">(<\/span>vertexs<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            _vIndexMap<span class=\"token punctuation\">[<\/span>vertexs<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> i<span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><br \/>\n        _linkTable<span class=\"token punctuation\">.<\/span><span class=\"token function\">resize<\/span><span class=\"token punctuation\">(<\/span>n<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">nullptr<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><\/p>\n<p>    size_t <span class=\"token function\">GetVertexIndex<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">const<\/span> V<span class=\"token operator\">&amp;<\/span> v<span class=\"token punctuation\">)<\/span><br \/>\n    <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token keyword\">auto<\/span> ret <span class=\"token operator\">&#061;<\/span> _vIndexMap<span class=\"token punctuation\">.<\/span><span class=\"token function\">find<\/span><span class=\"token punctuation\">(<\/span>v<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>ret <span class=\"token operator\">!&#061;<\/span> _vIndexMap<span class=\"token punctuation\">.<\/span><span class=\"token function\">end<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><br \/>\n        <span class=\"token punctuation\">{<\/span><br \/>\n            <span class=\"token keyword\">return<\/span> ret<span class=\"token operator\">-&gt;<\/span>second<span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><br \/>\n        <span class=\"token keyword\">else<\/span><br \/>\n        <span class=\"token punctuation\">{<\/span><br \/>\n            <span class=\"token keyword\">throw<\/span> <span class=\"token function\">invalid_argument<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string\">&#034;\u4e0d\u5b58\u5728\u7684\u9876\u70b9&#034;<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token keyword\">return<\/span> <span class=\"token operator\">&#8211;<\/span><span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><\/p>\n<p>    <span class=\"token keyword\">void<\/span> <span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">const<\/span> V<span class=\"token operator\">&amp;<\/span> src<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">const<\/span> V<span class=\"token operator\">&amp;<\/span> dst<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">const<\/span> W<span class=\"token operator\">&amp;<\/span> w<span class=\"token punctuation\">)<\/span><br \/>\n    <span class=\"token punctuation\">{<\/span><br \/>\n        size_t srcindex <span class=\"token operator\">&#061;<\/span> <span class=\"token function\">GetVertexIndex<\/span><span class=\"token punctuation\">(<\/span>src<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        size_t dstindex <span class=\"token operator\">&#061;<\/span> <span class=\"token function\">GetVertexIndex<\/span><span class=\"token punctuation\">(<\/span>dst<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>        <span class=\"token comment\">\/\/ 0 -&gt; 1<\/span><br \/>\n        Edge<span class=\"token operator\">*<\/span> sd_edge <span class=\"token operator\">&#061;<\/span> <span class=\"token keyword\">new<\/span> <span class=\"token function\">Edge<\/span><span class=\"token punctuation\">(<\/span>w<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        sd_edge<span class=\"token operator\">-&gt;<\/span>_srcIndex <span class=\"token operator\">&#061;<\/span> srcindex<span class=\"token punctuation\">;<\/span><br \/>\n        sd_edge<span class=\"token operator\">-&gt;<\/span>_dstIndex <span class=\"token operator\">&#061;<\/span> dstindex<span class=\"token punctuation\">;<\/span><br \/>\n        sd_edge<span class=\"token operator\">-&gt;<\/span>_next <span class=\"token operator\">&#061;<\/span> _linkTable<span class=\"token punctuation\">[<\/span>srcindex<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        _linkTable<span class=\"token punctuation\">[<\/span>srcindex<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> sd_edge<span class=\"token punctuation\">;<\/span><\/p>\n<p>        <span class=\"token comment\">\/\/ 1 -&gt; 0<\/span><br \/>\n        <span class=\"token comment\">\/\/ \u65e0\u5411\u56fe<\/span><br \/>\n        <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>Direction <span class=\"token operator\">&#061;&#061;<\/span> <span class=\"token boolean\">false<\/span><span class=\"token punctuation\">)<\/span><br \/>\n        <span class=\"token punctuation\">{<\/span><br \/>\n            Edge<span class=\"token operator\">*<\/span> ds_edge <span class=\"token operator\">&#061;<\/span> <span class=\"token keyword\">new<\/span> <span class=\"token function\">Edge<\/span><span class=\"token punctuation\">(<\/span>w<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            ds_edge<span class=\"token operator\">-&gt;<\/span>_srcIndex <span class=\"token operator\">&#061;<\/span> dstindex<span class=\"token punctuation\">;<\/span><br \/>\n            ds_edge<span class=\"token operator\">-&gt;<\/span>_dstIndex <span class=\"token operator\">&#061;<\/span> srcindex<span class=\"token punctuation\">;<\/span><br \/>\n            ds_edge<span class=\"token operator\">-&gt;<\/span>_next <span class=\"token operator\">&#061;<\/span> _linkTable<span class=\"token punctuation\">[<\/span>dstindex<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            _linkTable<span class=\"token punctuation\">[<\/span>dstindex<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> ds_edge<span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><\/p>\n<p><span class=\"token keyword\">private<\/span><span class=\"token operator\">:<\/span><br \/>\n    map<span class=\"token operator\">&lt;<\/span>string<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span><span class=\"token operator\">&gt;<\/span> _vIndexMap<span class=\"token punctuation\">;<\/span><br \/>\n    vector<span class=\"token operator\">&lt;<\/span>V<span class=\"token operator\">&gt;<\/span> _vertexs<span class=\"token punctuation\">;<\/span> <span class=\"token comment\">\/\/ \u9876\u70b9\u96c6\u5408<\/span><br \/>\n    vector<span class=\"token operator\">&lt;<\/span>Edge<span class=\"token operator\">*<\/span><span class=\"token operator\">&gt;<\/span> _linkTable<span class=\"token punctuation\">;<\/span> <span class=\"token comment\">\/\/ \u8fb9\u7684\u96c6\u5408\u7684\u90bb\u63a5\u8868<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p><span class=\"token keyword\">void<\/span> <span class=\"token function\">TestGraph<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token punctuation\">{<\/span><br \/>\n    string a<span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> <span class=\"token punctuation\">{<\/span> <span class=\"token string\">&#034;\u5f20\u4e09&#034;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token string\">&#034;\u674e\u56db&#034;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token string\">&#034;\u738b\u4e94&#034;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token string\">&#034;\u8d75\u516d&#034;<\/span> <span class=\"token punctuation\">}<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    Graph<span class=\"token operator\">&lt;<\/span>string<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span><span class=\"token operator\">&gt;<\/span> <span class=\"token function\">g1<\/span><span class=\"token punctuation\">(<\/span>a<span class=\"token punctuation\">,<\/span> <span class=\"token number\">4<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g1<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string\">&#034;\u5f20\u4e09&#034;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token string\">&#034;\u674e\u56db&#034;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">100<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g1<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string\">&#034;\u5f20\u4e09&#034;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token string\">&#034;\u738b\u4e94&#034;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">200<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g1<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string\">&#034;\u738b\u4e94&#034;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token string\">&#034;\u8d75\u516d&#034;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">30<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<h3>3. \u56fe\u7684\u904d\u5386<\/h3>\n<p>\u7ed9\u5b9a\u4e00\u4e2a\u56fe G \u548c\u5176\u4e2d\u4efb\u610f\u4e00\u4e2a\u9876\u70b9 v_0&#xff0c;\u4ece v_0 \u51fa\u53d1&#xff0c;\u6cbf\u7740\u56fe\u4e2d\u5404\u8fb9\u8bbf\u95ee\u56fe\u4e2d\u7684\u6240\u6709\u9876\u70b9&#xff0c;\u4e14\u6bcf\u4e2a\u9876\u70b9\u4ec5\u88ab\u904d\u5386\u4e00\u6b21\u3002\u201c\u904d\u5386\u201d\u5373\u5bf9\u7ed3\u70b9\u8fdb\u884c\u67d0\u79cd\u64cd\u4f5c\u7684\u610f\u601d\u3002<\/p>\n<p>\u8bf7\u601d\u8003\u6811\u4ee5\u524d\u662f\u600e\u4e48\u904d\u5386\u7684&#xff0c;\u6b64\u5904\u53ef\u4ee5\u76f4\u63a5\u7528\u6765\u904d\u5386\u56fe\u5417&#xff1f;\u4e3a\u4ec0\u4e48&#xff1f;<\/p>\n<h4>3.1 \u56fe\u7684\u5e7f\u5ea6\u4f18\u5148\u904d\u5386<\/h4>\n<p><img decoding=\"async\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/10\/20261021231745-6ad94819e68a9.png\" alt=\"\" \/><\/p>\n<p>\u6bd4\u5982\u73b0\u5728\u8981\u627e\u4e1c\u897f&#xff0c;\u5047\u8bbe\u6709\u4e09\u4e2a\u62bd\u5c49&#xff0c;\u4e1c\u897f\u5728\u54ea\u4e2a\u62bd\u5c49\u4e0d\u6e05\u695a&#xff0c;\u73b0\u5728\u8981\u5c06\u5176\u627e\u5230&#xff0c;\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u7684\u505a\u6cd5\u662f&#xff1a;<\/p>\n<li>\u5148\u5c06\u4e09\u4e2a\u62bd\u5c49\u6253\u5f00&#xff0c;\u5728\u6700\u5916\u5c42\u627e\u4e00\u904d\u3002<\/li>\n<li>\u5c06\u6bcf\u4e2a\u62bd\u5c49\u4e2d\u7ea2\u8272\u7684\u76d2\u5b50\u6253\u5f00&#xff0c;\u518d\u627e\u4e00\u904d\u3002<\/li>\n<li>\u5c06\u7ea2\u8272\u76d2\u5b50\u4e2d\u7eff\u8272\u76d2\u5b50\u6253\u5f00&#xff0c;\u518d\u627e\u4e00\u904d\u3002<\/li>\n<p>\u76f4\u5230\u627e\u5b8c\u6240\u6709\u7684\u76d2\u5b50\u3002\u6ce8\u610f&#xff1a;\u6bcf\u4e2a\u76d2\u5b50\u53ea\u80fd\u627e\u4e00\u6b21&#xff0c;\u4e0d\u80fd\u91cd\u590d\u627e\u3002<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/10\/20261021231746-6ad9481a4706f.png\" alt=\"\" \/> <img decoding=\"async\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/10\/20261021231746-6ad9481a9c92c.png\" alt=\"\" \/><\/p>\n<p>\u95ee\u9898&#xff1a; \u5982\u4f55\u9632\u6b62\u8282\u70b9\u88ab\u91cd\u590d\u904d\u5386&#xff1f;<\/p>\n<p><span class=\"token keyword\">void<\/span> <span class=\"token function\">BFS<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">const<\/span> V<span class=\"token operator\">&amp;<\/span> src<span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token punctuation\">{<\/span><br \/>\n    size_t srcindex <span class=\"token operator\">&#061;<\/span> <span class=\"token function\">GetVertexIndex<\/span><span class=\"token punctuation\">(<\/span>src<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    vector<span class=\"token operator\">&lt;<\/span><span class=\"token keyword\">bool<\/span><span class=\"token operator\">&gt;<\/span> visited<span class=\"token punctuation\">;<\/span><br \/>\n    visited<span class=\"token punctuation\">.<\/span><span class=\"token function\">resize<\/span><span class=\"token punctuation\">(<\/span>_vertexs<span class=\"token punctuation\">.<\/span><span class=\"token function\">size<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token boolean\">false<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>    queue<span class=\"token operator\">&lt;<\/span><span class=\"token keyword\">int<\/span><span class=\"token operator\">&gt;<\/span> q<span class=\"token punctuation\">;<\/span><br \/>\n    q<span class=\"token punctuation\">.<\/span><span class=\"token function\">push<\/span><span class=\"token punctuation\">(<\/span>srcindex<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    visited<span class=\"token punctuation\">[<\/span>srcindex<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> <span class=\"token boolean\">true<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>    size_t d <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    size_t dSize <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token keyword\">while<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token operator\">!<\/span>q<span class=\"token punctuation\">.<\/span><span class=\"token function\">empty<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><br \/>\n    <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token function\">printf<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string\">&#034;%s\u7684%d\u5ea6\u597d\u53cb:&#034;<\/span><span class=\"token punctuation\">,<\/span> src<span class=\"token punctuation\">.<\/span><span class=\"token function\">c_str<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">,<\/span> d<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>        <span class=\"token keyword\">while<\/span> <span class=\"token punctuation\">(<\/span>dSize<span class=\"token operator\">&#8212;<\/span><span class=\"token punctuation\">)<\/span><br \/>\n        <span class=\"token punctuation\">{<\/span><br \/>\n            size_t front <span class=\"token operator\">&#061;<\/span> q<span class=\"token punctuation\">.<\/span><span class=\"token function\">front<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            q<span class=\"token punctuation\">.<\/span><span class=\"token function\">pop<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>size_t i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> i <span class=\"token operator\">&lt;<\/span> _vertexs<span class=\"token punctuation\">.<\/span><span class=\"token function\">size<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span> <span class=\"token operator\">&#043;&#043;<\/span>i<span class=\"token punctuation\">)<\/span><br \/>\n            <span class=\"token punctuation\">{<\/span><br \/>\n                <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>visited<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;&#061;<\/span> <span class=\"token boolean\">false<\/span> <span class=\"token operator\">&amp;&amp;<\/span> _matrix<span class=\"token punctuation\">[<\/span>front<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">!&#061;<\/span> MAX_W<span class=\"token punctuation\">)<\/span><br \/>\n                <span class=\"token punctuation\">{<\/span><br \/>\n                    <span class=\"token function\">printf<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string\">&#034;[%d:%s] &#034;<\/span><span class=\"token punctuation\">,<\/span> i<span class=\"token punctuation\">,<\/span> _vertexs<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">.<\/span><span class=\"token function\">c_str<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n                    visited<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> <span class=\"token boolean\">true<\/span><span class=\"token punctuation\">;<\/span><br \/>\n                    q<span class=\"token punctuation\">.<\/span><span class=\"token function\">push<\/span><span class=\"token punctuation\">(<\/span>i<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n                <span class=\"token punctuation\">}<\/span><br \/>\n            <span class=\"token punctuation\">}<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><br \/>\n        cout <span class=\"token operator\">&lt;&lt;<\/span> endl<span class=\"token punctuation\">;<\/span><\/p>\n<p>        dSize <span class=\"token operator\">&#061;<\/span> q<span class=\"token punctuation\">.<\/span><span class=\"token function\">size<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token operator\">&#043;&#043;<\/span>d<span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><br \/>\n    cout <span class=\"token operator\">&lt;&lt;<\/span> endl<span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<p><span class=\"token keyword\">void<\/span> <span class=\"token function\">TestGraphDBFS<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token punctuation\">{<\/span><br \/>\n    string a<span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> <span class=\"token punctuation\">{<\/span> <span class=\"token string\">&#034;\u5f20\u4e09&#034;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token string\">&#034;\u674e\u56db&#034;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token string\">&#034;\u738b\u4e94&#034;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token string\">&#034;\u8d75\u516d&#034;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token string\">&#034;\u5468\u4e03&#034;<\/span> <span class=\"token punctuation\">}<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    Graph<span class=\"token operator\">&lt;<\/span>string<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span><span class=\"token operator\">&gt;<\/span> <span class=\"token function\">g1<\/span><span class=\"token punctuation\">(<\/span>a<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">sizeof<\/span><span class=\"token punctuation\">(<\/span>a<span class=\"token punctuation\">)<\/span> <span class=\"token operator\">\/<\/span> <span class=\"token keyword\">sizeof<\/span><span class=\"token punctuation\">(<\/span>string<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g1<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string\">&#034;\u5f20\u4e09&#034;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token string\">&#034;\u674e\u56db&#034;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">100<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g1<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string\">&#034;\u5f20\u4e09&#034;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token string\">&#034;\u738b\u4e94&#034;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">200<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g1<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string\">&#034;\u738b\u4e94&#034;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token string\">&#034;\u8d75\u516d&#034;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">30<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g1<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string\">&#034;\u738b\u4e94&#034;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token string\">&#034;\u5468\u4e03&#034;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">30<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>    g1<span class=\"token punctuation\">.<\/span><span class=\"token function\">BFS<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string\">&#034;\u5f20\u4e09&#034;<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g1<span class=\"token punctuation\">.<\/span><span class=\"token function\">DFS<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string\">&#034;\u5f20\u4e09&#034;<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<h4>3.2 \u56fe\u7684\u6df1\u5ea6\u4f18\u5148\u904d\u5386<\/h4>\n<p><img decoding=\"async\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/10\/20261021231747-6ad9481b05db7.png\" alt=\"\" \/><\/p>\n<p>\u6bd4\u5982\u73b0\u5728\u8981\u627e\u4e1c\u897f&#xff0c;\u5047\u8bbe\u6709\u4e09\u4e2a\u62bd\u5c49&#xff0c;\u4e1c\u897f\u5728\u54ea\u4e2a\u62bd\u5c49\u4e0d\u6e05\u695a&#xff0c;\u73b0\u5728\u8981\u5c06\u5176\u627e\u5230&#xff0c;\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u7684\u505a\u6cd5\u662f&#xff1a;<\/p>\n<li>\u5148\u5c06\u7b2c\u4e00\u4e2a\u62bd\u5c49\u6253\u5f00&#xff0c;\u5728\u6700\u5916\u5c42\u627e\u4e00\u904d\u3002<\/li>\n<li>\u5c06\u7b2c\u4e00\u4e2a\u62bd\u5c49\u4e2d\u7ea2\u8272\u7684\u76d2\u5b50\u6253\u5f00&#xff0c;\u5728\u7ea2\u8272\u76d2\u5b50\u4e2d\u627e\u4e00\u904d\u3002<\/li>\n<li>\u5c06\u7ea2\u8272\u76d2\u5b50\u4e2d\u7eff\u8272\u7684\u76d2\u5b50\u6253\u5f00&#xff0c;\u5728\u7eff\u8272\u76d2\u5b50\u4e2d\u627e\u4e00\u904d\u3002<\/li>\n<li>\u9012\u5f52\u67e5\u627e\u5269\u4f59\u7684\u4e24\u4e2a\u76d2\u5b50\u3002<\/li>\n<p>\u6df1\u5ea6\u4f18\u5148\u904d\u5386&#xff1a;\u4e00\u4e2a\u62bd\u5c49\u4e00\u6b21\u6027\u904d\u5386\u5b8c&#xff08;\u5305\u62ec\u8be5\u62bd\u5c49\u4e2d\u5305\u542b\u7684\u5c0f\u76d2\u5b50&#xff09;&#xff0c;\u518d\u53bb\u9012\u5f52\u904d\u5386\u5176\u4ed6\u76d2\u5b50\u3002<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/10\/20261021231747-6ad9481b693b5.png\" alt=\"\" \/><\/p>\n<p><span class=\"token keyword\">void<\/span> <span class=\"token function\">_DFS<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> index<span class=\"token punctuation\">,<\/span> vector<span class=\"token operator\">&lt;<\/span><span class=\"token keyword\">bool<\/span><span class=\"token operator\">&gt;<\/span><span class=\"token operator\">&amp;<\/span> visited<span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token operator\">!<\/span>visited<span class=\"token punctuation\">[<\/span>index<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span><br \/>\n    <span class=\"token punctuation\">{<\/span><br \/>\n        cout <span class=\"token operator\">&lt;&lt;<\/span> _v<span class=\"token punctuation\">[<\/span>index<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&lt;&lt;<\/span> <span class=\"token string\">&#034; &#034;<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        visited<span class=\"token punctuation\">[<\/span>index<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> <span class=\"token boolean\">true<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>        LinkEdge<span class=\"token operator\">*<\/span> pCur <span class=\"token operator\">&#061;<\/span> _linkEdges<span class=\"token punctuation\">[<\/span>index<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">while<\/span> <span class=\"token punctuation\">(<\/span>pCur<span class=\"token punctuation\">)<\/span><br \/>\n        <span class=\"token punctuation\">{<\/span><br \/>\n            <span class=\"token function\">_DFS<\/span><span class=\"token punctuation\">(<\/span>pCur<span class=\"token operator\">-&gt;<\/span>_dst<span class=\"token punctuation\">,<\/span> visited<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            pCur <span class=\"token operator\">&#061;<\/span> pCur<span class=\"token operator\">-&gt;<\/span>_pNext<span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<p><span class=\"token keyword\">void<\/span> <span class=\"token function\">DFS<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">const<\/span> V<span class=\"token operator\">&amp;<\/span> v<span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token punctuation\">{<\/span><br \/>\n    cout <span class=\"token operator\">&lt;&lt;<\/span> <span class=\"token string\">&#034;DFS:&#034;<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    vector<span class=\"token operator\">&lt;<\/span><span class=\"token keyword\">bool<\/span><span class=\"token operator\">&gt;<\/span> <span class=\"token function\">visited<\/span><span class=\"token punctuation\">(<\/span>_v<span class=\"token punctuation\">.<\/span><span class=\"token function\">size<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token boolean\">false<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token function\">_DFS<\/span><span class=\"token punctuation\">(<\/span><span class=\"token function\">GetIndexOfV<\/span><span class=\"token punctuation\">(<\/span>v<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">,<\/span> visited<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>    <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>size_t index <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> index <span class=\"token operator\">&lt;<\/span> _v<span class=\"token punctuation\">.<\/span><span class=\"token function\">size<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span> <span class=\"token operator\">&#043;&#043;<\/span>index<span class=\"token punctuation\">)<\/span><br \/>\n        <span class=\"token function\">_DFS<\/span><span class=\"token punctuation\">(<\/span>index<span class=\"token punctuation\">,<\/span> visited<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>    cout <span class=\"token operator\">&lt;&lt;<\/span> endl<span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<p><span class=\"token keyword\">void<\/span> <span class=\"token function\">TestGraphDBFS<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token punctuation\">{<\/span><br \/>\n    string a<span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> <span class=\"token punctuation\">{<\/span> <span class=\"token string\">&#034;\u5f20\u4e09&#034;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token string\">&#034;\u674e\u56db&#034;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token string\">&#034;\u738b\u4e94&#034;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token string\">&#034;\u8d75\u516d&#034;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token string\">&#034;\u5468\u4e03&#034;<\/span> <span class=\"token punctuation\">}<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    Graph<span class=\"token operator\">&lt;<\/span>string<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span><span class=\"token operator\">&gt;<\/span> <span class=\"token function\">g1<\/span><span class=\"token punctuation\">(<\/span>a<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">sizeof<\/span><span class=\"token punctuation\">(<\/span>a<span class=\"token punctuation\">)<\/span> <span class=\"token operator\">\/<\/span> <span class=\"token keyword\">sizeof<\/span><span class=\"token punctuation\">(<\/span>string<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g1<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string\">&#034;\u5f20\u4e09&#034;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token string\">&#034;\u674e\u56db&#034;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">100<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g1<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string\">&#034;\u5f20\u4e09&#034;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token string\">&#034;\u738b\u4e94&#034;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">200<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g1<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string\">&#034;\u738b\u4e94&#034;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token string\">&#034;\u8d75\u516d&#034;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">30<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g1<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string\">&#034;\u738b\u4e94&#034;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token string\">&#034;\u5468\u4e03&#034;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">30<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<h3>4. \u6700\u5c0f\u751f\u6210\u6811<\/h3>\n<p>\u8fde\u901a\u56fe\u4e2d\u7684\u6bcf\u4e00\u68f5\u751f\u6210\u6811&#xff0c;\u90fd\u662f\u539f\u56fe\u7684\u4e00\u4e2a\u6781\u5927\u65e0\u73af\u5b50\u56fe&#xff0c;\u5373&#xff1a;\u4ece\u5176\u4e2d\u5220\u53bb\u4efb\u4f55\u4e00\u6761\u8fb9&#xff0c;\u751f\u6210\u6811\u5c31\u4e0d\u5728\u8fde\u901a&#xff1b;\u53cd\u4e4b&#xff0c;\u5728\u5176\u4e2d\u5f15\u5165\u4efb\u4f55\u4e00\u6761\u65b0\u8fb9&#xff0c;\u90fd\u4f1a\u5f62\u6210\u4e00\u6761\u56de\u8def\u3002<\/p>\n<p>\u82e5\u8fde\u901a\u56fe\u7531 n \u4e2a\u9876\u70b9\u7ec4\u6210&#xff0c;\u5219\u5176\u751f\u6210\u6811\u5fc5\u542b n \u4e2a\u9876\u70b9\u548c n &#8211; 1 \u6761\u8fb9\u3002\u56e0\u6b64\u6784\u9020\u6700\u5c0f\u751f\u6210\u6811\u7684\u51c6\u5219\u6709\u4e09\u6761&#xff1a;<\/p>\n<li>\u53ea\u80fd\u4f7f\u7528\u56fe\u4e2d\u7684\u8fb9\u6765\u6784\u9020\u6700\u5c0f\u751f\u6210\u6811\u3002<\/li>\n<li>\u53ea\u80fd\u4f7f\u7528\u6070\u597d n &#8211; 1 \u6761\u8fb9\u6765\u8fde\u63a5\u56fe\u4e2d\u7684 n \u4e2a\u9876\u70b9\u3002<\/li>\n<li>\u9009\u7528\u7684 n &#8211; 1 \u6761\u8fb9\u4e0d\u80fd\u6784\u6210\u56de\u8def\u3002<\/li>\n<p>\u6784\u9020\u6700\u5c0f\u751f\u6210\u6811\u7684\u65b9\u6cd5&#xff1a;Kruskal \u7b97\u6cd5\u548c Prim \u7b97\u6cd5\u3002\u8fd9\u4e24\u4e2a\u7b97\u6cd5\u90fd\u91c7\u7528\u4e86\u9010\u6b65\u6c42\u89e3\u7684\u8d2a\u5fc3\u7b56\u7565\u3002<\/p>\n<p>\u8d2a\u5fc3\u7b97\u6cd5&#xff1a; \u662f\u6307\u5728\u95ee\u9898\u6c42\u89e3\u65f6&#xff0c;\u603b\u662f\u505a\u51fa\u5f53\u524d\u770b\u8d77\u6765\u6700\u597d\u7684\u9009\u62e9\u3002\u4e5f\u5c31\u662f\u8bf4&#xff0c;\u8d2a\u5fc3\u7b97\u6cd5\u505a\u51fa\u7684\u4e0d\u662f\u6574\u4f53\u6700\u4f18\u7684\u9009\u62e9&#xff0c;\u800c\u662f\u67d0\u79cd\u610f\u4e49\u4e0a\u7684\u5c40\u90e8\u6700\u4f18\u89e3\u3002\u8d2a\u5fc3\u7b97\u6cd5\u4e0d\u662f\u5bf9\u6240\u6709\u7684\u95ee\u9898\u90fd\u80fd\u5f97\u5230\u6574\u4f53\u6700\u4f18\u89e3\u3002<\/p>\n<h4>4.1 Kruskal \u7b97\u6cd5<\/h4>\n<p>\u4efb\u7ed9\u4e00\u4e2a\u6709 n \u4e2a\u9876\u70b9\u7684\u8fde\u901a\u7f51\u7edc N &#061; {V, E}&#xff0c;\u9996\u5148\u6784\u9020\u4e00\u4e2a\u7531\u8fd9 n \u4e2a\u9876\u70b9\u7ec4\u6210\u3001\u4e0d\u542b\u4efb\u4f55\u8fb9\u7684\u56fe G &#061; {V, NULL}&#xff0c;\u5176\u4e2d\u6bcf\u4e2a\u9876\u70b9\u81ea\u6210\u4e00\u4e2a\u8fde\u901a\u5206\u91cf\u3002<\/p>\n<p>\u5176\u6b21\u4e0d\u65ad\u4ece E \u4e2d\u53d6\u51fa\u6743\u503c\u6700\u5c0f\u7684\u4e00\u6761\u8fb9&#xff08;\u82e5\u6709\u591a\u6761\u4efb\u53d6\u5176\u4e00&#xff09;&#xff0c;\u82e5\u8be5\u8fb9\u7684\u4e24\u4e2a\u9876\u70b9\u6765\u81ea\u4e0d\u540c\u7684\u8fde\u901a\u5206\u91cf&#xff0c;\u5219\u5c06\u6b64\u8fb9\u52a0\u5165\u5230 G \u4e2d\u3002\u5982\u6b64\u91cd\u590d&#xff0c;\u76f4\u5230\u6240\u6709\u9876\u70b9\u5728\u540c\u4e00\u4e2a\u8fde\u901a\u5206\u91cf\u4e0a\u4e3a\u6b62\u3002<\/p>\n<p>\u6838\u5fc3&#xff1a; \u6bcf\u6b21\u8fed\u4ee3\u65f6&#xff0c;\u9009\u51fa\u4e00\u6761\u5177\u6709\u6700\u5c0f\u6743\u503c&#xff0c;\u4e14\u4e24\u7aef\u70b9\u4e0d\u5728\u540c\u4e00\u8fde\u901a\u5206\u91cf\u4e0a\u7684\u8fb9&#xff0c;\u52a0\u5165\u751f\u6210\u6811\u3002<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/10\/20261021231747-6ad9481bd0439.png\" alt=\"\" \/><\/p>\n<p>W <span class=\"token function\">Kruskal<\/span><span class=\"token punctuation\">(<\/span>Self<span class=\"token operator\">&amp;<\/span> minTree<span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token punctuation\">{<\/span><br \/>\n    minTree<span class=\"token punctuation\">.<\/span>_vertexs <span class=\"token operator\">&#061;<\/span> _vertexs<span class=\"token punctuation\">;<\/span><br \/>\n    minTree<span class=\"token punctuation\">.<\/span>_vIndexMap <span class=\"token operator\">&#061;<\/span> _vIndexMap<span class=\"token punctuation\">;<\/span><br \/>\n    minTree<span class=\"token punctuation\">.<\/span>_matrix<span class=\"token punctuation\">.<\/span><span class=\"token function\">resize<\/span><span class=\"token punctuation\">(<\/span>_vertexs<span class=\"token punctuation\">.<\/span><span class=\"token function\">size<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">auto<\/span><span class=\"token operator\">&amp;<\/span> e <span class=\"token operator\">:<\/span> minTree<span class=\"token punctuation\">.<\/span>_matrix<span class=\"token punctuation\">)<\/span><br \/>\n    <span class=\"token punctuation\">{<\/span><br \/>\n        e<span class=\"token punctuation\">.<\/span><span class=\"token function\">resize<\/span><span class=\"token punctuation\">(<\/span>_vertexs<span class=\"token punctuation\">.<\/span><span class=\"token function\">size<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">,<\/span> MAX_W<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><\/p>\n<p>    priority_queue<span class=\"token operator\">&lt;<\/span>Edge<span class=\"token punctuation\">,<\/span> vector<span class=\"token operator\">&lt;<\/span>Edge<span class=\"token operator\">&gt;<\/span><span class=\"token punctuation\">,<\/span> greater<span class=\"token operator\">&lt;<\/span>Edge<span class=\"token operator\">&gt;&gt;<\/span> pq<span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>size_t i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> i <span class=\"token operator\">&lt;<\/span> _matrix<span class=\"token punctuation\">.<\/span><span class=\"token function\">size<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span> <span class=\"token operator\">&#043;&#043;<\/span>i<span class=\"token punctuation\">)<\/span><br \/>\n    <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>size_t j <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> j <span class=\"token operator\">&lt;<\/span> _matrix<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">.<\/span><span class=\"token function\">size<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span> <span class=\"token operator\">&#043;&#043;<\/span>j<span class=\"token punctuation\">)<\/span><br \/>\n        <span class=\"token punctuation\">{<\/span><br \/>\n            <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>i <span class=\"token operator\">&lt;<\/span> j <span class=\"token operator\">&amp;&amp;<\/span> _matrix<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">!&#061;<\/span> MAX_W<span class=\"token punctuation\">)<\/span><br \/>\n            <span class=\"token punctuation\">{<\/span><br \/>\n                pq<span class=\"token punctuation\">.<\/span><span class=\"token function\">push<\/span><span class=\"token punctuation\">(<\/span><span class=\"token function\">Edge<\/span><span class=\"token punctuation\">(<\/span>i<span class=\"token punctuation\">,<\/span> j<span class=\"token punctuation\">,<\/span> _matrix<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token punctuation\">}<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><\/p>\n<p>    W total <span class=\"token operator\">&#061;<\/span> <span class=\"token function\">W<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>    <span class=\"token comment\">\/\/ \u8d2a\u5fc3\u7b97\u6cd5&#xff0c;\u4ece\u6700\u5c0f\u7684\u8fb9\u5f00\u59cb\u9009<\/span><br \/>\n    size_t i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    UnionFindSet <span class=\"token function\">ufs<\/span><span class=\"token punctuation\">(<\/span>_vertexs<span class=\"token punctuation\">.<\/span><span class=\"token function\">size<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token keyword\">while<\/span> <span class=\"token punctuation\">(<\/span>i <span class=\"token operator\">&lt;<\/span> _vertexs<span class=\"token punctuation\">.<\/span><span class=\"token function\">size<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token operator\">&amp;&amp;<\/span> <span class=\"token operator\">!<\/span>pq<span class=\"token punctuation\">.<\/span><span class=\"token function\">empty<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><br \/>\n    <span class=\"token punctuation\">{<\/span><br \/>\n        Edge min <span class=\"token operator\">&#061;<\/span> pq<span class=\"token punctuation\">.<\/span><span class=\"token function\">top<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        pq<span class=\"token punctuation\">.<\/span><span class=\"token function\">pop<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>        <span class=\"token comment\">\/\/ \u8fb9\u4e0d\u5728\u4e00\u4e2a\u96c6\u5408&#xff0c;\u8bf4\u660e\u4e0d\u4f1a\u6784\u6210\u73af&#xff0c;\u5219\u6dfb\u52a0\u5230\u6700\u5c0f\u751f\u6210\u6811<\/span><br \/>\n        <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>ufs<span class=\"token punctuation\">.<\/span><span class=\"token function\">FindRoot<\/span><span class=\"token punctuation\">(<\/span>min<span class=\"token punctuation\">.<\/span>_srci<span class=\"token punctuation\">)<\/span> <span class=\"token operator\">!&#061;<\/span> ufs<span class=\"token punctuation\">.<\/span><span class=\"token function\">FindRoot<\/span><span class=\"token punctuation\">(<\/span>min<span class=\"token punctuation\">.<\/span>_dsti<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><br \/>\n        <span class=\"token punctuation\">{<\/span><br \/>\n            <span class=\"token comment\">\/\/ cout &lt;&lt; _vertexs[min._srci] &lt;&lt; &#034;-&#034; &lt;&lt; _vertexs[min._dsti]<\/span><br \/>\n            <span class=\"token comment\">\/\/      &lt;&lt; &#034;:&#034; &lt;&lt; _matrix[min._srci][min._dsti] &lt;&lt; endl;<\/span><br \/>\n            minTree<span class=\"token punctuation\">.<\/span><span class=\"token function\">_AddEdge<\/span><span class=\"token punctuation\">(<\/span>min<span class=\"token punctuation\">.<\/span>_srci<span class=\"token punctuation\">,<\/span> min<span class=\"token punctuation\">.<\/span>_dsti<span class=\"token punctuation\">,<\/span> min<span class=\"token punctuation\">.<\/span>_w<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            total <span class=\"token operator\">&#043;&#061;<\/span> min<span class=\"token punctuation\">.<\/span>_w<span class=\"token punctuation\">;<\/span><br \/>\n            ufs<span class=\"token punctuation\">.<\/span><span class=\"token function\">Union<\/span><span class=\"token punctuation\">(<\/span>min<span class=\"token punctuation\">.<\/span>_srci<span class=\"token punctuation\">,<\/span> min<span class=\"token punctuation\">.<\/span>_dsti<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token operator\">&#043;&#043;<\/span>i<span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><\/p>\n<p>    <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>i <span class=\"token operator\">&#061;&#061;<\/span> _vertexs<span class=\"token punctuation\">.<\/span><span class=\"token function\">size<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><br \/>\n    <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token keyword\">return<\/span> total<span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><br \/>\n    <span class=\"token keyword\">else<\/span><br \/>\n    <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token keyword\">return<\/span> <span class=\"token function\">W<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<p><span class=\"token keyword\">void<\/span> <span class=\"token function\">TestGraphMinTree<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token keyword\">const<\/span> <span class=\"token keyword\">char<\/span><span class=\"token operator\">*<\/span> str <span class=\"token operator\">&#061;<\/span> <span class=\"token string\">&#034;abcdefghi&#034;<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    Graph<span class=\"token operator\">&lt;<\/span><span class=\"token keyword\">char<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span><span class=\"token operator\">&gt;<\/span> <span class=\"token function\">g<\/span><span class=\"token punctuation\">(<\/span>str<span class=\"token punctuation\">,<\/span> <span class=\"token function\">strlen<\/span><span class=\"token punctuation\">(<\/span>str<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;a&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;b&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">4<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;a&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;h&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">8<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token comment\">\/\/ g.AddEdge(&#039;a&#039;, &#039;h&#039;, 9);<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;b&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;c&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">8<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;b&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;h&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">11<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;c&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;i&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">2<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;c&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;f&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">4<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;c&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;d&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">7<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;d&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;f&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">14<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;d&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;e&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">9<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;e&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;f&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">10<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;f&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;g&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">2<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;g&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;h&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;g&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;i&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">6<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;h&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;i&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">7<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>    Graph<span class=\"token operator\">&lt;<\/span><span class=\"token keyword\">char<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span><span class=\"token operator\">&gt;<\/span> kminTree<span class=\"token punctuation\">;<\/span><br \/>\n    cout <span class=\"token operator\">&lt;&lt;<\/span> <span class=\"token string\">&#034;Kruskal:&#034;<\/span> <span class=\"token operator\">&lt;&lt;<\/span> g<span class=\"token punctuation\">.<\/span><span class=\"token function\">Kruskal<\/span><span class=\"token punctuation\">(<\/span>kminTree<span class=\"token punctuation\">)<\/span> <span class=\"token operator\">&lt;&lt;<\/span> endl<span class=\"token punctuation\">;<\/span><br \/>\n    kminTree<span class=\"token punctuation\">.<\/span><span class=\"token function\">Print<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>    Graph<span class=\"token operator\">&lt;<\/span><span class=\"token keyword\">char<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span><span class=\"token operator\">&gt;<\/span> pminTree<span class=\"token punctuation\">;<\/span><br \/>\n    cout <span class=\"token operator\">&lt;&lt;<\/span> <span class=\"token string\">&#034;Prim:&#034;<\/span> <span class=\"token operator\">&lt;&lt;<\/span> g<span class=\"token punctuation\">.<\/span><span class=\"token function\">Prim<\/span><span class=\"token punctuation\">(<\/span>pminTree<span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;a&#039;<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token operator\">&lt;&lt;<\/span> endl<span class=\"token punctuation\">;<\/span><br \/>\n    pminTree<span class=\"token punctuation\">.<\/span><span class=\"token function\">Print<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<h4>4.2 Prim \u7b97\u6cd5<\/h4>\n<p><img decoding=\"async\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/10\/20261021231748-6ad9481cea067.png\" alt=\"\" \/><\/p>\n<p>W <span class=\"token function\">Prim<\/span><span class=\"token punctuation\">(<\/span>Self<span class=\"token operator\">&amp;<\/span> minTree<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">const<\/span> V<span class=\"token operator\">&amp;<\/span> src<span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token punctuation\">{<\/span><br \/>\n    minTree<span class=\"token punctuation\">.<\/span>_vertexs <span class=\"token operator\">&#061;<\/span> _vertexs<span class=\"token punctuation\">;<\/span><br \/>\n    minTree<span class=\"token punctuation\">.<\/span>_vIndexMap <span class=\"token operator\">&#061;<\/span> _vIndexMap<span class=\"token punctuation\">;<\/span><br \/>\n    minTree<span class=\"token punctuation\">.<\/span>_matrix<span class=\"token punctuation\">.<\/span><span class=\"token function\">resize<\/span><span class=\"token punctuation\">(<\/span>_vertexs<span class=\"token punctuation\">.<\/span><span class=\"token function\">size<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">auto<\/span><span class=\"token operator\">&amp;<\/span> e <span class=\"token operator\">:<\/span> minTree<span class=\"token punctuation\">.<\/span>_matrix<span class=\"token punctuation\">)<\/span><br \/>\n    <span class=\"token punctuation\">{<\/span><br \/>\n        e<span class=\"token punctuation\">.<\/span><span class=\"token function\">resize<\/span><span class=\"token punctuation\">(<\/span>_vertexs<span class=\"token punctuation\">.<\/span><span class=\"token function\">size<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">,<\/span> MAX_W<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><\/p>\n<p>    size_t srci <span class=\"token operator\">&#061;<\/span> <span class=\"token function\">GetVertexIndex<\/span><span class=\"token punctuation\">(<\/span>src<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>    set<span class=\"token operator\">&lt;<\/span>size_t<span class=\"token operator\">&gt;<\/span> inSet<span class=\"token punctuation\">;<\/span><br \/>\n    inSet<span class=\"token punctuation\">.<\/span><span class=\"token function\">insert<\/span><span class=\"token punctuation\">(<\/span>srci<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>    priority_queue<span class=\"token operator\">&lt;<\/span>Edge<span class=\"token punctuation\">,<\/span> vector<span class=\"token operator\">&lt;<\/span>Edge<span class=\"token operator\">&gt;<\/span><span class=\"token punctuation\">,<\/span> greater<span class=\"token operator\">&lt;<\/span>Edge<span class=\"token operator\">&gt;&gt;<\/span> pq<span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>size_t i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> i <span class=\"token operator\">&lt;<\/span> _vertexs<span class=\"token punctuation\">.<\/span><span class=\"token function\">size<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span> <span class=\"token operator\">&#043;&#043;<\/span>i<span class=\"token punctuation\">)<\/span><br \/>\n    <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>_matrix<span class=\"token punctuation\">[<\/span>srci<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">!&#061;<\/span> MAX_W<span class=\"token punctuation\">)<\/span><br \/>\n        <span class=\"token punctuation\">{<\/span><br \/>\n            pq<span class=\"token punctuation\">.<\/span><span class=\"token function\">push<\/span><span class=\"token punctuation\">(<\/span><span class=\"token function\">Edge<\/span><span class=\"token punctuation\">(<\/span>srci<span class=\"token punctuation\">,<\/span> i<span class=\"token punctuation\">,<\/span> _matrix<span class=\"token punctuation\">[<\/span>srci<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><\/p>\n<p>    W total <span class=\"token operator\">&#061;<\/span> <span class=\"token function\">W<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token keyword\">while<\/span> <span class=\"token punctuation\">(<\/span>inSet<span class=\"token punctuation\">.<\/span><span class=\"token function\">size<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token operator\">&lt;<\/span> _vertexs<span class=\"token punctuation\">.<\/span><span class=\"token function\">size<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token operator\">&amp;&amp;<\/span> <span class=\"token operator\">!<\/span>pq<span class=\"token punctuation\">.<\/span><span class=\"token function\">empty<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><br \/>\n    <span class=\"token punctuation\">{<\/span><br \/>\n        Edge min <span class=\"token operator\">&#061;<\/span> pq<span class=\"token punctuation\">.<\/span><span class=\"token function\">top<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        pq<span class=\"token punctuation\">.<\/span><span class=\"token function\">pop<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>        <span class=\"token comment\">\/\/ \u9632\u6b62\u73af\u7684\u95ee\u9898<\/span><br \/>\n        <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>inSet<span class=\"token punctuation\">.<\/span><span class=\"token function\">find<\/span><span class=\"token punctuation\">(<\/span>min<span class=\"token punctuation\">.<\/span>_srci<span class=\"token punctuation\">)<\/span> <span class=\"token operator\">&#061;&#061;<\/span> inSet<span class=\"token punctuation\">.<\/span><span class=\"token function\">end<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token operator\">||<\/span><br \/>\n            inSet<span class=\"token punctuation\">.<\/span><span class=\"token function\">find<\/span><span class=\"token punctuation\">(<\/span>min<span class=\"token punctuation\">.<\/span>_dsti<span class=\"token punctuation\">)<\/span> <span class=\"token operator\">&#061;&#061;<\/span> inSet<span class=\"token punctuation\">.<\/span><span class=\"token function\">end<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><br \/>\n        <span class=\"token punctuation\">{<\/span><br \/>\n            <span class=\"token comment\">\/\/ cout &lt;&lt; _vertexs[min._srci] &lt;&lt; &#034;-&#034; &lt;&lt; _vertexs[min._dsti]<\/span><br \/>\n            <span class=\"token comment\">\/\/      &lt;&lt; &#034;:&#034; &lt;&lt; _matrix[min._srci][min._dsti] &lt;&lt; endl;<\/span><br \/>\n            minTree<span class=\"token punctuation\">.<\/span><span class=\"token function\">_AddEdge<\/span><span class=\"token punctuation\">(<\/span>min<span class=\"token punctuation\">.<\/span>_srci<span class=\"token punctuation\">,<\/span> min<span class=\"token punctuation\">.<\/span>_dsti<span class=\"token punctuation\">,<\/span> min<span class=\"token punctuation\">.<\/span>_w<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            total <span class=\"token operator\">&#043;&#061;<\/span> min<span class=\"token punctuation\">.<\/span>_w<span class=\"token punctuation\">;<\/span><\/p>\n<p>            <span class=\"token comment\">\/\/ \u65b0\u5165\u9876\u70b9\u7684\u8fde\u63a5\u8fb9\u8fdb\u5165\u961f\u5217<\/span><br \/>\n            <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>size_t i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> i <span class=\"token operator\">&lt;<\/span> _vertexs<span class=\"token punctuation\">.<\/span><span class=\"token function\">size<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span> <span class=\"token operator\">&#043;&#043;<\/span>i<span class=\"token punctuation\">)<\/span><br \/>\n            <span class=\"token punctuation\">{<\/span><br \/>\n                <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>_matrix<span class=\"token punctuation\">[<\/span>min<span class=\"token punctuation\">.<\/span>_dsti<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">!&#061;<\/span> MAX_W <span class=\"token operator\">&amp;&amp;<\/span><br \/>\n                    inSet<span class=\"token punctuation\">.<\/span><span class=\"token function\">find<\/span><span class=\"token punctuation\">(<\/span>i<span class=\"token punctuation\">)<\/span> <span class=\"token operator\">&#061;&#061;<\/span> inSet<span class=\"token punctuation\">.<\/span><span class=\"token function\">end<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><br \/>\n                <span class=\"token punctuation\">{<\/span><br \/>\n                    pq<span class=\"token punctuation\">.<\/span><span class=\"token function\">push<\/span><span class=\"token punctuation\">(<\/span><span class=\"token function\">Edge<\/span><span class=\"token punctuation\">(<\/span>min<span class=\"token punctuation\">.<\/span>_dsti<span class=\"token punctuation\">,<\/span> i<span class=\"token punctuation\">,<\/span> _matrix<span class=\"token punctuation\">[<\/span>min<span class=\"token punctuation\">.<\/span>_dsti<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n                <span class=\"token punctuation\">}<\/span><br \/>\n            <span class=\"token punctuation\">}<\/span><br \/>\n            inSet<span class=\"token punctuation\">.<\/span><span class=\"token function\">insert<\/span><span class=\"token punctuation\">(<\/span>min<span class=\"token punctuation\">.<\/span>_dsti<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><\/p>\n<p>    <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>inSet<span class=\"token punctuation\">.<\/span><span class=\"token function\">size<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token operator\">&#061;&#061;<\/span> _vertexs<span class=\"token punctuation\">.<\/span><span class=\"token function\">size<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><br \/>\n    <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token keyword\">return<\/span> total<span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><br \/>\n    <span class=\"token keyword\">else<\/span><br \/>\n    <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token keyword\">return<\/span> <span class=\"token function\">W<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<p><span class=\"token keyword\">void<\/span> <span class=\"token function\">TestGraphMinTree<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token keyword\">const<\/span> <span class=\"token keyword\">char<\/span><span class=\"token operator\">*<\/span> str <span class=\"token operator\">&#061;<\/span> <span class=\"token string\">&#034;abcdefghi&#034;<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    Graph<span class=\"token operator\">&lt;<\/span><span class=\"token keyword\">char<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span><span class=\"token operator\">&gt;<\/span> <span class=\"token function\">g<\/span><span class=\"token punctuation\">(<\/span>str<span class=\"token punctuation\">,<\/span> <span class=\"token function\">strlen<\/span><span class=\"token punctuation\">(<\/span>str<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;a&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;b&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">4<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;a&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;h&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">8<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token comment\">\/\/ g.AddEdge(&#039;a&#039;, &#039;h&#039;, 9);<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;b&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;c&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">8<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;b&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;h&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">11<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;c&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;i&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">2<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;c&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;f&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">4<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;c&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;d&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">7<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;d&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;f&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">14<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;d&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;e&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">9<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;e&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;f&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">10<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;f&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;g&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">2<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;g&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;h&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;g&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;i&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">6<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;h&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;i&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">7<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>    Graph<span class=\"token operator\">&lt;<\/span><span class=\"token keyword\">char<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span><span class=\"token operator\">&gt;<\/span> kminTree<span class=\"token punctuation\">;<\/span><br \/>\n    cout <span class=\"token operator\">&lt;&lt;<\/span> <span class=\"token string\">&#034;Kruskal:&#034;<\/span> <span class=\"token operator\">&lt;&lt;<\/span> g<span class=\"token punctuation\">.<\/span><span class=\"token function\">Kruskal<\/span><span class=\"token punctuation\">(<\/span>kminTree<span class=\"token punctuation\">)<\/span> <span class=\"token operator\">&lt;&lt;<\/span> endl<span class=\"token punctuation\">;<\/span><br \/>\n    kminTree<span class=\"token punctuation\">.<\/span><span class=\"token function\">Print<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>    Graph<span class=\"token operator\">&lt;<\/span><span class=\"token keyword\">char<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span><span class=\"token operator\">&gt;<\/span> pminTree<span class=\"token punctuation\">;<\/span><br \/>\n    cout <span class=\"token operator\">&lt;&lt;<\/span> <span class=\"token string\">&#034;Prim:&#034;<\/span> <span class=\"token operator\">&lt;&lt;<\/span> g<span class=\"token punctuation\">.<\/span><span class=\"token function\">Prim<\/span><span class=\"token punctuation\">(<\/span>pminTree<span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;a&#039;<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token operator\">&lt;&lt;<\/span> endl<span class=\"token punctuation\">;<\/span><br \/>\n    pminTree<span class=\"token punctuation\">.<\/span><span class=\"token function\">Print<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<h3>5. \u6700\u77ed\u8def\u5f84\u95ee\u9898<\/h3>\n<p>\u6700\u77ed\u8def\u5f84\u95ee\u9898&#xff1a; \u4ece\u5e26\u6743\u6709\u5411\u56fe G \u4e2d\u7684\u67d0\u4e00\u9876\u70b9\u51fa\u53d1&#xff0c;\u627e\u51fa\u4e00\u6761\u901a\u5f80\u53e6\u4e00\u9876\u70b9\u7684\u6700\u77ed\u8def\u5f84&#xff0c;\u6700\u77ed\u4e5f\u5c31\u662f\u6cbf\u8def\u5f84\u5404\u8fb9\u7684\u6743\u503c\u603b\u548c\u8fbe\u5230\u6700\u5c0f\u3002<\/p>\n<h4>5.1 \u5355\u6e90\u6700\u77ed\u8def\u5f84\u2013Dijkstra \u7b97\u6cd5<\/h4>\n<p>\u5355\u6e90\u6700\u77ed\u8def\u5f84\u95ee\u9898&#xff1a; \u7ed9\u5b9a\u4e00\u4e2a\u56fe G &#061; (V, E)&#xff0c;\u6c42\u6e90\u7ed3\u70b9 s \u2208 V \u5230\u56fe\u4e2d\u6bcf\u4e2a\u7ed3\u70b9 v \u2208 V \u7684\u6700\u77ed\u8def\u5f84\u3002Dijkstra \u7b97\u6cd5\u9002\u7528\u4e8e\u89e3\u51b3\u5e26\u6743\u91cd\u7684\u6709\u5411\u56fe\u4e0a\u7684\u5355\u6e90\u6700\u77ed\u8def\u5f84\u95ee\u9898&#xff0c;\u540c\u65f6\u7b97\u6cd5\u8981\u6c42\u56fe\u4e2d\u6240\u6709\u8fb9\u7684\u6743\u91cd\u975e\u8d1f\u3002\u4e00\u822c\u5728\u6c42\u89e3\u6700\u77ed\u8def\u5f84\u7684\u65f6\u5019\u90fd\u662f\u5df2\u77e5\u4e00\u4e2a\u8d77\u70b9\u548c\u4e00\u4e2a\u7ec8\u70b9&#xff0c;\u6240\u4ee5\u4f7f\u7528 Dijkstra \u7b97\u6cd5\u6c42\u89e3\u8fc7\u540e\u4e5f\u5c31\u5f97\u5230\u4e86\u6240\u9700\u8d77\u70b9\u5230\u7ec8\u70b9\u7684\u6700\u77ed\u8def\u5f84\u3002<\/p>\n<p>\u9488\u5bf9\u4e00\u4e2a\u5e26\u6743\u6709\u5411\u56fe G&#xff0c;\u5c06\u6240\u6709\u7ed3\u70b9\u5206\u4e3a\u4e24\u7ec4 S \u548c Q&#xff0c;S \u662f\u5df2\u7ecf\u786e\u5b9a\u6700\u77ed\u8def\u5f84\u7684\u7ed3\u70b9\u96c6\u5408&#xff0c;\u5728\u521d\u59cb\u65f6\u4e3a\u7a7a&#xff08;\u521d\u59cb\u65f6\u4e5f\u53ef\u4ee5\u5c06\u6e90\u8282\u70b9 s \u653e\u5165&#xff0c;\u6bd5\u7adf\u6e90\u8282\u70b9\u5230\u81ea\u5df1\u7684\u4ee3\u4ef7\u662f 0&#xff09;&#xff0c;Q \u4e3a\u5176\u4f59\u672a\u786e\u5b9a\u6700\u77ed\u8def\u5f84\u7684\u7ed3\u70b9\u96c6\u5408\u3002\u6bcf\u6b21\u4ece Q \u4e2d\u627e\u51fa\u4e00\u4e2a\u8d77\u70b9\u5230\u8be5\u7ed3\u70b9\u4ee3\u4ef7\u6700\u5c0f\u7684\u7ed3\u70b9 u&#xff0c;\u5c06 u \u4ece Q \u4e2d\u79fb\u51fa&#xff0c;\u5e76\u653e\u5165 S \u4e2d&#xff0c;\u5bf9 u \u7684\u6bcf\u4e00\u4e2a\u76f8\u90bb\u7ed3\u70b9 v \u8fdb\u884c\u677e\u5f1b\u64cd\u4f5c\u3002<\/p>\n<p>\u677e\u5f1b\u5373\u5bf9\u6bcf\u4e00\u4e2a\u76f8\u90bb\u7ed3\u70b9 v&#xff0c;\u5224\u65ad\u6e90\u8282\u70b9 s \u5230\u7ed3\u70b9 u \u7684\u4ee3\u4ef7\u4e0e u \u5230 v \u7684\u4ee3\u4ef7\u4e4b\u548c\u662f\u5426\u6bd4\u539f\u6765 s \u5230 v \u7684\u4ee3\u4ef7\u66f4\u5c0f&#xff0c;\u82e5\u4ee3\u4ef7\u6bd4\u539f\u6765\u5c0f\u5219\u8981\u5c06 s \u5230 v \u7684\u4ee3\u4ef7\u66f4\u65b0\u4e3a s \u5230 u \u4e0e u \u5230 v \u7684\u4ee3\u4ef7\u4e4b\u548c&#xff0c;\u5426\u5219\u7ef4\u6301\u539f\u6837\u3002\u5982\u6b64\u4e00\u76f4\u5faa\u73af\u76f4\u81f3\u96c6\u5408 Q \u4e3a\u7a7a&#xff0c;\u5373\u6240\u6709\u8282\u70b9\u90fd\u5df2\u7ecf\u67e5\u627e\u8fc7\u4e00\u904d\u5e76\u786e\u5b9a\u4e86\u6700\u77ed\u8def\u5f84&#xff0c;\u81f3\u4e8e\u4e00\u4e9b\u8d77\u70b9\u5230\u8fbe\u4e0d\u4e86\u7684\u7ed3\u70b9\u5728\u7b97\u6cd5\u5faa\u73af\u540e\u5176\u4ee3\u4ef7\u4ecd\u4e3a\u521d\u59cb\u8bbe\u5b9a\u7684\u503c&#xff0c;\u4e0d\u53d1\u751f\u53d8\u5316\u3002Dijkstra \u7b97\u6cd5\u6bcf\u6b21\u90fd\u662f\u9009\u62e9 V &#8211; S \u4e2d\u6700\u5c0f\u7684\u8def\u5f84\u8282\u70b9\u6765\u8fdb\u884c\u66f4\u65b0&#xff0c;\u5e76\u52a0\u5165 S \u4e2d&#xff0c;\u6240\u4ee5\u8be5\u7b97\u6cd5\u4f7f\u7528\u7684\u662f\u8d2a\u5fc3\u7b56\u7565\u3002<\/p>\n<p>Dijkstra \u7b97\u6cd5\u5b58\u5728\u7684\u95ee\u9898\u662f\u4e0d\u652f\u6301\u56fe\u4e2d\u5e26\u8d1f\u6743\u8def\u5f84&#xff0c;\u5982\u679c\u5e26\u6709\u8d1f\u6743\u8def\u5f84&#xff0c;\u5219\u53ef\u80fd\u4f1a\u627e\u4e0d\u5230\u4e00\u4e9b\u8def\u5f84\u7684\u6700\u77ed\u8def\u5f84\u3002<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/10\/20261021231749-6ad9481d9e992.png\" alt=\"\" \/><\/p>\n<p><span class=\"token keyword\">void<\/span> <span class=\"token function\">Dijkstra<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">const<\/span> V<span class=\"token operator\">&amp;<\/span> src<span class=\"token punctuation\">,<\/span> vector<span class=\"token operator\">&lt;<\/span>W<span class=\"token operator\">&gt;<\/span><span class=\"token operator\">&amp;<\/span> dist<span class=\"token punctuation\">,<\/span> vector<span class=\"token operator\">&lt;<\/span><span class=\"token keyword\">int<\/span><span class=\"token operator\">&gt;<\/span><span class=\"token operator\">&amp;<\/span> parentPath<span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token punctuation\">{<\/span><br \/>\n    size_t N <span class=\"token operator\">&#061;<\/span> _vertexs<span class=\"token punctuation\">.<\/span><span class=\"token function\">size<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    size_t srci <span class=\"token operator\">&#061;<\/span> <span class=\"token function\">GetVertexIndex<\/span><span class=\"token punctuation\">(<\/span>src<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>    <span class=\"token comment\">\/\/ vector&lt;W&gt; dist&#xff0c;\u8bb0\u5f55 srci-\u5176\u4ed6\u9876\u70b9\u6700\u77ed\u8def\u5f84\u6743\u503c\u6570\u7ec4<\/span><br \/>\n    dist<span class=\"token punctuation\">.<\/span><span class=\"token function\">resize<\/span><span class=\"token punctuation\">(<\/span>N<span class=\"token punctuation\">,<\/span> MAX_W<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token comment\">\/\/ vector&lt;int&gt; parentPath&#xff0c;\u8bb0\u5f55 srci-\u5176\u4ed6\u9876\u70b9\u6700\u77ed\u8def\u5f84\u7236\u9876\u70b9\u6570\u7ec4<\/span><br \/>\n    parentPath<span class=\"token punctuation\">.<\/span><span class=\"token function\">resize<\/span><span class=\"token punctuation\">(<\/span>N<span class=\"token punctuation\">,<\/span> <span class=\"token operator\">&#8211;<\/span><span class=\"token number\">1<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>    <span class=\"token comment\">\/\/ \u6807\u8bb0\u662f\u5426\u627e\u5230\u6700\u77ed\u8def\u5f84\u7684\u9876\u70b9\u96c6\u5408 S<\/span><br \/>\n    vector<span class=\"token operator\">&lt;<\/span><span class=\"token keyword\">bool<\/span><span class=\"token operator\">&gt;<\/span> S<span class=\"token punctuation\">;<\/span><br \/>\n    S<span class=\"token punctuation\">.<\/span><span class=\"token function\">resize<\/span><span class=\"token punctuation\">(<\/span>N<span class=\"token punctuation\">,<\/span> <span class=\"token boolean\">false<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>    <span class=\"token comment\">\/\/ srci \u7684\u6743\u503c\u7ed9\u4e00\u4e2a\u6700\u5c0f\u503c&#xff0c;\u65b9\u4fbf\u8d2a\u5fc3\u7b2c\u4e00\u6b21\u627e\u5230\u8fd9\u4e2a\u8282\u70b9<\/span><br \/>\n    dist<span class=\"token punctuation\">[<\/span>srci<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> <span class=\"token function\">W<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>    <span class=\"token comment\">\/\/ N \u4e2a\u9876\u70b9\u66f4\u65b0 N \u6b21<\/span><br \/>\n    <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>size_t i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> i <span class=\"token operator\">&lt;<\/span> N<span class=\"token punctuation\">;<\/span> <span class=\"token operator\">&#043;&#043;<\/span>i<span class=\"token punctuation\">)<\/span><br \/>\n    <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token comment\">\/\/ \u8d2a\u5fc3\u7b97\u6cd5&#xff1a;srci \u5230\u4e0d\u5728 S \u4e2d\u8def\u5f84\u6700\u77ed\u7684\u90a3\u4e2a\u9876\u70b9 u<\/span><br \/>\n        W min <span class=\"token operator\">&#061;<\/span> MAX_W<span class=\"token punctuation\">;<\/span><br \/>\n        size_t u <span class=\"token operator\">&#061;<\/span> srci<span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>size_t j <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> j <span class=\"token operator\">&lt;<\/span> N<span class=\"token punctuation\">;<\/span> <span class=\"token operator\">&#043;&#043;<\/span>j<span class=\"token punctuation\">)<\/span><br \/>\n        <span class=\"token punctuation\">{<\/span><br \/>\n            <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>S<span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;&#061;<\/span> <span class=\"token boolean\">false<\/span> <span class=\"token operator\">&amp;&amp;<\/span> dist<span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&lt;<\/span> min<span class=\"token punctuation\">)<\/span><br \/>\n            <span class=\"token punctuation\">{<\/span><br \/>\n                min <span class=\"token operator\">&#061;<\/span> dist<span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span><br \/>\n                u <span class=\"token operator\">&#061;<\/span> j<span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token punctuation\">}<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><br \/>\n        S<span class=\"token punctuation\">[<\/span>u<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> <span class=\"token boolean\">true<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>        <span class=\"token comment\">\/\/ \u677e\u5f1b\u7b97\u6cd5&#xff1a;\u66f4\u65b0\u4e00\u904d u \u8fde\u63a5\u7684\u6240\u6709\u8fb9&#xff0c;\u770b\u662f\u5426\u80fd\u66f4\u65b0\u51fa\u66f4\u77ed\u8fde\u63a5\u8def\u5f84<\/span><br \/>\n        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>size_t k <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> k <span class=\"token operator\">&lt;<\/span> N<span class=\"token punctuation\">;<\/span> <span class=\"token operator\">&#043;&#043;<\/span>k<span class=\"token punctuation\">)<\/span><br \/>\n        <span class=\"token punctuation\">{<\/span><br \/>\n            <span class=\"token comment\">\/\/ \u5982\u679c srci-&gt;u &#043; u-&gt;k \u6bd4 srci-&gt;k \u66f4\u77ed\u5219\u8fdb\u884c\u66f4\u65b0<\/span><br \/>\n            <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>S<span class=\"token punctuation\">[<\/span>k<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;&#061;<\/span> <span class=\"token boolean\">false<\/span> <span class=\"token operator\">&amp;&amp;<\/span> _matrix<span class=\"token punctuation\">[<\/span>u<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>k<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">!&#061;<\/span> MAX_W <span class=\"token operator\">&amp;&amp;<\/span><br \/>\n                dist<span class=\"token punctuation\">[<\/span>u<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#043;<\/span> _matrix<span class=\"token punctuation\">[<\/span>u<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>k<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&lt;<\/span> dist<span class=\"token punctuation\">[<\/span>k<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span><br \/>\n            <span class=\"token punctuation\">{<\/span><br \/>\n                dist<span class=\"token punctuation\">[<\/span>k<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> dist<span class=\"token punctuation\">[<\/span>u<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#043;<\/span> _matrix<span class=\"token punctuation\">[<\/span>u<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>k<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span><br \/>\n                parentPath<span class=\"token punctuation\">[<\/span>k<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> u<span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token punctuation\">}<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<p><span class=\"token comment\">\/\/ \u6253\u5370\u6700\u77ed\u8def\u5f84\u7684\u903b\u8f91\u7b97\u6cd5<\/span><br \/>\n<span class=\"token keyword\">void<\/span> <span class=\"token function\">PrinrtShotPath<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">const<\/span> V<span class=\"token operator\">&amp;<\/span> src<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">const<\/span> vector<span class=\"token operator\">&lt;<\/span>W<span class=\"token operator\">&gt;<\/span><span class=\"token operator\">&amp;<\/span> dist<span class=\"token punctuation\">,<\/span><br \/>\n                    <span class=\"token keyword\">const<\/span> vector<span class=\"token operator\">&lt;<\/span><span class=\"token keyword\">int<\/span><span class=\"token operator\">&gt;<\/span><span class=\"token operator\">&amp;<\/span> parentPath<span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token punctuation\">{<\/span><br \/>\n    size_t N <span class=\"token operator\">&#061;<\/span> _vertexs<span class=\"token punctuation\">.<\/span><span class=\"token function\">size<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    size_t srci <span class=\"token operator\">&#061;<\/span> <span class=\"token function\">GetVertexIndex<\/span><span class=\"token punctuation\">(<\/span>src<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>size_t i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> i <span class=\"token operator\">&lt;<\/span> N<span class=\"token punctuation\">;<\/span> <span class=\"token operator\">&#043;&#043;<\/span>i<span class=\"token punctuation\">)<\/span><br \/>\n    <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>i <span class=\"token operator\">&#061;&#061;<\/span> srci<span class=\"token punctuation\">)<\/span><br \/>\n            <span class=\"token keyword\">continue<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>        vector<span class=\"token operator\">&lt;<\/span><span class=\"token keyword\">int<\/span><span class=\"token operator\">&gt;<\/span> path<span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">int<\/span> parenti <span class=\"token operator\">&#061;<\/span> i<span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">while<\/span> <span class=\"token punctuation\">(<\/span>parenti <span class=\"token operator\">!&#061;<\/span> srci<span class=\"token punctuation\">)<\/span><br \/>\n        <span class=\"token punctuation\">{<\/span><br \/>\n            path<span class=\"token punctuation\">.<\/span><span class=\"token function\">push_back<\/span><span class=\"token punctuation\">(<\/span>parenti<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            parenti <span class=\"token operator\">&#061;<\/span> parentPath<span class=\"token punctuation\">[<\/span>parenti<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><br \/>\n        path<span class=\"token punctuation\">.<\/span><span class=\"token function\">push_back<\/span><span class=\"token punctuation\">(<\/span>srci<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token function\">reverse<\/span><span class=\"token punctuation\">(<\/span>path<span class=\"token punctuation\">.<\/span><span class=\"token function\">begin<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">,<\/span> path<span class=\"token punctuation\">.<\/span><span class=\"token function\">end<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">auto<\/span> pos <span class=\"token operator\">:<\/span> path<span class=\"token punctuation\">)<\/span><br \/>\n        <span class=\"token punctuation\">{<\/span><br \/>\n            cout <span class=\"token operator\">&lt;&lt;<\/span> _vertexs<span class=\"token punctuation\">[<\/span>pos<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&lt;&lt;<\/span> <span class=\"token string\">&#034;-&gt;&#034;<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><br \/>\n        cout <span class=\"token operator\">&lt;&lt;<\/span> dist<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&lt;&lt;<\/span> endl<span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<p><span class=\"token keyword\">void<\/span> <span class=\"token function\">TestGraphDijkstra<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token keyword\">const<\/span> <span class=\"token keyword\">char<\/span><span class=\"token operator\">*<\/span> str <span class=\"token operator\">&#061;<\/span> <span class=\"token string\">&#034;syztx&#034;<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    Graph<span class=\"token operator\">&lt;<\/span><span class=\"token keyword\">char<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span><span class=\"token punctuation\">,<\/span> INT_MAX<span class=\"token punctuation\">,<\/span> <span class=\"token boolean\">true<\/span><span class=\"token operator\">&gt;<\/span> <span class=\"token function\">g<\/span><span class=\"token punctuation\">(<\/span>str<span class=\"token punctuation\">,<\/span> <span class=\"token function\">strlen<\/span><span class=\"token punctuation\">(<\/span>str<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;s&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;t&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">10<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;s&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;y&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">5<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;y&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;t&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">3<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;y&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;x&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">9<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;y&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;z&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">2<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;z&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;s&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">7<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;z&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;x&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">6<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;t&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;y&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">2<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;t&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;x&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;x&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;z&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">4<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>    vector<span class=\"token operator\">&lt;<\/span><span class=\"token keyword\">int<\/span><span class=\"token operator\">&gt;<\/span> dist<span class=\"token punctuation\">;<\/span><br \/>\n    vector<span class=\"token operator\">&lt;<\/span><span class=\"token keyword\">int<\/span><span class=\"token operator\">&gt;<\/span> parentPath<span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">Dijkstra<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;s&#039;<\/span><span class=\"token punctuation\">,<\/span> dist<span class=\"token punctuation\">,<\/span> parentPath<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">PrinrtShotPath<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;s&#039;<\/span><span class=\"token punctuation\">,<\/span> dist<span class=\"token punctuation\">,<\/span> parentPath<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>    <span class=\"token comment\">\/\/ \u56fe\u4e2d\u5e26\u6709\u8d1f\u6743\u8def\u5f84\u65f6&#xff0c;\u8d2a\u5fc3\u7b56\u7565\u5219\u5931\u6548\u4e86\u3002<\/span><br \/>\n    <span class=\"token comment\">\/\/ \u6d4b\u8bd5\u7ed3\u679c\u53ef\u4ee5\u770b\u5230 s-&gt;t-&gt;y \u4e4b\u95f4\u7684\u6700\u77ed\u8def\u5f84\u6ca1\u66f4\u65b0\u51fa\u6765<\/span><br \/>\n    <span class=\"token comment\">\/*<br \/>\n    const char* str &#061; &#034;sytx&#034;;<br \/>\n    Graph&lt;char, int, INT_MAX, true&gt; g(str, strlen(str));<br \/>\n    g.AddEdge(&#039;s&#039;, &#039;t&#039;, 10);<br \/>\n    g.AddEdge(&#039;s&#039;, &#039;y&#039;, 5);<br \/>\n    g.AddEdge(&#039;t&#039;, &#039;y&#039;, -7);<br \/>\n    g.AddEdge(&#039;y&#039;, &#039;x&#039;, 3);<\/p>\n<p>    vector&lt;int&gt; dist;<br \/>\n    vector&lt;int&gt; parentPath;<br \/>\n    g.Dijkstra(&#039;s&#039;, dist, parentPath);<br \/>\n    g.PrinrtShotPath(&#039;s&#039;, dist, parentPath);<br \/>\n    *\/<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<h4>5.2 \u5355\u6e90\u6700\u77ed\u8def\u5f84\u2013Bellman-Ford \u7b97\u6cd5<\/h4>\n<p>Dijkstra \u7b97\u6cd5\u53ea\u80fd\u7528\u6765\u89e3\u51b3\u6b63\u6743\u56fe\u7684\u5355\u6e90\u6700\u77ed\u8def\u5f84\u95ee\u9898&#xff0c;\u4f46\u6709\u4e9b\u9898\u76ee\u4f1a\u51fa\u73b0\u8d1f\u6743\u56fe\u3002\u8fd9\u65f6\u8fd9\u4e2a\u7b97\u6cd5\u5c31\u4e0d\u80fd\u5e2e\u52a9\u6211\u4eec\u89e3\u51b3\u95ee\u9898\u4e86&#xff0c;\u800c Bellman-Ford \u7b97\u6cd5\u53ef\u4ee5\u89e3\u51b3\u8d1f\u6743\u56fe\u7684\u5355\u6e90\u6700\u77ed\u8def\u5f84\u95ee\u9898\u3002<\/p>\n<p>\u5b83\u7684\u4f18\u70b9\u662f\u53ef\u4ee5\u89e3\u51b3\u5e26\u6709\u8d1f\u6743\u8fb9\u7684\u5355\u6e90\u6700\u77ed\u8def\u5f84\u95ee\u9898&#xff0c;\u800c\u4e14\u53ef\u4ee5\u7528\u6765\u5224\u65ad\u662f\u5426\u6709\u8d1f\u6743\u56de\u8def\u3002\u5b83\u4e5f\u6709\u660e\u663e\u7684\u7f3a\u70b9&#xff0c;\u5b83\u7684\u65f6\u95f4\u590d\u6742\u5ea6 O(N * E)&#xff08;N \u662f\u70b9\u6570&#xff0c;E \u662f\u8fb9\u6570&#xff09;\u666e\u904d\u662f\u8981\u9ad8\u4e8e Dijkstra \u7b97\u6cd5 O(N\u00b2) \u7684\u3002\u50cf\u8fd9\u91cc\u5982\u679c\u6211\u4eec\u4f7f\u7528\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0&#xff0c;\u90a3\u4e48\u904d\u5386\u6240\u6709\u8fb9\u7684\u6570\u91cf\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5c31\u662f O(N^3)&#xff0c;\u8fd9\u91cc\u4e5f\u53ef\u4ee5\u770b\u51fa\u6765 Bellman-Ford \u5c31\u662f\u4e00\u79cd\u66b4\u529b\u6c42\u89e3\u66f4\u65b0\u3002<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/10\/20261021231750-6ad9481e17cde.png\" alt=\"\" \/><\/p>\n<p><span class=\"token keyword\">bool<\/span> <span class=\"token function\">BellmanFord<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">const<\/span> V<span class=\"token operator\">&amp;<\/span> src<span class=\"token punctuation\">,<\/span> vector<span class=\"token operator\">&lt;<\/span>W<span class=\"token operator\">&gt;<\/span><span class=\"token operator\">&amp;<\/span> dist<span class=\"token punctuation\">,<\/span> vector<span class=\"token operator\">&lt;<\/span><span class=\"token keyword\">int<\/span><span class=\"token operator\">&gt;<\/span><span class=\"token operator\">&amp;<\/span> parentPath<span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token punctuation\">{<\/span><br \/>\n    size_t N <span class=\"token operator\">&#061;<\/span> _vertexs<span class=\"token punctuation\">.<\/span><span class=\"token function\">size<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    size_t srci <span class=\"token operator\">&#061;<\/span> <span class=\"token function\">GetVertexIndex<\/span><span class=\"token punctuation\">(<\/span>src<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>    <span class=\"token comment\">\/\/ vector&lt;W&gt; dist&#xff0c;\u8bb0\u5f55 srci-\u5176\u4ed6\u9876\u70b9\u6700\u77ed\u8def\u5f84\u6743\u503c\u6570\u7ec4<\/span><br \/>\n    dist<span class=\"token punctuation\">.<\/span><span class=\"token function\">resize<\/span><span class=\"token punctuation\">(<\/span>N<span class=\"token punctuation\">,<\/span> MAX_W<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token comment\">\/\/ vector&lt;int&gt; parentPath&#xff0c;\u8bb0\u5f55 srci-\u5176\u4ed6\u9876\u70b9\u6700\u77ed\u8def\u5f84\u7236\u9876\u70b9\u6570\u7ec4<\/span><br \/>\n    parentPath<span class=\"token punctuation\">.<\/span><span class=\"token function\">resize<\/span><span class=\"token punctuation\">(<\/span>N<span class=\"token punctuation\">,<\/span> <span class=\"token operator\">&#8211;<\/span><span class=\"token number\">1<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>    <span class=\"token comment\">\/\/ \u5148\u66f4\u65b0 srci-&gt;srci \u4e3a\u6700\u5c0f\u503c<\/span><br \/>\n    dist<span class=\"token punctuation\">[<\/span>srci<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> <span class=\"token function\">W<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>    <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>size_t k <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> k <span class=\"token operator\">&lt;<\/span> N <span class=\"token operator\">&#8211;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span> <span class=\"token operator\">&#043;&#043;<\/span>k<span class=\"token punctuation\">)<\/span><br \/>\n    <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token keyword\">bool<\/span> exchange <span class=\"token operator\">&#061;<\/span> <span class=\"token boolean\">false<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>size_t i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> i <span class=\"token operator\">&lt;<\/span> N<span class=\"token punctuation\">;<\/span> <span class=\"token operator\">&#043;&#043;<\/span>i<span class=\"token punctuation\">)<\/span><br \/>\n        <span class=\"token punctuation\">{<\/span><br \/>\n            <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>size_t j <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> j <span class=\"token operator\">&lt;<\/span> N<span class=\"token punctuation\">;<\/span> <span class=\"token operator\">&#043;&#043;<\/span>j<span class=\"token punctuation\">)<\/span><br \/>\n            <span class=\"token punctuation\">{<\/span><br \/>\n                <span class=\"token comment\">\/\/ srci-&gt;i &#043; i-&gt;j &lt; srci-&gt;j \u5219\u66f4\u65b0\u8def\u5f84\u53ca\u6743\u503c<\/span><br \/>\n                <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>_matrix<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">!&#061;<\/span> MAX_W <span class=\"token operator\">&amp;&amp;<\/span><br \/>\n                    dist<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#043;<\/span> _matrix<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&lt;<\/span> dist<span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span><br \/>\n                <span class=\"token punctuation\">{<\/span><br \/>\n                    dist<span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> dist<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#043;<\/span> _matrix<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span><br \/>\n                    parentPath<span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> i<span class=\"token punctuation\">;<\/span><br \/>\n                    exchange <span class=\"token operator\">&#061;<\/span> <span class=\"token boolean\">true<\/span><span class=\"token punctuation\">;<\/span><br \/>\n                <span class=\"token punctuation\">}<\/span><br \/>\n            <span class=\"token punctuation\">}<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><\/p>\n<p>        <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>exchange <span class=\"token operator\">&#061;&#061;<\/span> <span class=\"token boolean\">false<\/span><span class=\"token punctuation\">)<\/span><br \/>\n            <span class=\"token keyword\">break<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><\/p>\n<p>    <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>size_t i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> i <span class=\"token operator\">&lt;<\/span> N<span class=\"token punctuation\">;<\/span> <span class=\"token operator\">&#043;&#043;<\/span>i<span class=\"token punctuation\">)<\/span><br \/>\n    <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>size_t j <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> j <span class=\"token operator\">&lt;<\/span> N<span class=\"token punctuation\">;<\/span> <span class=\"token operator\">&#043;&#043;<\/span>j<span class=\"token punctuation\">)<\/span><br \/>\n        <span class=\"token punctuation\">{<\/span><br \/>\n            <span class=\"token comment\">\/\/ \u68c0\u67e5\u6709\u6ca1\u6709\u8d1f\u6743\u56de\u8def<\/span><br \/>\n            <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>_matrix<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">!&#061;<\/span> MAX_W <span class=\"token operator\">&amp;&amp;<\/span><br \/>\n                dist<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#043;<\/span> _matrix<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&lt;<\/span> dist<span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span><br \/>\n            <span class=\"token punctuation\">{<\/span><br \/>\n                <span class=\"token keyword\">return<\/span> <span class=\"token boolean\">false<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token punctuation\">}<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><br \/>\n    <span class=\"token keyword\">return<\/span> <span class=\"token boolean\">true<\/span><span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<p><span class=\"token keyword\">void<\/span> <span class=\"token function\">TestGraphBellmanFord<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token keyword\">const<\/span> <span class=\"token keyword\">char<\/span><span class=\"token operator\">*<\/span> str <span class=\"token operator\">&#061;<\/span> <span class=\"token string\">&#034;syztx&#034;<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    Graph<span class=\"token operator\">&lt;<\/span><span class=\"token keyword\">char<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span><span class=\"token punctuation\">,<\/span> INT_MAX<span class=\"token punctuation\">,<\/span> <span class=\"token boolean\">true<\/span><span class=\"token operator\">&gt;<\/span> <span class=\"token function\">g<\/span><span class=\"token punctuation\">(<\/span>str<span class=\"token punctuation\">,<\/span> <span class=\"token function\">strlen<\/span><span class=\"token punctuation\">(<\/span>str<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;s&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;t&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">6<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;s&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;y&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">7<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;y&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;z&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">9<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;y&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;x&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token operator\">&#8211;<\/span><span class=\"token number\">3<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;z&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;s&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">2<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;z&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;x&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">7<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;t&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;x&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">5<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;t&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;y&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">8<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;t&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;z&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token operator\">&#8211;<\/span><span class=\"token number\">4<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;x&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;t&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token operator\">&#8211;<\/span><span class=\"token number\">2<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>    vector<span class=\"token operator\">&lt;<\/span><span class=\"token keyword\">int<\/span><span class=\"token operator\">&gt;<\/span> dist<span class=\"token punctuation\">;<\/span><br \/>\n    vector<span class=\"token operator\">&lt;<\/span><span class=\"token keyword\">int<\/span><span class=\"token operator\">&gt;<\/span> parentPath<span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>g<span class=\"token punctuation\">.<\/span><span class=\"token function\">BellmanFord<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;s&#039;<\/span><span class=\"token punctuation\">,<\/span> dist<span class=\"token punctuation\">,<\/span> parentPath<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><br \/>\n    <span class=\"token punctuation\">{<\/span><br \/>\n        g<span class=\"token punctuation\">.<\/span><span class=\"token function\">PrinrtShotPath<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;s&#039;<\/span><span class=\"token punctuation\">,<\/span> dist<span class=\"token punctuation\">,<\/span> parentPath<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><br \/>\n    <span class=\"token keyword\">else<\/span><br \/>\n    <span class=\"token punctuation\">{<\/span><br \/>\n        cout <span class=\"token operator\">&lt;&lt;<\/span> <span class=\"token string\">&#034;\u5b58\u5728\u8d1f\u6743\u56de\u8def&#034;<\/span> <span class=\"token operator\">&lt;&lt;<\/span> endl<span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><\/p>\n<p>    <span class=\"token comment\">\/\/ \u5fae\u8c03\u56fe\u7ed3\u6784&#xff0c;\u5e26\u6709\u8d1f\u6743\u56de\u8def\u7684\u6d4b\u8bd5<\/span><br \/>\n    <span class=\"token comment\">\/*<br \/>\n    const char* str &#061; &#034;syztx&#034;;<br \/>\n    Graph&lt;char, int, INT_MAX, true&gt; g(str, strlen(str));<br \/>\n    g.AddEdge(&#039;s&#039;, &#039;t&#039;, 6);<br \/>\n    g.AddEdge(&#039;s&#039;, &#039;y&#039;, 7);<br \/>\n    g.AddEdge(&#039;y&#039;, &#039;x&#039;, -3);<br \/>\n    g.AddEdge(&#039;y&#039;, &#039;z&#039;, 9);<br \/>\n    g.AddEdge(&#039;y&#039;, &#039;x&#039;, -3);<br \/>\n    g.AddEdge(&#039;y&#039;, &#039;s&#039;, 1); \/\/ \u65b0\u589e<br \/>\n    g.AddEdge(&#039;z&#039;, &#039;s&#039;, 2);<br \/>\n    g.AddEdge(&#039;z&#039;, &#039;x&#039;, 7);<br \/>\n    g.AddEdge(&#039;t&#039;, &#039;x&#039;, 5);<br \/>\n    g.AddEdge(&#039;t&#039;, &#039;y&#039;, -8); \/\/ \u66f4\u6539<br \/>\n    g.AddEdge(&#039;t&#039;, &#039;z&#039;, -4);<br \/>\n    g.AddEdge(&#039;x&#039;, &#039;t&#039;, -2);<\/p>\n<p>    vector&lt;int&gt; dist;<br \/>\n    vector&lt;int&gt; parentPath;<br \/>\n    if (g.BellmanFord(&#039;s&#039;, dist, parentPath))<br \/>\n    {<br \/>\n        g.PrinrtShotPath(&#039;s&#039;, dist, parentPath);<br \/>\n    }<br \/>\n    else<br \/>\n    {<br \/>\n        cout &lt;&lt; &#034;\u5b58\u5728\u8d1f\u6743\u56de\u8def&#034; &lt;&lt; endl;<br \/>\n    }<br \/>\n    *\/<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<h4>5.3 \u591a\u6e90\u6700\u77ed\u8def\u5f84\u2013Floyd-Warshall \u7b97\u6cd5<\/h4>\n<p>Floyd-Warshall \u7b97\u6cd5\u662f\u89e3\u51b3\u4efb\u610f\u4e24\u70b9\u95f4\u6700\u77ed\u8def\u5f84\u7684\u4e00\u79cd\u7b97\u6cd5\u3002<\/p>\n<p>Floyd \u7b97\u6cd5\u8003\u8651\u7684\u662f\u4e00\u6761\u6700\u77ed\u8def\u5f84\u7684\u4e2d\u95f4\u8282\u70b9&#xff0c;\u5373\u7b80\u5355\u8def\u5f84 p &#061; {v_1, v_2, &#8230;, v_n} \u4e0a\u9664 v_1 \u548c v_n \u5916\u7684\u4efb\u610f\u8282\u70b9\u3002<\/p>\n<p>\u8bbe k \u662f p \u7684\u4e00\u4e2a\u4e2d\u95f4\u8282\u70b9&#xff0c;\u90a3\u4e48\u4ece i \u5230 j \u7684\u6700\u77ed\u8def\u5f84 p \u5c31\u88ab\u5206\u6210 i \u5230 k \u548c k \u5230 j \u7684\u4e24\u6bb5\u6700\u77ed\u8def\u5f84 p_1\u3001p_2\u3002p_1 \u662f\u4ece i \u5230 k \u4e14\u4e2d\u95f4\u8282\u70b9\u5c5e\u4e8e {1, 2, &#8230;, k &#8211; 1} \u53d6\u5f97\u7684\u4e00\u6761\u6700\u77ed\u8def\u5f84\u3002p_2 \u662f\u4ece k \u5230 j \u4e14\u4e2d\u95f4\u8282\u70b9\u5c5e\u4e8e {1, 2, &#8230;, k &#8211; 1} \u53d6\u5f97\u7684\u4e00\u6761\u6700\u77ed\u8def\u5f84\u3002\u4ece\u7ed3\u70b9 k \u5230\u7ed3\u70b9 j \u7684\u8def\u5f84\u4e5f\u9075\u5b88\u540c\u6837\u7684\u89c4\u5219\u3002<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/10\/20261021231750-6ad9481e7f98b.png\" alt=\"\" \/><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/10\/20261021231750-6ad9481ebe3df.png\" alt=\"\" \/><\/p>\n<p>\u5373 Floyd \u7b97\u6cd5\u672c\u8d28\u662f\u4e09\u7ef4\u52a8\u6001\u89c4\u5212&#xff0c;D[i][j][k] \u8868\u793a\u4ece\u70b9 i \u5230\u70b9 j \u53ea\u7ecf\u8fc7 0 \u5230 k \u4e2a\u70b9\u7684\u6700\u77ed\u8def\u5f84&#xff0c;\u7136\u540e\u5efa\u7acb\u8f6c\u79fb\u65b9\u7a0b\u3002\u901a\u8fc7\u7a7a\u95f4\u4f18\u5316&#xff0c;\u4f18\u5316\u6389\u6700\u540e\u4e00\u7ef4&#xff0c;\u53d8\u6210\u4e00\u4e2a\u6700\u77ed\u8def\u5f84\u7684\u8fed\u4ee3\u7b97\u6cd5&#xff0c;\u6700\u540e\u5373\u53ef\u5f97\u5230\u6240\u6709\u70b9\u7684\u6700\u77ed\u8def\u3002<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/10\/20261021231751-6ad9481f093cc.png\" alt=\"\" \/><\/p>\n<p><span class=\"token keyword\">void<\/span> <span class=\"token function\">FloydWarShall<\/span><span class=\"token punctuation\">(<\/span>vector<span class=\"token operator\">&lt;<\/span>vector<span class=\"token operator\">&lt;<\/span>W<span class=\"token operator\">&gt;&gt;<\/span><span class=\"token operator\">&amp;<\/span> vvDist<span class=\"token punctuation\">,<\/span><br \/>\n                   vector<span class=\"token operator\">&lt;<\/span>vector<span class=\"token operator\">&lt;<\/span><span class=\"token keyword\">int<\/span><span class=\"token operator\">&gt;&gt;<\/span><span class=\"token operator\">&amp;<\/span> vvParentPath<span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token punctuation\">{<\/span><br \/>\n    size_t N <span class=\"token operator\">&#061;<\/span> _vertexs<span class=\"token punctuation\">.<\/span><span class=\"token function\">size<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    vvDist<span class=\"token punctuation\">.<\/span><span class=\"token function\">resize<\/span><span class=\"token punctuation\">(<\/span>N<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    vvParentPath<span class=\"token punctuation\">.<\/span><span class=\"token function\">resize<\/span><span class=\"token punctuation\">(<\/span>N<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>    <span class=\"token comment\">\/\/ \u521d\u59cb\u5316\u6743\u503c\u548c\u8def\u5f84\u77e9\u9635<\/span><br \/>\n    <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>size_t i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> i <span class=\"token operator\">&lt;<\/span> N<span class=\"token punctuation\">;<\/span> <span class=\"token operator\">&#043;&#043;<\/span>i<span class=\"token punctuation\">)<\/span><br \/>\n    <span class=\"token punctuation\">{<\/span><br \/>\n        vvDist<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">.<\/span><span class=\"token function\">resize<\/span><span class=\"token punctuation\">(<\/span>N<span class=\"token punctuation\">,<\/span> MAX_W<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        vvParentPath<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">.<\/span><span class=\"token function\">resize<\/span><span class=\"token punctuation\">(<\/span>N<span class=\"token punctuation\">,<\/span> <span class=\"token operator\">&#8211;<\/span><span class=\"token number\">1<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><\/p>\n<p>    <span class=\"token comment\">\/\/ \u5c06\u76f4\u63a5\u76f8\u8fde\u7684\u8def\u5f84\u521d\u59cb\u5316<\/span><br \/>\n    <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>size_t i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> i <span class=\"token operator\">&lt;<\/span> N<span class=\"token punctuation\">;<\/span> <span class=\"token operator\">&#043;&#043;<\/span>i<span class=\"token punctuation\">)<\/span><br \/>\n    <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>size_t j <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> j <span class=\"token operator\">&lt;<\/span> N<span class=\"token punctuation\">;<\/span> <span class=\"token operator\">&#043;&#043;<\/span>j<span class=\"token punctuation\">)<\/span><br \/>\n        <span class=\"token punctuation\">{<\/span><br \/>\n            <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>_matrix<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">!&#061;<\/span> MAX_W<span class=\"token punctuation\">)<\/span><br \/>\n            <span class=\"token punctuation\">{<\/span><br \/>\n                vvDist<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> _matrix<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span><br \/>\n                vvParentPath<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> i<span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token punctuation\">}<\/span><br \/>\n            <span class=\"token keyword\">else<\/span><br \/>\n            <span class=\"token punctuation\">{<\/span><br \/>\n                vvParentPath<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> <span class=\"token operator\">&#8211;<\/span><span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token punctuation\">}<\/span><\/p>\n<p>            <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>i <span class=\"token operator\">&#061;&#061;<\/span> j<span class=\"token punctuation\">)<\/span><br \/>\n            <span class=\"token punctuation\">{<\/span><br \/>\n                vvDist<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><br \/>\n                vvParentPath<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> <span class=\"token operator\">&#8211;<\/span><span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span><br \/>\n            <span class=\"token punctuation\">}<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><\/p>\n<p>    <span class=\"token comment\">\/\/ \u4f9d\u6b21\u7528\u9876\u70b9 k \u4f5c\u4e3a\u4e2d\u8f6c\u70b9\u66f4\u65b0\u6700\u77ed\u8def\u5f84<\/span><br \/>\n    <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>size_t k <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> k <span class=\"token operator\">&lt;<\/span> N<span class=\"token punctuation\">;<\/span> <span class=\"token operator\">&#043;&#043;<\/span>k<span class=\"token punctuation\">)<\/span><br \/>\n    <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>size_t i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> i <span class=\"token operator\">&lt;<\/span> N<span class=\"token punctuation\">;<\/span> <span class=\"token operator\">&#043;&#043;<\/span>i<span class=\"token punctuation\">)<\/span><br \/>\n        <span class=\"token punctuation\">{<\/span><br \/>\n            <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>size_t j <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> j <span class=\"token operator\">&lt;<\/span> N<span class=\"token punctuation\">;<\/span> <span class=\"token operator\">&#043;&#043;<\/span>j<span class=\"token punctuation\">)<\/span><br \/>\n            <span class=\"token punctuation\">{<\/span><br \/>\n                <span class=\"token comment\">\/\/ i-&gt;k &#043; k-&gt;j \u6bd4 i-&gt;j \u524d\u9762\u66f4\u65b0\u7684\u8ddd\u79bb\u66f4\u77ed&#xff0c;\u5219\u66f4\u65b0<\/span><br \/>\n                <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>vvDist<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>k<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">!&#061;<\/span> MAX_W <span class=\"token operator\">&amp;&amp;<\/span> vvDist<span class=\"token punctuation\">[<\/span>k<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">!&#061;<\/span> MAX_W <span class=\"token operator\">&amp;&amp;<\/span><br \/>\n                    vvDist<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>k<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#043;<\/span> vvDist<span class=\"token punctuation\">[<\/span>k<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&lt;<\/span> vvDist<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span><br \/>\n                <span class=\"token punctuation\">{<\/span><br \/>\n                    vvDist<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> vvDist<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>k<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#043;<\/span> vvDist<span class=\"token punctuation\">[<\/span>k<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span><br \/>\n                    vvParentPath<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> vvParentPath<span class=\"token punctuation\">[<\/span>k<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span><br \/>\n                <span class=\"token punctuation\">}<\/span><br \/>\n            <span class=\"token punctuation\">}<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><\/p>\n<p>        <span class=\"token comment\">\/\/ \u6253\u5370\u6743\u503c\u548c\u8def\u5f84\u77e9\u9635\u89c2\u5bdf\u6570\u636e<\/span><br \/>\n        <span class=\"token comment\">\/\/ for (size_t i &#061; 0; i &lt; N; &#043;&#043;i)<\/span><br \/>\n        <span class=\"token comment\">\/\/ {<\/span><br \/>\n        <span class=\"token comment\">\/\/     for (size_t j &#061; 0; j &lt; N; &#043;&#043;j)<\/span><br \/>\n        <span class=\"token comment\">\/\/     {<\/span><br \/>\n        <span class=\"token comment\">\/\/         if (vvDist[i][j] &#061;&#061; MAX_W)<\/span><br \/>\n        <span class=\"token comment\">\/\/         {<\/span><br \/>\n        <span class=\"token comment\">\/\/             \/\/ cout &lt;&lt; &#034;*&#034; &lt;&lt; &#034; &#034;;<\/span><br \/>\n        <span class=\"token comment\">\/\/             printf(&#034;%3c&#034;, &#039;*&#039;);<\/span><br \/>\n        <span class=\"token comment\">\/\/         }<\/span><br \/>\n        <span class=\"token comment\">\/\/         else<\/span><br \/>\n        <span class=\"token comment\">\/\/         {<\/span><br \/>\n        <span class=\"token comment\">\/\/             \/\/ cout &lt;&lt; vvDist[i][j] &lt;&lt; &#034; &#034;;<\/span><br \/>\n        <span class=\"token comment\">\/\/             printf(&#034;%3d&#034;, vvDist[i][j]);<\/span><br \/>\n        <span class=\"token comment\">\/\/         }<\/span><br \/>\n        <span class=\"token comment\">\/\/     }<\/span><br \/>\n        <span class=\"token comment\">\/\/     cout &lt;&lt; endl;<\/span><br \/>\n        <span class=\"token comment\">\/\/ }<\/span><br \/>\n        <span class=\"token comment\">\/\/ cout &lt;&lt; endl;<\/span><br \/>\n        <span class=\"token comment\">\/\/ for (size_t i &#061; 0; i &lt; N; &#043;&#043;i)<\/span><br \/>\n        <span class=\"token comment\">\/\/ {<\/span><br \/>\n        <span class=\"token comment\">\/\/     for (size_t j &#061; 0; j &lt; N; &#043;&#043;j)<\/span><br \/>\n        <span class=\"token comment\">\/\/     {<\/span><br \/>\n        <span class=\"token comment\">\/\/         \/\/ cout &lt;&lt; vvParentPath[i][j] &lt;&lt; &#034; &#034;;<\/span><br \/>\n        <span class=\"token comment\">\/\/         printf(&#034;%3d&#034;, vvParentPath[i][j]);<\/span><br \/>\n        <span class=\"token comment\">\/\/     }<\/span><br \/>\n        <span class=\"token comment\">\/\/     cout &lt;&lt; endl;<\/span><br \/>\n        <span class=\"token comment\">\/\/ }<\/span><br \/>\n        <span class=\"token comment\">\/\/ cout &lt;&lt; &#034;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#061;&#034; &lt;&lt; endl;<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<p><span class=\"token keyword\">void<\/span> <span class=\"token function\">TestFloydWarShall<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token keyword\">const<\/span> <span class=\"token keyword\">char<\/span><span class=\"token operator\">*<\/span> str <span class=\"token operator\">&#061;<\/span> <span class=\"token string\">&#034;12345&#034;<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    Graph<span class=\"token operator\">&lt;<\/span><span class=\"token keyword\">char<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span><span class=\"token punctuation\">,<\/span> INT_MAX<span class=\"token punctuation\">,<\/span> <span class=\"token boolean\">true<\/span><span class=\"token operator\">&gt;<\/span> <span class=\"token function\">g<\/span><span class=\"token punctuation\">(<\/span>str<span class=\"token punctuation\">,<\/span> <span class=\"token function\">strlen<\/span><span class=\"token punctuation\">(<\/span>str<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;1&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;2&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">3<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;1&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;3&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">8<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;1&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;5&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token operator\">&#8211;<\/span><span class=\"token number\">4<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;2&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;4&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;2&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;5&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">7<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;3&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;2&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">4<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;4&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;1&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">2<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;4&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;3&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token operator\">&#8211;<\/span><span class=\"token number\">5<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">AddEdge<\/span><span class=\"token punctuation\">(<\/span><span class=\"token char\">&#039;5&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token char\">&#039;4&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">6<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>    vector<span class=\"token operator\">&lt;<\/span>vector<span class=\"token operator\">&lt;<\/span><span class=\"token keyword\">int<\/span><span class=\"token operator\">&gt;&gt;<\/span> vvDist<span class=\"token punctuation\">;<\/span><br \/>\n    vector<span class=\"token operator\">&lt;<\/span>vector<span class=\"token operator\">&lt;<\/span><span class=\"token keyword\">int<\/span><span class=\"token operator\">&gt;&gt;<\/span> vvParentPath<span class=\"token punctuation\">;<\/span><br \/>\n    g<span class=\"token punctuation\">.<\/span><span class=\"token function\">FloydWarShall<\/span><span class=\"token punctuation\">(<\/span>vvDist<span class=\"token punctuation\">,<\/span> vvParentPath<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>    <span class=\"token comment\">\/\/ \u6253\u5370\u4efb\u610f\u4e24\u70b9\u4e4b\u95f4\u7684\u6700\u77ed\u8def\u5f84<\/span><br \/>\n    <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>size_t i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> i <span class=\"token operator\">&lt;<\/span> <span class=\"token function\">strlen<\/span><span class=\"token punctuation\">(<\/span>str<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span> <span class=\"token operator\">&#043;&#043;<\/span>i<span class=\"token punctuation\">)<\/span><br \/>\n    <span class=\"token punctuation\">{<\/span><br \/>\n        g<span class=\"token punctuation\">.<\/span><span class=\"token function\">PrinrtShotPath<\/span><span class=\"token punctuation\">(<\/span>str<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">,<\/span> vvDist<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">,<\/span> vvParentPath<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        cout <span class=\"token operator\">&lt;&lt;<\/span> endl<span class=\"token punctuation\">;<\/span><br 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