{"id":119131,"date":"2026-10-28T23:56:31","date_gmt":"2026-10-28T15:56:31","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/119131.html"},"modified":"2026-10-28T23:56:31","modified_gmt":"2026-10-28T15:56:31","slug":"%e5%88%86%e6%b2%bb%e7%ae%97%e6%b3%95%e7%9a%84%e8%af%a6%e8%a7%a3","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/119131.html","title":{"rendered":"\u5206\u6cbb\u7b97\u6cd5\u7684\u8be6\u89e3"},"content":{"rendered":"<p>\u5206\u6cbb\u7b97\u6cd5&#xff08;Divide and Conquer&#xff09;\u662f\u7b97\u6cd5\u8bbe\u8ba1\u4e2d\u6700\u6838\u5fc3\u7684\u601d\u60f3\u4e4b\u4e00\u3002\u5b83\u7684\u903b\u8f91\u975e\u5e38\u7b26\u5408\u76f4\u89c9&#xff1a;\u628a\u4e00\u4e2a\u5927\u95ee\u9898\u62c6\u6210\u51e0\u4e2a\u5c0f\u95ee\u9898&#xff0c;\u5206\u522b\u89e3\u51b3\u5c0f\u95ee\u9898&#xff0c;\u518d\u628a\u5c0f\u95ee\u9898\u7684\u7ed3\u679c\u5408\u5e76\u8d77\u6765\u5f97\u5230\u5927\u95ee\u9898\u7684\u89e3\u3002<\/p>\n<p>\u4e0b\u9762\u4ece\u539f\u7406\u3001\u9002\u7528\u6761\u4ef6\u3001\u7ecf\u5178\u5b9e\u4f8b\u5230\u7f16\u7801\u5b9e\u73b0&#xff0c;\u5b8c\u6574\u68b3\u7406\u4e00\u904d\u3002<\/p>\n<p>\u4e00\u3001\u6838\u5fc3\u601d\u60f3\u4e0e\u4e09\u4e2a\u6b65\u9aa4<\/p>\n<p>\u5206\u6cbb\u7b97\u6cd5\u7684\u6267\u884c\u8fc7\u7a0b\u56fa\u5b9a\u4e3a\u4e09\u6b65&#xff1a;<\/p>\n<p>\u5206\u89e3&#xff08;Divide&#xff09;&#xff1a;\u5c06\u539f\u95ee\u9898\u5206\u89e3\u4e3a\u82e5\u5e72\u4e2a\u89c4\u6a21\u8f83\u5c0f\u3001\u76f8\u4e92\u72ec\u7acb\u3001\u4e0e\u539f\u95ee\u9898\u5f62\u5f0f\u76f8\u540c\u7684\u5b50\u95ee\u9898\u3002<br \/>\n\u89e3\u51b3&#xff08;Conquer&#xff09;&#xff1a;\u9012\u5f52\u5730\u6c42\u89e3\u5404\u4e2a\u5b50\u95ee\u9898\u3002\u82e5\u5b50\u95ee\u9898\u89c4\u6a21\u8db3\u591f\u5c0f&#xff0c;\u5219\u76f4\u63a5\u6c42\u89e3\u3002<br \/>\n\u5408\u5e76&#xff08;Merge&#xff09;&#xff1a;\u5c06\u5404\u5b50\u95ee\u9898\u7684\u89e3\u5408\u5e76\u4e3a\u539f\u95ee\u9898\u7684\u89e3\u3002<br \/>\n\u7528\u9012\u5f52\u4f2a\u4ee3\u7801\u8868\u8fbe\u5c31\u662f&#xff1a;<\/p>\n<p>text<br \/>\ndivide_and_conquer(problem):<br \/>\nif problem \u8db3\u591f\u5c0f:<br \/>\nreturn \u76f4\u63a5\u6c42\u89e3(problem)<br \/>\nsub_problems &#061; \u5206\u89e3(problem)<br \/>\nsub_results &#061; [divide_and_conquer\u00a7 for p in sub_problems]<br \/>\nreturn \u5408\u5e76(sub_results)<br \/>\n\u4e8c\u3001\u9002\u7528\u6761\u4ef6&#xff08;\u4ec0\u4e48\u65f6\u5019\u80fd\u7528\u5206\u6cbb&#xff09;<\/p>\n<p>\u4e0d\u662f\u6240\u6709\u95ee\u9898\u90fd\u80fd\u5206\u6cbb&#xff0c;\u5fc5\u987b\u540c\u65f6\u6ee1\u8db3\u4ee5\u4e0b\u6761\u4ef6&#xff1a;<\/p>\n<p>\u53ef\u5206\u89e3\u6027&#xff1a;\u95ee\u9898\u80fd\u5206\u89e3\u4e3a\u89c4\u6a21\u66f4\u5c0f\u7684\u540c\u7c7b\u5b50\u95ee\u9898\u3002<br \/>\n\u5b50\u95ee\u9898\u72ec\u7acb&#xff1a;\u5b50\u95ee\u9898\u4e4b\u95f4\u4e0d\u5305\u542b\u516c\u5171\u7684\u5b50\u5b50\u95ee\u9898&#xff08;\u5426\u5219\u4f1a\u4ea7\u751f\u5927\u91cf\u91cd\u590d\u8ba1\u7b97&#xff0c;\u8fd9\u65f6\u66f4\u9002\u5408\u7528\u52a8\u6001\u89c4\u5212&#xff09;\u3002<br \/>\n\u53ef\u5408\u5e76\u6027&#xff1a;\u5b50\u95ee\u9898\u7684\u89e3\u80fd\u6709\u6548\u5408\u5e76\u6210\u539f\u95ee\u9898\u7684\u89e3\u3002<br \/>\n\u89c4\u6a21\u7f29\u5c0f&#xff1a;\u5206\u89e3\u540e\u95ee\u9898\u89c4\u6a21\u786e\u5b9e\u51cf\u5c0f&#xff0c;\u4e14\u80fd\u9012\u5f52\u5230\u8fb9\u754c\u6761\u4ef6\u3002<br \/>\n\u5173\u952e\u533a\u5206&#xff1a;\u5206\u6cbb vs \u52a8\u6001\u89c4\u5212\u3002\u5206\u6cbb\u7684\u5b50\u95ee\u9898\u76f8\u4e92\u72ec\u7acb&#xff08;\u5982\u5f52\u5e76\u6392\u5e8f\u5de6\u53f3\u4e24\u534a\u4e92\u4e0d\u5f71\u54cd&#xff09;&#xff1b;\u52a8\u6001\u89c4\u5212\u7684\u5b50\u95ee\u9898\u76f8\u4e92\u91cd\u53e0&#xff08;\u5982\u6590\u6ce2\u90a3\u5951\u6570\u5217&#xff0c;\u9700\u8981\u8bb0\u5fc6\u5316\u907f\u514d\u91cd\u590d\u8ba1\u7b97&#xff09;\u3002<br \/>\n\u4e09\u3001\u7ecf\u5178\u5e94\u7528\u5b9e\u4f8b<\/p>\n<li>\u5f52\u5e76\u6392\u5e8f&#xff08;Merge Sort&#xff09;\u2014\u2014 \u6700\u5178\u578b\u7684\u5206\u6cbb<\/li>\n<p>\u5206\u89e3&#xff1a;\u628a\u6570\u7ec4\u4ece\u4e2d\u95f4\u5207\u6210\u4e24\u534a\u3002<br \/>\n\u89e3\u51b3&#xff1a;\u9012\u5f52\u5730\u5bf9\u5de6\u53f3\u4e24\u534a\u5206\u522b\u6392\u5e8f\u3002<br \/>\n\u5408\u5e76&#xff1a;\u5c06\u4e24\u4e2a\u6709\u5e8f\u5b50\u6570\u7ec4\u5408\u5e76\u6210\u4e00\u4e2a\u6709\u5e8f\u6570\u7ec4\u3002<br \/>\n\u65f6\u95f4\u590d\u6742\u5ea6&#xff1a;O(n log n)&#xff0c;\u7a7a\u95f4\u590d\u6742\u5ea6 O(n)&#xff08;\u5408\u5e76\u65f6\u9700\u8981\u8f85\u52a9\u6570\u7ec4&#xff09;\u3002<\/p>\n<li>\u5feb\u901f\u6392\u5e8f&#xff08;Quick Sort&#xff09;<\/li>\n<p>\u5206\u89e3&#xff1a;\u9009\u4e00\u4e2a\u57fa\u51c6\u503c&#xff08;pivot&#xff09;&#xff0c;\u628a\u6570\u7ec4\u5206\u6210&#034;\u5c0f\u4e8e\u57fa\u51c6&#034;\u548c&#034;\u5927\u4e8e\u57fa\u51c6&#034;\u4e24\u90e8\u5206\u3002<br \/>\n\u89e3\u51b3&#xff1a;\u9012\u5f52\u6392\u5e8f\u4e24\u90e8\u5206\u3002<br \/>\n\u5408\u5e76&#xff1a;\u65e0\u9700\u5408\u5e76&#xff0c;\u539f\u5730\u5206\u533a\u540e\u81ea\u7136\u6709\u5e8f\u3002<br \/>\n\u6ce8\u610f&#xff1a;\u5feb\u6392\u7684&#034;\u5206\u89e3&#034;\u4f9d\u8d56\u57fa\u51c6\u9009\u62e9&#xff0c;\u6700\u574f\u60c5\u51b5 O(n\u00b2)&#xff0c;\u5e73\u5747 O(n log n)\u3002\u5b83\u4e0e\u5f52\u5e76\u7684\u533a\u522b\u5728\u4e8e\u5206\u89e3\u65f6\u505a\u4e86\u5927\u91cf\u5de5\u4f5c&#xff0c;\u5408\u5e76\u65f6\u51e0\u4e4e\u4e0d\u505a\u4e8b\u3002<\/p>\n<li>\u4e8c\u5206\u67e5\u627e&#xff08;Binary Search&#xff09;<\/li>\n<p>\u5206\u89e3&#xff1a;\u4e0e\u4e2d\u95f4\u5143\u7d20\u6bd4\u8f83&#xff0c;\u5c06\u641c\u7d22\u8303\u56f4\u7f29\u5c0f\u4e00\u534a\u3002<br \/>\n\u89e3\u51b3&#xff1a;\u9012\u5f52\u5728\u67d0\u4e00\u534a\u4e2d\u67e5\u627e\u3002<br \/>\n\u5408\u5e76&#xff1a;\u65e0\u9700\u5408\u5e76&#xff0c;\u76f4\u63a5\u8fd4\u56de\u7ed3\u679c\u3002<br \/>\n\u8fd9\u662f&#034;\u5206\u89e3\u540e\u53ea\u89e3\u51b3\u4e00\u4e2a\u5b50\u95ee\u9898&#034;\u7684\u7279\u4f8b&#xff0c;\u65f6\u95f4\u590d\u6742\u5ea6 O(log n)\u3002<\/p>\n<li>\u5927\u6574\u6570\u4e58\u6cd5&#xff08;Karatsuba \u7b97\u6cd5&#xff09;<\/li>\n<p>\u5c06 n \u4f4d\u5927\u6574\u6570\u4e58\u6cd5\u5206\u89e3\u4e3a 3 \u6b21 n\/2 \u4f4d\u4e58\u6cd5&#xff08;\u800c\u975e\u6734\u7d20\u7b97\u6cd5\u7684 4 \u6b21&#xff09;&#xff0c;\u628a\u590d\u6742\u5ea6\u4ece O(n\u00b2) \u964d\u5230 O(n^1.585)\u3002<\/p>\n<li>\u6700\u8fd1\u70b9\u5bf9\u95ee\u9898<\/li>\n<p>\u5c06\u70b9\u96c6\u6309 x \u5750\u6807\u5206\u6210\u5de6\u53f3\u4e24\u534a&#xff0c;\u5206\u522b\u6c42\u6700\u8fd1\u70b9\u5bf9&#xff0c;\u518d\u5904\u7406\u8de8\u8d8a\u4e2d\u7ebf\u7684\u70b9\u5bf9&#xff0c;\u5408\u5e76\u5f97\u5230\u5168\u5c40\u6700\u8fd1\u70b9\u5bf9\u3002\u590d\u6742\u5ea6 O(n log n)\u3002<\/p>\n<p>\u56db\u3001\u7f16\u7801\u5b9e\u4f8b<\/p>\n<p>\u5b9e\u4f8b 1&#xff1a;\u5f52\u5e76\u6392\u5e8f&#xff08;Python&#xff09;<\/p>\n<p>python<br \/>\ndef merge_sort(arr):<br \/>\n# \u8fb9\u754c\u6761\u4ef6&#xff1a;\u957f\u5ea6 0 \u6216 1 \u76f4\u63a5\u8fd4\u56de<br \/>\nif len(arr) &lt;&#061; 1:<br \/>\nreturn arr<\/p>\n<p># 1. \u5206\u89e3&#xff1a;\u4ece\u4e2d\u95f4\u5207\u5206<br \/>\nmid &#061; len(arr) \/\/ 2<br \/>\nleft &#061; merge_sort(arr[:mid])    # 2. \u89e3\u51b3&#xff1a;\u9012\u5f52\u6392\u5e8f\u5de6\u534a<br \/>\nright &#061; merge_sort(arr[mid:])   #    \u9012\u5f52\u6392\u5e8f\u53f3\u534a<\/p>\n<p># 3. \u5408\u5e76&#xff1a;\u5c06\u4e24\u4e2a\u6709\u5e8f\u6570\u7ec4\u5408\u5e76<br \/>\nreturn merge(left, right)<\/p>\n<p>def merge(left, right):<br \/>\nresult &#061; []<br \/>\ni &#061; j &#061; 0<br \/>\nwhile i &lt; len(left) and j &lt; len(right):<br \/>\nif left[i] &lt;&#061; right[j]:<br \/>\nresult.append(left[i])<br \/>\ni &#043;&#061; 1<br \/>\nelse:<br \/>\nresult.append(right[j])<br \/>\nj &#043;&#061; 1<br \/>\n# \u628a\u5269\u4f59\u90e8\u5206\u8ffd\u52a0\u8fdb\u53bb<br \/>\nresult.extend(left[i:])<br \/>\nresult.extend(right[j:])<br \/>\nreturn result<\/p>\n<h2>\u6d4b\u8bd5<\/h2>\n<p>if name &#061;&#061; \u201cmain\u201d:<br \/>\ndata &#061; [38, 27, 43, 3, 9, 82, 10]<br \/>\nprint(merge_sort(data))  # [3, 9, 10, 27, 38, 43, 82]<br \/>\n\u6267\u884c\u8fc7\u7a0b\u793a\u610f&#xff08;\u4ee5 [38, 27, 43, 3] \u4e3a\u4f8b&#xff09;&#xff1a;<\/p>\n<p>text<br \/>\n[38, 27, 43, 3]<br \/>\n\/<br \/>\n[38,27]    [43,3]<br \/>\n\/  \\\\       \/<br \/>\n[38][27]  [43][3]<br \/>\n\\\\  \/       \\\\  \/<br \/>\n[27,38]   [3,43]<br \/>\n\\\\        \/<br \/>\n[3,27,38,43]<br \/>\n\u5b9e\u4f8b 2&#xff1a;\u5feb\u901f\u6392\u5e8f&#xff08;Python&#xff09;<\/p>\n<p>python<br \/>\ndef quick_sort(arr):<br \/>\nif len(arr) &lt;&#061; 1:<br \/>\nreturn arr<\/p>\n<p># 1. \u5206\u89e3&#xff1a;\u9009\u57fa\u51c6&#xff0c;\u5206\u533a<br \/>\npivot &#061; arr[len(arr) \/\/ 2]<br \/>\nleft &#061; [x for x in arr if x &lt; pivot]      # \u5c0f\u4e8e\u57fa\u51c6<br \/>\nmiddle &#061; [x for x in arr if x &#061;&#061; pivot]   # \u7b49\u4e8e\u57fa\u51c6<br \/>\nright &#061; [x for x in arr if x &gt; pivot]     # \u5927\u4e8e\u57fa\u51c6<\/p>\n<p># 2. \u89e3\u51b3 &#043; 3. \u5408\u5e76&#xff1a;\u9012\u5f52\u6392\u5e8f\u5de6\u53f3\u5e76\u62fc\u63a5<br \/>\nreturn quick_sort(left) &#043; middle &#043; quick_sort(right)<\/p>\n<h2>\u6d4b\u8bd5<\/h2>\n<p>data &#061; [38, 27, 43, 3, 9, 82, 10]<br \/>\nprint(quick_sort(data))  # [3, 9, 10, 27, 38, 43, 82]<br \/>\n\u4e0a\u9762\u8fd9\u4e2a\u5199\u6cd5\u53ef\u8bfb\u6027\u5f3a\u4f46\u989d\u5916\u7a7a\u95f4\u591a\u3002\u5de5\u7a0b\u4e2d\u5e38\u7528\u539f\u5730\u5206\u533a&#xff08;Lomuto \u6216 Hoare \u5206\u533a&#xff09;\u7248\u672c&#xff0c;\u7a7a\u95f4\u590d\u6742\u5ea6 O(log n)&#xff08;\u9012\u5f52\u6808&#xff09;\u3002<br \/>\n\u5b9e\u4f8b 3&#xff1a;\u4e8c\u5206\u67e5\u627e&#xff08;Python&#xff09;<\/p>\n<p>python<br \/>\ndef binary_search(arr, target):<br \/>\n# arr \u5fc5\u987b\u662f\u6709\u5e8f\u6570\u7ec4<br \/>\nif not arr:<br \/>\nreturn -1<\/p>\n<p>mid &#061; len(arr) \/\/ 2<br \/>\nif arr[mid] &#061;&#061; target:<br \/>\n    return mid                       # \u627e\u5230&#xff0c;\u8fd4\u56de\u7d22\u5f15<br \/>\nelif arr[mid] &lt; target:<br \/>\n    # \u53ea\u89e3\u51b3\u53f3\u534a\u90e8\u5206\u8fd9\u4e00\u4e2a\u5b50\u95ee\u9898<br \/>\n    res &#061; binary_search(arr[mid&#043;1:], target)<br \/>\n    return mid &#043; 1 &#043; res if res !&#061; -1 else -1<br \/>\nelse:<br \/>\n    # \u53ea\u89e3\u51b3\u5de6\u534a\u90e8\u5206<br \/>\n    return binary_search(arr[:mid], target)<\/p>\n<h2>\u6d4b\u8bd5<\/h2>\n<p>data &#061; [3, 9, 10, 27, 38, 43, 82]<br \/>\nprint(binary_search(data, 27))   # 3<br \/>\nprint(binary_search(data, 100))  # -1<br \/>\n\u5de5\u7a0b\u4e2d\u66f4\u63a8\u8350\u8fed\u4ee3\u7248\u4e8c\u5206\u67e5\u627e&#xff0c;\u907f\u514d\u5207\u7247\u62f7\u8d1d\u548c\u9012\u5f52\u5f00\u9500&#xff1a;<\/p>\n<p>python<br \/>\ndef binary_search_iter(arr, target):<br \/>\nlo, hi &#061; 0, len(arr) &#8211; 1<br \/>\nwhile lo &lt;&#061; hi:<br \/>\nmid &#061; 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