![\"\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250815180030-689f75be8ef99.png\")
<\/p>\nPractice Day eight:Leetcode T1584<\/p>\n
<\/p>\n
\u4e0a\u4e00\u6b21，\u6211\u4eec\u4f7f\u7528\u4e86Prim\u7b97\u6cd5\u89e3\u51b3\u4e86\u6700\u5c0f\u751f\u6210\u6811\u7684\u95ee\u9898，\u4e8b\u5b9e\u4e0aKruskal\u7b97\u6cd5\u4e5f\u662f\u89e3\u51b3\u6700\u5c0f\u751f\u6210\u6811\u7684\u4e00\u4e2a\u6709\u6548\u65b9\u6cd5\u3002\u6211\u4eec\u53ef\u4e0d\u80fd\u539a\u6b64\u8584\u5f7c，\u53ea\u8bb2\u89e3Prim\u7b97\u6cd5\u800c\u5bf9Kruskal\u7b97\u6cd5\u7f6e\u4e4b\u4e0d\u7406\u3002<\/p>\n
\u73b0\u5728，\u6211\u4eec\u5148\u6765\u8bb2\u89e3Kruskal\u7b97\u6cd5\u7684\u57fa\u672c\u539f\u7406:<\/p>\n
1 \u83b7\u53d6\u6240\u6709\u8282\u70b9\u8fde\u6210\u8fb9\u7684\u6570\u503c，\u5982\u679c\u6570\u503c\u4e3a0，\u76f4\u63a5\u7ec8\u6b62\u7a0b\u5e8f，\u8f93\u51fa\u7ed3\u679c\u3002\u5426\u5219，\u8fdb\u5165\u4e0b\u4e00\u6b65\u64cd\u4f5c\u3002<\/p>\n
2 \u4ee5\u4ece\u5927\u5230\u5c0f，\u597d\u5427，\u5176\u5b9e\u4ece\u5c0f\u5230\u5927\u4e5f\u884c，\u603b\u4e4b，\u7ed9\u4ed6\u4eec\u6392\u5e8f<\/p>\n
3 \u9009\u53d6\u5176\u4e2d\u6700\u5c0f\u7684\u8fb9<\/p>\n
4 \u5224\u65ad\u8fd9\u6761\u8fb9\u88ab\u5b9e\u88c5\u540e，\u662f\u5426\u4f1a\u5f62\u6210\u591a\u4e2a\u8fde\u901a\u5206\u91cf，\u7528\u4eba\u8bdd\u8bf4，\u5c31\u662f\u662f\u5426\u4f1a\u6210\u73af<\/p>\n
5 \u6210\u73af：\u90a3\u5c31\u6eda\u86cb，\u6211\u4eec\u8981\u6700\u5c0f\u751f\u6210\u6811，\u6210\u73af\u663e\u7136\u4e0d\u7b26\u5408\u6211\u4eec\u7684\u6807\u51c6\u3002<\/p>\n
\u4e0d\u6210\u73af：\u4f60\u8fc7\u5173\u3002\u5b9e\u88c5\u8fd9\u6761\u8fb9\u3002<\/p>\n
6 \u5224\u65ad\u6b64\u65f6\u6240\u6709\u8282\u70b9\u662f\u5426\u5df2\u7136\u88ab\u94fe\u63a5\u3002\u82e5\u5df2\u7136\u5b8c\u5168\u94fe\u63a5，\u7ec8\u6b62\u7a0b\u5e8f\u3002\u82e5\u672a\u94fe\u63a5\u5168\u90e8\u8282\u70b9，\u5219\u5faa\u73af\u6b65\u9aa43，4，5，6<\/p>\n
\u5b8c\u6210\u5bf9Kruskal\u7b97\u6cd5\u7684\u521d\u6b65\u4e86\u89e3\u540e，\u6211\u4eec\u6765\u770b\u770b\u9898\u76ee\u8981\u6c42：\u5f97\u5230\u6700\u5c0f\u751f\u6210\u6811\u3002<\/p>\n
\u6ca1\u4ec0\u4e48\u597d\u8bf4\u7684，\u5c31\u7528Kruksal\u7b97\u6cd5\u3002<\/p>\n
\u4f46\u662f\u9898\u76ee\u8fd9\u4e2a\u56fe，\u6bd4\u8f83\u6709\u610f\u601d\u7684\u662f，\u4ed6\u662f\u4ee5\u5750\u6807\u7684\u5f62\u5f0f\u7ed9\u51fa\u4f4d\u7f6e\u7684，\u5e76\u4e0d\u662f\u76f4\u63a5\u628a\u4e00\u4e2a\u8fde\u901a\u56fe\u7684\u6743\u91cd\u544a\u8bc9\u4f60\u3002<\/p>\n
\u6211\u60f3\u8bf4\u7684\u662f，\u8fd9\u6837\u4e00\u6765，\u6211\u4eec\u7684\u64cd\u4f5c\u7a7a\u95f4\u5c31\u53d8\u5927\u4e86\u3002\u4e3a\u4ec0\u4e48\u8fd9\u4e48\u8bf4？\u56e0\u4e3a\u6211\u4eec\u83b7\u5f97\u4e86\u4e00\u4e2a\u65b0\u7684\u4fe1\u606f：\u89d2\u5ea6\u3002<\/p>\n
\u82e5\u662f\u5355\u7eaf\u7ed9\u51fa\u4e00\u4e2a\u65e0\u5411\u94fe\u63a5\u56fe\u7ed9\u4f60，\u4f60\u53ea\u80fd\u77e5\u9053\u5404\u4e2a\u8282\u70b9\u4e4b\u95f4\u7684\u6743\u91cd，\u4f60\u53ea\u6709\u8fd9\u4e2a\u4fe1\u606f\u3002<\/p>\n
\u4f46\u662f\u4ee5\u5750\u6807\u5f62\u5f0f\u7ed9\u51fa，\u4f60\u5c31\u53ef\u4ee5\u83b7\u77e5：\u8282\u70b9\u4e4b\u95f4\u7684\u6743\u91cd\u3001\u6743\u91cd\u95f4\u7684\u5939\u89d2<\/p>\n
\u6765\u601d\u8003\u4e00\u4e0b，\u8fd9\u4e2a\u5939\u89d2，\u4ed6\u786e\u5b9e\u53ef\u4ee5\u5e2e\u52a9\u6211\u4eec\u7b80\u5316\u4ee3\u7801\u5417？<\/p>\n
\u66f4\u591a\u65f6\u5019，\u589e\u52a0\u4e00\u4e2a\u5f71\u54cd\u56e0\u7d20，\u4ed6\u5f80\u5f80\u4f1a\u8ba9\u4e8b\u60c5\u53d8\u5f97\u66f4\u590d\u6742，\u8fd9\u91cc\u4e5f\u4e0d\u5217\u5916\u3002<\/p>\n
\u4f46\u662f，\u53ef\u4ee5\u8ba9\u6211\u4eec\u7684\u7a0b\u5e8f\u66f4\u52a0\u5feb\u901f！\u54c8\u54c8\u54c8\u54c8！<\/p>\n
\u6765\u770b\u770b\u8fd9\u662f\u600e\u4e48\u4e00\u56de\u4e8b\u5427！<\/p>\n
\u867d\u7136Kruskal\u7b97\u6cd5\u7684\u6838\u5fc3\u662f\u4ee5\u8fb9\u4e3a\u57fa\u7840\u5c55\u5f00\u7684，\u6240\u4ee5，\u6211\u4eec\u53ef\u4ee5\u5f88\u8f7b\u6613\u7684\u77e5\u9053，\u5f53\u8fb9\u8d8a\u591a，\u8d8a\u590d\u6742\u7684\u65f6\u5019，\u7b97\u6cd5\u5b9e\u73b0\u65f6\u95f4\u4f1a\u53d8\u5f97\u5f88\u957f\u3002\u5bf9\u4e8e\u8fd9\u7c7b\u56fe，\u6211\u4eec\u7ba1\u4ed6\u53eb\u7a20\u5bc6\u56fe\u3002<\/p>\n
\u5bf9\u4e8e\u7a20\u5bc6\u56fe，\u6211\u4eec\u66f4\u559c\u6b22\u7528Prim\u7b97\u6cd5\u8fdb\u884c\u89e3\u51b3\u3002\u56e0\u4e3aPrim\u7b97\u6cd5\u662f\u4ee5\u70b9\u4e3a\u57fa\u7840\u7684，\u8fb9\u7684\u591a\u4e0e\u5be1，\u5b9e\u9645\u5bf9\u7b97\u6cd5\u7684\u5b9e\u73b0\u7684\u65f6\u95f4\u5f71\u54cd\u5e76\u4e0d\u5927\u3002<\/p>\n
\u4e8b\u5b9e\u4e0a，Kruskal\u66f4\u9002\u5408\u89e3\u51b3\u8fb9\u66f4\u5c11\u7684\u7a00\u758f\u56fe，\u4ed6\u5728\u7a00\u758f\u56fe\u4e0a\u89e3\u51b3\u95ee\u9898\u7684\u65f6\u95f4\u8fdc\u6bd4Prim\u7b97\u6cd5\u597d\u5f97\u591a\u3002\u533a\u5206\u7a20\u5bc6\u56fe\u548c\u7a00\u758f\u56fe\u7684\u4e34\u754c\u6761\u4ef6，\u5c31\u662f\u8fb9\u7684\u6570\u76ee\u63a5\u8fd1n^2\/2\u3002<\/p>\n
\u8fd9\u65f6\u4f60\u53ef\u80fd\u5bdf\u89c9\u51fa\u95ee\u9898\u6240\u5728\u4e86：\u9898\u76ee\u53ea\u7ed9\u4e86\u5750\u6807，\u5de6\u8fb9\u4e24\u4e24\u76f8\u8fde\u53ef\u662f\u6709n（n-1）\/2\u6761\u8fb9，\u8fd9\u6beb\u65e0\u7591\u95ee\u662f\u7a20\u5bc6\u56fe，Kruskal\u7b97\u6cd5\u5b8c\u5168\u4e0d\u9002\u7528\u4e8e\u8fd9\u4e2a\u9898\u76ee\u3002<\/p>\n
\u6ca1\u6709Prim\u5927\u4eba\u7684\u529b\u91cf，\u6211\u4eec\u8be5\u5982\u4f55\u6218\u80dc\u7a20\u5bc6\u56fe？！<\/p>\n
\u4f5c\u8005\u6de1\u7136\u4e00\u7b11，\u4e00\u628a\u6293\u4f4f\u7a20\u5bc6\u56fe，\u9877\u523b\u70bc\u5316！<\/p>\n
\u6211\u5c31\u662f\u975e\u5f97\u8981\u7528Kruskal，\u4e3a\u4ec0\u4e48\u975e\u8981\u7b97\u6cd5\u9002\u5e94\u9898\u76ee？\u6211\u5b8c\u5168\u53ef\u4ee5\u6539\u9020\u9898\u76ee，\u8ba9\u4ed6\u53d8\u5f97\u9002\u5e94\u7b97\u6cd5！<\/p>\n
\u8fd9\u65f6\u5019\u5c31\u9700\u8981\u7528\u5230\u5f3a\u5927\u7684\u51e0\u4f55\u6b66\u5668\u4e86！<\/p>\n
\u5728\u4e00\u4e2a\u5e73\u9762\u4e09\u89d2\u5f62\u4e2d：<\/p>\n
\u53e4\u8bed\u6709\u66f0：\u5927\u89d2\u5bf9\u5927\u8fb9\u3002<\/p>\n
\u7b80\u800c\u8a00\u4e4b，\u5c31\u662f\u89d2\u8d8a\u5927，\u8fb9\u5c31\u8d8a\u5927\u3002\u8fd9\u4e2a\u53ef\u4ee5\u901a\u8fc7\u4f59\u5f26\u5b9a\u7406\u4e25\u683c\u8bc1\u660e\u3002\u53cd\u6b63\u4f60\u9ad8\u4e2d\u5b66\u8fc7\u4e86，\u5bf9\u5427？<\/p>\n
\u6784\u6210\u4e00\u4e2a\u4e09\u89d2\u5f62，\u6211\u4eec\u9700\u8981\u4e09\u4e2a\u9876\u70b9，\u600e\u4e48\u6392\u5217\u90fd\u53ef\u4ee5\u3002<\/p>\n
\u6211\u4eec\u77e5\u9053，\u82e5\u60f3\u8fde\u540c\u4e09\u4e2a\u70b9，\u5b9e\u9645\u4e0a\u6211\u4eec\u53ea\u9700\u9009\u62e9\u4e09\u6761\u8fb9\u4e2d\u7684\u5176\u4e2d\u4e24\u6761\u5373\u53ef\u3002<\/p>\n
\u82e5\u662f\u9700\u8981\u6700\u5c0f\u751f\u6210\u6811，\u5219\u80af\u5b9a\u4e0d\u80fd\u9009\u5176\u4e2d\u6700\u5927\u7684\u8fd9\u6761\u8fb9\u3002<\/p>\n
\u8fd9\u6837，\u6211\u4eec\u5c31\u901a\u8fc7\u51e0\u4f55\u5173\u7cfb，\u51cf\u5c11\u4e86\u53ef\u4ee5\u9009\u62e9\u7684\u8fb9，\u7a20\u5bc6\u56fe，\u6e10\u6e10\u53d8\u6210\u4e86\u7a00\u758f\u56fe\u3002<\/p>\n
\u5f88\u660e\u4e86\u4e86，\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u3010\u89d2\u5ea6\u3011\u6765\u9009\u62e9\u8fb9\u3002<\/p>\n
\u600e\u4e48\u5e94\u7528\u4ed6？\u8fd9\u597d\u50cf\u5f88\u9ebb\u70e6\u6b38？\u5939\u89d2\u662f\u76f8\u5bf9\u800c\u8a00\u7684，\u4e5f\u5c31\u662f\u8bf4，\u5982\u679c\u6211\u8981\u4f7f\u7528\u5939\u89d2\u8fd9\u4e2a\u4fe1\u606f，\u6211\u4e0d\u80fd\u5b64\u7acb\u7684\u4f7f\u7528\u5b83，\u6211\u4eec\u5fc5\u987b\u6709\u4e24\u6761\u8fb9\u624d\u80fd\u4f7f\u7528\u4ed6\u3002<\/p>\n
\u5c31\u7b97\u77e5\u9053\u4e86\u8fb9\u957f，\u6211\u4eec\u4ecd\u9700\u901a\u8fc7\u4e09\u89d2\u5f62\u7684\u4f59\u5f26\u5b9a\u7406\u624d\u80fd\u77e5\u9053\u89d2\u5ea6，md，\u8fd9\u4e0d\u662f\u6ca1\u4e8b\u627e\u4e8b\u5417，\u6211\u8981\u77e5\u9053\u957f\u5ea6\u4e86，\u8fd8\u8981\u4ec0\u4e48\u89d2\u5ea6\u3002<\/p>\n
\u6240\u4ee5\u89d2\u5ea6\u7684\u5e94\u7528\u65b9\u5f0f\u4e0d\u662f\u8fd9\u6837\u7684，\u6211\u4eec\u5982\u679c\u80fd\u627e\u5230\u4e00\u4e2a\u7279\u6b8a\u7684\u89d2\u5ea6，\u8ba9\u4ed6\u548c\u8fb9\u5bf9\u5e94\u8d77\u6765，\u90a3\u8be5\u6709\u591a\u597d\u3002\u8fb9，\u5b9e\u9645\u4e0a\u53c8\u662f\u5750\u6807\u7684\u4e8c\u7ea7\u4ea7\u7269，\u800c\u9898\u76ee\u53c8\u662f\u4ee5\u5750\u6807\u7684\u5f62\u5f0f\u7ed9\u51fa，\u8981\u6c42\u6211\u4eec\u8ba1\u7b97\u7684\u662f\u66fc\u54c8\u987f\u8ddd\u79bb，\u800c\u4e0d\u662f\u5b9e\u9645\u610f\u4e49\u4e0a\u7684\u8fb9\u957f，\u90a3\u6211\u4eec\u4e3a\u4ec0\u4e48\u4e0d\u76f4\u63a5\u4ee5\u66fc\u54c8\u987f\u8ddd\u79bb\u7684\u5f62\u5f0f\u6765\u7b49\u6548\u5904\u7406\u8fd9\u4e2a\u8fb9\u957f\u5462？\u6bd5\u7adf\u8fb9\u957f\u7684\u5927\u5c0f\u5b9e\u9645\u4e0a\u4e0e\u66fc\u54c8\u987f\u8ddd\u79bb（\u5b9e\u9645\u5e94\u63cf\u8ff0\u4e3a\u6a2a\u5411\u8ddd\u79bb\u548c\u7eb5\u5411\u8ddd\u79bb）\u662f\u4e00 \u4e00\u5bf9\u5e94\u7684\u3002<\/p>\n
\u7b54\u6848\u5df2\u7ecf\u5f88\u660e\u663e\u4e86！\u806a\u660e\u7684\u4f60\u5df2\u7ecf\u60f3\u5230，\u8fd9\u4e2a\u89d2\u5ea6，\u4ed6\u5c31\u662f45\u00b0，\u53ea\u6709\u4ed6，\u624d\u53ef\u4ee5\u5f88\u8f7b\u6613\u7684\u628a\u6a2a\u7eb5\u5750\u6807\u7ed9\u7b80\u5355\u8054\u7cfb\u8d77\u6765，\u56e0\u4e3a\u8fd9\u662f\u4e00\u4e2a\u53ef\u4ee5\u901a\u8fc7\u6570\u503c\u7b80\u5355\u4e8c\u5206\u7c7b\u7684\u6a21\u578b，\u82e5\u6a2a\u5411\u8ddd\u79bb\u4e0e\u7eb5\u5411\u8ddd\u79bb\u76f8\u7b49，\u90a3\u5c31\u662f45\u00b0，\u82e5\u4e0d\u7b49，\u5c31\u4e0d\u662f45\u00b0，\u5b8c\u5168\u4e0d\u9700\u8981\u8fdb\u884c\u522b\u7684\u5904\u7406\u3002<\/p>\n
\u5148\u501f\u529b\u6263\u5b98\u65b9\u7684\u56fe\u7528\u4e00\u4e0b<\/p>\n
![\"\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250815180032-689f75c0e7917.png\")
<\/p>\n
\u518d\u7ed3\u5408\u6211\u4eec\u524d\u9762\u7684\u7406\u8bba，\u6211\u4eec\u53ef\u4ee5\u5f97\u77e5，45\u00b0\u5bf9\u5e94\u7684\u89d2\u4e00\u5b9a\u4e0d\u4f1a\u662f\u6700\u5927\u89d2，\u4e5f\u5c31\u662f，\u8fd9\u4e2a\u89d2\u6240\u5bf9\u8fb9\u4e00\u5b9a\u4e0d\u662f\u6700\u5927\u8fb9，\u8fd9\u5c31\u5f3a\u8feb\u6211\u4eec\u5fc5\u987b\u8981\u5728\u7b80\u5355\u8ba1\u7b97\u5c31\u77e5\u9053\u4e86，\u4e0d\u518d\u8fc7\u591a\u63cf\u8ff0\u3002<\/p>\n
\u53ef\u4ee5\u770b\u5230，\u6574\u4e2a\u5e73\u9762，\u88ab\u6211\u4eec\u5206\u4e3a\u4e86\u516b\u4e2a\u90e8\u5206\u3002\u73b0\u5728，\u6211\u4eec\u5c31\u53ef\u4ee5\u5206\u533a\u57df\u8fdb\u884c\u5904\u7406\u4e86\u3002<\/p>\n
1\u00a0\u4ee5\u9009\u4e2d\u7684\u8282\u70b9\u4f5c\u4e3a\u539f\u70b9，\u5b9e\u73b0\u8fd9\u4e2a\u975e\u5e38\u7b80\u5355，\u53ea\u9700\u8981\u628a\u4e24\u4e2a\u8282\u70b9\u7684\u5750\u6807\u76f8\u51cf\u5373\u53ef，\u8fd9\u76f8\u5f53\u4e8e\u5e73\u79fb\u5750\u6807\u7cfb，\u73b0\u5728，\u6211\u4eec\u7ba1\u8fd9\u4e2a\u5904\u4e8e\u539f\u70b9\u5750\u6807\u53eb\u505aa<\/p>\n
2 \u6211\u4eec\u5148\u4ee5P1\u533a\u4e3a\u4f8b，\u6211\u4eec\u518d\u968f\u4fbf\u53d6\u4e00\u4e2a\u5750\u6807，\u4e5f\u8bb8\u6ca1\u8fd9\u4e48\u5e78\u8fd0，\u4e00\u4e0b\u5c31\u53d6\u5230\u4e86P1\u533a，\u6211\u7ba1\u4ed6\u53ebb<\/p>\n
3 \u6211\u4eec\u5982\u4f55\u5224\u5b9a，b\u4ed6\u786e\u5b9e\u5728P1\u533a\u95f4\u91cc\u9762\u5462？\u518d\u8fdb\u4e00\u6b65，\u8fd9\u4e2aP1\u533a\u5230\u5e95\u662f\u600e\u4e48\u5212\u5206\u7684\u5462？\u6211\u4eec\u77e5\u9053，P1\u533a\u4e0d\u8fc7\u662fa\u70b9\u7684\u5f15\u7533，\u6240\u4ee5\u5bf9\u4e8e\u533a\u57df\u5212\u5206\u7684\u95ee\u9898，\u6211\u4eec\u9700\u8981\u4ecea\u70b9\u5165\u624b\u3002<\/p>\n
4 \u6211\u4eec\u77e5\u9053，\u5bf9\u4e8e45\u00b0，\u5176\u659c\u7387\u4e3aK=1，\u4e5f\u5c31\u662f\u8bf4，\u5bf9\u4e8e\u6070\u597d\u5904\u4e8e\u70b9a\u768445\u00b0\u65b9\u5411\u7684\u70b9（\u6ce8\u610f，\u8fd9\u91cc\u63cf\u8ff0\u7684\u4e0d\u662f\u5904\u4e8eP1\u533a\u57df\u7684\u70b9），\u5e94\u8be5\u6ee1\u8db3Xz-Xa>0\u00a0 \u8fd9\u662f\u4e3a\u4e86\u4fdd\u8bc1b\u5728a\u7684\u53f3\u534a\u533a\u95f4\u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/p>\n
Yz-Xz=Ya-Xa\u00a0 \u00a0 \u8fd9\u662f\u56e0\u4e3a\u5904\u4e8ea\u7684P1\u533a\u57df\u7684\u70b9，\u5e94\u8be5\u6ee1\u8db3\u65b9\u7a0b\u5f0fY-X=C(C\u662f\u4e00\u4e2a\u5e38\u6570，\u8868\u793a\u5bf9\u4ee5\u539f\u70b9\u51fa\u53d1\u768445\u00b0\u7684\u6821\u6b63\u6570\u503c，\u6bd5\u7adf\u4e0d\u7ba1\u600e\u4e48\u8bf4，\u6211\u4eec\u768445\u00b0\u4ecd\u7136\u4ee5\u771f\u6b63\u7684\u539f\u70b9\u53ca\u5176\u5750\u6807\u7cfb\u4e3a\u6807\u51c6）<\/p>\n
5 \u90a3\u4e48，\u5f88\u6e05\u6670\u4e86，\u82e5\u662f\u70b9b\u5904\u4e8eP1\u533a\u57df，\u5c31\u5e94\u8be5\u6ee1\u8db3Xb-Xa>0，\u4ee5\u53caYb-Xb>C<\/p>\n
6 \u5728\u5b8c\u6210\u533a\u57df\u7684\u5224\u522b\u4e4b\u540e，\u6211\u4eec\u5c31\u9700\u8981\u8fdb\u884c\u6700\u77ed\u8fb9\u7684\u5224\u65ad\u3002\u6211\u4eec\u77e5\u9053，\u5bf9\u4e8e\u5904\u4e8eP1\u533a\u57df\u7684\u70b9\u800c\u8a00，\u66fc\u54c8\u987f\u8ddd\u79bb\u53ef\u4ee5\u88ab\u8868\u793a\u4e3aD=（Xz-Xa）+（Yz-Ya）=（Xz+Yz）-（Xa+Ya），\u6709\u610f\u601d\u7684\u662f，Xa+Ya\u5728\u6b64\u5904\u5b9e\u9645\u4e0a\u662f\u4e2a\u5b9a\u503c，\u5728\u5904\u7406\u7684\u8fc7\u7a0b\u4e2d，\u4f1a\u6539\u53d8\u7684\u53ea\u6709Xz+Yz\u3002\u4e5f\u5c31\u662f\u8bf4：（Xz+Yz）\u8d8a\u5927，\u66fc\u54c8\u987f\u8ddd\u79bb\u5c31\u8d8a\u5927<\/p>\n
7 \u90a3\u4e48，\u5bf9\u4e8e\u66fc\u54c8\u987f\u8ddd\u79bb\u7684\u5224\u65ad，\u5c31\u7b80\u5316\u4e3a\u4e86\u5bf9\u6240\u9009\u8282\u70b9（Xz+Yz）\u503c\u7684\u5224\u65ad，\u8fd9\u4e2a\u503c\u53ef\u4e0d\u4f1a\u56e0\u4e3a\u53c2\u8003\u7cfb\u7684\u6539\u53d8\u800c\u6539\u53d8，\u51cf\u5c11\u4e86\u6211\u4eec\u5927\u91cf\u7684\u5b58\u50a8\u5de5\u4f5c\u4e0e\u5b58\u50a8\u7a7a\u95f4，\u5e76\u4e14\u8fd9\u4e2a\u662f\u6bcf\u4e00\u4e2a\u8282\u70b9\u72ec\u6709\u7684\u503c<\/p>\n
8 \u8bb0\u5f55\u5904\u4e8eP1\u533a\u57df\u7684\u6240\u6709\u503c，\u901a\u8fc7\u4e8c\u53c9\u641c\u7d22\u6811（BIT），\u9009\u51fa\u5176\u4e2d\u7684\u6700\u5c0f\u503c，\u653e\u5165\u961f\u5217，\u4fdd\u7559\u4ed6\u3002<\/p>\n
9 \u90a3\u4e48\u5269\u4e0b\u7684P2，P3，P4\u4e5f\u662f\u540c\u6837\u7684\u64cd\u4f5c，\u8bb0\u5f55\u4e0b\u672c\u533a\u95f4\u7684\u6700\u5c0f\u503c\u3002\u4e3a\u4ec0\u4e48\u4e0d\u628a\u5de6\u4fa7\u4e5f\u4e00\u8d77\u8ba1\u7b97\u4e86，\u4e3a\u4ec0\u4e48\u53ea\u8ba1\u7b97\u53f3\u4fa7？\u56e0\u4e3a\u8fd9\u6837\u5c31\u4f1a\u9020\u6210\u91cd\u590d\u3002\u6765\u60f3\u4e00\u60f3\u5427！\u6211\u4eec\u662f\u9700\u8981\u5bf9\u6240\u6709\u8282\u70b9\u90fd\u8fdb\u884c\u8fd9\u4e2a\u64cd\u4f5c\u7684！\u8fd9\u610f\u5473\u7740\u4ec0\u4e48？！<\/p>\n
10 \u610f\u5473\u7740，\u5904\u4e8e\u6700\u5de6\u4fa7\u7684\u8282\u70b9\u4e00\u5b9a\u4f1a\u88ab\u9009\u4e2d！\u53ea\u8981\u4ed6\u88ab\u9009\u4e2d，\u5c31\u610f\u5473\u7740\u9664\u4e86\u4ed6\u4e4b\u5916\u7684\u8282\u70b9\u5c31\u4e00\u5b9a\u4f1a\u4e0e\u4ed6\u4ea7\u751f\u8054\u7cfb，\u56e0\u4e3a\u6211\u4eec\u7684\u8ba1\u7b97\u6d41\u7a0b\u5c31\u662f\u8fd9\u4e48\u8bbe\u8ba1\u7684，\u7531\u4e8e\u8fd9\u662f\u4e00\u4e2a\u65e0\u5411\u56fe，\u63cf\u8ff0\u4e3aa\u4e0eb\u8fde\u63a5，\u4e0e\u63cf\u8ff0\u4e3ab\u4e0ea\u8fde\u63a5，\u5b9e\u9645\u4e0a\u5730\u4f4d\u662f\u7b49\u4ef7\u7684，a\u7684\u53f3\u4fa7，\u5b9e\u9645\u4e0a\u5c31\u662fb\u7684\u5de6\u4fa7，\u53ea\u4e0d\u8fc7\u4e3b\u4f53\u4e0d\u540c，\u63cf\u8ff0\u65b9\u5f0f\u4e0d\u540c\u800c\u5df2：\u6211\u7684\u8fb9，\u5c31\u662f\u4f60\u7684\u8fb9<\/p>\n
11 \u6240\u4ee5，\u6211\u4eec\u53ea\u9700\u8981\u8ba1\u7b97\u4e00\u4fa7，\u5c31\u8db3\u4ee5\u8ba1\u7b97\u6240\u6709\u8282\u70b9\u4e4b\u95f4\u7684\u8054\u7cfb\u4e86，\u5982\u679c\u5168\u65b9\u4f4d\u8fdb\u884c\u8ba1\u7b97，\u5c31\u4f1a\u6d6a\u8d39\u8ba1\u7b97\u8d44\u6e90，\u6ca1\u6709\u8fd9\u4e2a\u5fc5\u8981<\/p>\n
12 \u4e00\u4e2a\u8282\u70b9，\u6700\u591a\u5f15\u7533\u51fa\u516b\u6761\u8fb9\u4e0e\u5176\u4ed6\u8282\u70b9\u8fde\u63a5，\u56e0\u4e3a\u6bcf\u4e2a\u533a\u57df，\u53ea\u6709\u4e00\u6761\u6700\u77ed\u8fb9\u3002\u8fd9\u5f53\u7136\u4e0d\u662f\u4ec0\u4e48\u4e8c\u7ea7\u7ed3\u8bba，\u8fd9\u53ea\u662f\u6211\u4eec\u7531\u533a\u57df\u5212\u5206\u800c\u5f97\u51fa\u6765\u7684\u6570\u5b57\u3002<\/p>\n
13 \u6b38，\u6709\u4eba\u53ef\u80fd\u4f1a\u8bf4：\u4f60\u600e\u4e48\u8bc1\u660e\u8fd9\u516b\u6761\u8fb9\u80af\u5b9a\u80fd\u8fde\u63a5\u6240\u6709\u8282\u70b9？\u5176\u5b9e\u4f60\u53ef\u4ee5\u8fd9\u6837\u60f3：\u5176\u5b9e，\u8fd9\u5c31\u662f\u6bcf\u4e2a\u8282\u70b9\u81ea\u53d1\u7684\u627e\u79bb\u81ea\u5df1\u6700\u8fd1\u7684\u8282\u70b9\u3002\u867d\u7136\u8fd9\u6709\u8fddKruskal\u7b97\u6cd5\u7684\u601d\u60f3，\u53cd\u800c\u4e0ePrim\u7b97\u6cd5\u7684\u601d\u60f3\u66f4\u63a5\u8fd1，\u4f46\u5b9e\u9645\u4e0a，\u7b5b\u67e5\u7684\u65f6\u5019\u6211\u4eec\u5229\u7528\u7684\u662fKruskal\u7b97\u6cd5，\u6240\u4ee5\u4ed6\u662f\u5f53\u4e4b\u65e0\u6127\u7684Kruskal\u7b97\u6cd5\u3002<\/p>\n
13 \u6211\u4eec\u5b8c\u6210\u4e86\u5bf9\u8fb9\u7684\u51cf\u5c11\u540e ，\u8fd9\u5b8c\u5168\u5c31\u662f\u4e00\u4e2a\u7a00\u758f\u56fe\u4e86\u3002\u628a\u8fd9\u4e9b\u7b5b\u9009\u540e\u7684\u8fb9\u6254\u8fdb\u961f\u5217\u4e2d，\u7136\u540e\u4e3a\u4ed6\u4eec\u6392\u5e8f\u3002<\/p>\n
14 \u53d6\u51fa\u6700\u5c0f\u8fb9，\u5224\u65ad\u5b83\u662f\u5426\u6210\u73af，\u8fd9\u4e00\u70b9，\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u5e76\u67e5\u96c6\u6765\u5b9e\u73b0，\u53ea\u9700\u8981\u5224\u65ad\u5176\u662f\u5426\u5c5e\u4e8e\u540c\u4e00\u4e2a\u7236\u8282\u70b9\u5c31\u53ef\u4ee5\u4e86\u3002<\/p>\n
15 \u6210\u73af：\u820d\u5f03，\u9009\u53d6\u4e0b\u4e00\u6761\u8fb9\u957f\u7ee7\u7eed\u5de5\u4f5c<\/p>\n
\u00a0\u4e0d\u6210\u73af：\u9009\u53d6\u4ed6，\u5c06\u5176\u653e\u5165\u56fe\u4e2d，\u7136\u540e\u8bb0\u5f55\u8fb9\u957f<\/p>\n
16 \u5faa\u73af\u4ee5\u4e0a\u6b65\u9aa414\u4e0e\u6b65\u9aa415\u3002<\/p>\n
17 \u5f53\u65e0\u8fb9\u53ef\u9009\u65f6，\u7a0b\u5e8f\u7ec8\u6b62<\/p>\n
\u6d41\u7a0b\u5df2\u7136\u5927\u81f4\u68b3\u7406\u4e86\u4e00\u904d，\u4e0b\u9762，\u5c31\u662f\u7a0b\u5e8f\u7684\u6784\u5efa\u4e86\u3002<\/p>\n<\/p>\n
\u5e76\u67e5\u96c6（Disjoint-Set）<\/p>\n
\u6bd4\u8f83\u62b1\u6b49\u7684\u662f，\u6211\u5e76\u6ca1\u6709\u5728\u524d\u51e0\u5929\u7684\u6bcf\u65e5\u4e00\u7ec3\u4e2d\u53d1\u5e03\u6709\u5173\u5e76\u67e5\u96c6\u7684\u5185\u5bb9，\u6240\u4ee5，\u5728\u6b64\u5904\u5361\u4f4f\u7684\u5c0f\u4f19\u4f34\u4eec\u53ef\u4ee5\u5148\u53bbB\u7ad9\u67e5\u8be2\u76f8\u5173\u5185\u5bb9，\u51e0\u5206\u949f\u5c31\u53ef\u4ee5\u641e\u5b9a\u3002<\/p>\n
\u7b80\u5355\u6765\u8bf4，\u5e76\u67e5\u96c6\u5c31\u662f\u67e5\u8be2\u6bcf\u4e00\u4e2a\u8282\u70b9\u5f52\u5c5e\u7684\u4e00\u79cd\u6811\u72b6\u7ed3\u6784\u3002<\/p>\n
\u5b66\u4e60\u4e86\u5e76\u67e5\u96c6\u540e，\u6211\u4eec\u6765\u6784\u5efa\u4e00\u4e2a\u5e76\u67e5\u96c6，\u6765\u5e2e\u52a9\u6211\u4eec\u5224\u65ad\u6240\u9009\u8282\u70b9\u662f\u5426\u6210\u73af\u3002<\/p>\n
1 \u5e76\u67e5\u96c6\u6709\u56db\u4e2a\u8981\u7d20：\u8282\u70b9\u4e2a\u6570\u3001\u6bcf\u4e2a\u8282\u70b9\u7684\u79e9（\u6811\u7684\u9ad8\u5ea6）\u3001\u7236\u8282\u70b9\u3001\u5b50\u8282\u70b9<\/p>\n
2 \u4e3a\u4e86\u66f4\u5feb\u4e00\u4e9b，\u4e5f\u8bb8\u6211\u4eec\u7684\u5143\u7d20\u5e76\u4e0d\u662f\u53ea\u67e5\u8be2\u8fd9\u4e48\u4e00\u6b21，\u6211\u4eec\u5e0c\u671b\u540e\u7eed\u7684\u67e5\u8be2\u4e2d\u80fd\u5feb\u4e00\u4e9b\u3002\u60f3\u60f3\u770b\u5427，\u8fd9\u4e2a\u7ed3\u6784\u6811\u662f\u600e\u4e48\u6784\u6210\u7684？\u6bd4\u8f83\u574f\u7684\u60c5\u51b5\u662f\u4e00\u7ea7\u4e00\u7ea7\u5411\u4e0a\u67e5\u8be2，\u90a3\u6211\u5e0c\u671b\u5728\u8fd9\u6b21\u4e00\u67e5\u8be2\u540e，\u4ee5\u540e\u5c31\u4e0d\u8981\u8fd9\u6837\u8fd9\u4e48\u6162\u7684\u67e5\u8be2\u4e86，\u6240\u4ee5\u6211\u76f4\u63a5\u628a\u8fd9\u4e00\u7ea7\u8282\u70b9\u6302\u5230\u7236\u8282\u70b9\u4e0a\u9762，\u8fd9\u6837，\u6211\u5728\u4e0b\u6b21\u67e5\u8be2\u540e，\u53ea\u9700\u8981\u518d\u5411\u4e0a\u67e5\u4e00\u7ea7，\u6211\u5c31\u77e5\u9053\u672c\u8282\u70b9\u662f\u4ece\u5c5e\u4e8e\u8fd9\u4e2a\u7236\u8282\u70b9\u3002\u4eba\u4eec\u7ed9\u8fd9\u79cd\u64cd\u4f5c\u53d6\u4e86\u4e00\u4e2a\u52a8\u4eba\u7684\u540d\u5b57，\u8def\u5f84\u538b\u7f29\u3002<\/p>\n
3 \u4e0a\u9762\u8bf4\u7684\u662f\u5df2\u7ecf\u8fde\u63a5\u597d\u7684\u60c5\u51b5，\u5982\u679c，\u662f\u65b0\u8fde\u63a5\u7684\u5462？\u8fd9\u5c31\u610f\u5473\u7740，\u5408\u5e76\u3002\u5408\u5e76\u4e5f\u662f\u6709\u8bb2\u7a76\u7684，\u54ea\u4e24\u4e2a\u8282\u70b9\u5408\u5e76？\u4ee5\u4ec0\u4e48\u5f62\u5f0f\u5408\u5e76？<\/p>\n
4 \u7b2c\u4e00，\u5408\u5e76\u901a\u5e38\u662f\u7236\u8282\u70b9\u4e4b\u95f4\u8fdb\u884c\u5408\u5e76，\u5982\u679c\u4f60\u60f3\u628a\u4e00\u4e2a\u8282\u70b9\u5408\u5e76\u5230\u53e6\u4e00\u4e2a\u8282\u70b9\u4e0a，\u8bf7\u52a1\u5fc5\u627e\u5230\u8fd9\u4e24\u4e2a\u8282\u70b9\u7684\u7236\u8282\u70b9，\u7136\u540e\u518d\u5408\u5e76\u7236\u8282\u70b9<\/p>\n
5 \u7b2c\u4e8c，\u5c31\u5408\u5e76\u7684\u65b9\u5f0f\u800c\u8a00， \u901a\u5e38\u6709\u4e24\u79cd，\u4e0d\u8fc7\u4ed6\u4eec\u7edf\u79f0\u4e3a\u201c\u6309\u79e9\u5408\u5e76\u201d，\u7b2c\u4e00\u79cd\u662f\u6309\u6811\u9ad8\u5408\u5e76，\u4e5f\u5c31\u662f\u54ea\u68f5\u6811\u66f4\u9ad8，\u77ee\u7684\u90a3\u68f5\u6811\u7684\u7236\u8282\u70b9\u5c31\u8fde\u63a5\u5230\u9ad8\u7684\u90a3\u9897\u6811\u7684\u7236\u8282\u70b9\u4e0a\u3002\u4e0d\u8fc7\u8fd9\u4f1a\u4ea7\u751f\u4e00\u4e9b\u5e7b\u89c9\u3002\u56e0\u4e3a\u6d89\u53ca\u5230\u8def\u5f84\u538b\u7f29\u7684\u8bdd，\u8fdb\u884c\u8def\u5f84\u538b\u7f29\u540e，\u6811\u7684\u5b9e\u9645\u9ad8\u5ea6\u53ef\u80fd\u4f1a\u53d8\u77ee，\u5bfc\u81f4\u4f1a\u51fa\u73b0\u4e24\u68f5\u6811\u7b49\u9ad8，\u6216\u8005\u9ad8\u4e00\u4e9b\u7684\u6811\u5408\u5e76\u5230\u77ee\u4e00\u4e9b\u7684\u6811\u4e0a\u9762，\u4e0d\u8fc7\u65e0\u4f24\u5927\u96c5，\u8fd9\u4e2a\u6cd5\u5b50\u8fd8\u662f\u5f88\u597d\u7528\u7684，\u5e76\u4e0d\u4f1a\u56e0\u4e3a\u4e9b\u8bb8\u5e7b\u89c9\u4ea7\u751f\u4e0d\u7a33\u5b9a\u6027\u3002<\/p>\n
6 \u4e3a\u4e86\u89e3\u51b3\u8fd9\u4e2a\u5e7b\u89c9，\u6211\u4eec\u6709\u7b2c\u4e8c\u79cd\u529e\u6cd5，\u6309\u8282\u70b9\u4e2a\u6570\u5408\u5e76，\u8c01\u8282\u70b9\u66f4\u591a，\u8c01\u5c31\u505a\u7238\u7238，\u56e0\u4e3a\u513f\u5b50\u7684\u4e2a\u6570\u4e0d\u4f1a\u56e0\u8def\u5f84\u538b\u7f29\u800c\u53d1\u751f\u6539\u53d8\u3002<\/p>\n
\u4ee5\u4e0b\u662f\u4ee3\u7801\u5b9e\u73b0：<\/p>\n
class DisjointSetUnion {
\nprivate:
\n vector<int> f, rank;
\n int n;<\/p>\n
public:
\n DisjointSetUnion(int _n) {
\n n = _n;
\n rank.resize(n, 1);
\n f.resize(n);
\n for (int i = 0; i < n; i++) {
\n f[i] = i;
\n }
\n }<\/p>\n
int find(int j) {
\n return f[j] == j ? j : f[j] = find(f[j]);
\n }<\/p>\n
int unionSet(int j, int k) {
\n int fj = find(j), fk = find(k);
\n if (fj == fk) {
\n return false;
\n }
\n if (rank[fj] < rank[fk]) {
\n swap(fj, fk);
\n }
\n rank[fj] += rank[fk];
\n f[fj] = fj;
\n return true;
\n }
\n}; <\/p>\n
\u5176\u4e2d，<\/p>\n
find（）\u51fd\u6570\u5c31\u662f\u8def\u5f84\u538b\u7f29，\u4ee5\u8fed\u4ee3\u7684\u65b9\u5f0f\u5b8c\u6210\u3002<\/p>\n
unionSet（）\u51fd\u6570\u5c31\u662f\u68c0\u67e5\u662f\u5426\u6210\u73af，\u4ee5\u53ca\u6309\u79e9\u5408\u5e76\u3002<\/p>\n<\/p>\n
\u63a5\u7740，\u6211\u4eec\u6765\u6784\u5efa\u4e8c\u53c9\u7d22\u5f15\u6811（Binary Indexed Tree）\u7684\u7c7b，\u53c8\u540d\u6811\u72b6\u6570\u7ec4，\u6ca1\u5b66\u8fc7\u7684\u5c0f\u4f19\u4f34\u53ef\u4ee5\u770b\u770b\u8fd9\u4e2a\u89c6\u9891https:\/\/www.bilibili.com\/video\/BV1pE41197Qj\/?spm_id_from=333.1387.homepage.video_card.click&vd_source=f8b81fa5a70ab21de75192e5c7ec720c<\/span> \u597d\u5427，\u6211\u627f\u8ba4\u6709\u4e9b\u64cd\u4e4b\u8fc7\u6025\u3002\u6784\u5efa\u4ed6\u7684\u539f\u56e0\u662f：\u5feb\u901f\u627e\u5230\u5206\u9694\u7684\u56db\u5757P1，2，3，4\u4e2d，\u6bcf\u4e2a\u533a\u57df\u7684\u6700\u5c0f\u8fb9<\/p>\n 1 \u9996\u5148，\u6784\u6210\u6811\u72b6\u6570\u7ec4\u7684\u8981\u7d20\u6709\u4e09\u4e2a：<\/p>\n \u2460\u7528\u4e8e\u8bb0\u5f55\u6570\u503c\u7684\u6570\u7ec4：\u8fd9\u91cc\u662f\u8bb0\u5f55\u6bcf\u4e2a\u8282\u70b9\u672c\u8eab\u7684\u7279\u5f81\u503c，\u8fd9\u91cc\u7684\u7279\u5f81\u503c\u662f\u4e00\u4e2a\u7edf\u79f0\u3002\u5148\u7ed9\u8fd9\u4e2a\u6570\u7ec4\u547d\u4e2a\u540d\u5427，\u65e2\u7136\u4ed6\u53eb\u6811\u72b6\u6570\u7ec4，\u90a3\u6211\u4eec\u5c31\u7ba1\u4ed6\u53ebtree\u3002\u5bf9\u4e8etree\u800c\u8a00，\u5b9e\u9645\u4e0a，\u5b83\u8bb0\u5f55\u4e86\u4e24\u79cd\u4fe1\u606f\u3002\u770b\u770b\u4ed6\u7684\u5f62\u6001tree[pos]=val<\/p>\n \u6211\u4eec\u53d1\u73b0，\u5b83\u53ef\u4ee5\u5728\u4e24\u4e2a\u5730\u65b9\u8bb0\u5f55\u4fe1\u606f：\u6570\u7ec4\u7d22\u5f15pos，\u5b58\u50a8\u7684\u503cval\u3002<\/p>\n \u8fd9\u5206\u522b\u7528\u4e8e\u5e72\u4ec0\u4e48\u5462？<\/p>\n \u5bf9\u4e8e\u6570\u7ec4\u7d22\u5f15pos，\u7528\u6765\u5212\u5b9a\u533a\u57df，\u4e5f\u5c31\u662f\u6211\u4eec\u7684P1，2，3，4<\/p>\n \u800c\u5b58\u50a8\u7684\u503cval，\u5219\u662f\u7528\u6765\u5224\u65ad\u6700\u77ed\u8fb9<\/p>\n \u6211\u4eec\u4ecd\u65e7\u7528P1\u8fdb\u884c\u4e3e\u4f8b\u3002<\/p>\n<\/p>\n \u6211\u4eec\u6765\u770b\u770bpos\u662f\u600e\u4e48\u6765\u7684，\u53c8\u662f\u600e\u4e48\u8fd0\u4f5c，\u624d\u62e5\u6709\u4e86\u5212\u5206\u533a\u57df\u7684\u80fd\u529b<\/p>\n \u56de\u60f3\u4ee5\u4e0b\u5212\u5206\u533a\u57df\u7684\u6761\u4ef6：Xb-Xa>0，\u4ee5\u53caYb-Xb>C=Ya-Xa<\/p>\n \u9996\u5148，\u6211\u4eec\u9700\u8981\u786e\u4fdd\u53d6\u5230\u7684\u70b9，\u597d\u5427，\u5176\u5b9e\u6211\u4eec\u9700\u8981\u53d6\u4e24\u4e2a\u70b9\u3002\u5982\u4f55\u786e\u4fdd\u4f60\u5728\u4ee5\u5176\u4e2d\u4e00\u4e2a\u70b9\u4e3a\u57fa\u51c6\u70b9\u65f6，\u53e6\u4e00\u4e2a\u70b9\u4e00\u5b9a\u4f1a\u5728\u5176\u53f3\u4fa7，\u4e5f\u5c31\u662fXb-Xa>0？\u5f88\u7b80\u5355，\u3010\u6392\u5e8f\u3011\u3002\u53ea\u8981\u6309\u7167X\u503c\u7684\u5927\u5c0f\u6392\u5e8f，\u6211\u4eec\u5148\u53d6X\u503c\u6700\u5c0f\u7684\u70b9\u4f5c\u4e3a\u57fa\u51c6\u70b9，\u90a3\u4e48\u5373\u4fbf\u53e6\u5916\u4e00\u4e2a\u70b9\u662f\u968f\u4fbf\u53d6\u7684，\u4f9d\u7136\u53ef\u4ee5Xb-Xa>0\u3002<\/p>\n \u5bf9\u4e8e\u6392\u5e8f，\u5b9e\u73b0\u65b9\u6cd5\u662f\u6784\u5efa\u4e00\u4e2a\u53ef\u4ee5\u540c\u65f6\u5b58\u50a8x，y\u503c\u7684\u5bf9\u8c61，\u4f60\u53ef\u80fd\u4f1a\u60f3\u5230\u6bcf\u65e5\u4e00\u7ec38\u7684\u952e\u503c\u5bf9，\u4f46\u8fd9\u91cc\u6211\u4eec\u4f7f\u7528\u7ed3\u6784\u4f53\u66f4\u5408\u9002\u4e00\u4e9b，\u56e0\u4e3a\u6211\u4eec\u8fd8\u6709\u522b\u7684\u4fe1\u606f\u9700\u8981\u5b58\u50a8，\u6bd4\u5982\u2014\u2014\u7d22\u5f15\u3002\u6240\u4ee5\u952e\u503c\u5bf9\u662f\u4e0d\u5408\u9002\u7684\u3002<\/p>\n struct Pos { \u6211\u4eec\u6709\u8bf8\u591a\u8282\u70b9，\u663e\u7136\u4e00\u4e2a\u7ed3\u6784\u4f53\u662f\u4e0d\u591f\u7684，\u6240\u4ee5\u6211\u4eec\u9700\u8981\u4e00\u4e2a\u5b58\u653e\u7ed3\u6784\u4f53\u7684\u5bb9\u5668\u3002<\/p>\n vector<Pos> pos <\/p>\n \u6709\u4e86\u8fd9\u4e2a，\u6211\u4eec\u5c31\u53ef\u4ee5\u8fdb\u884c\u6392\u5e8f\u7684\u64cd\u4f5c\u4e86<\/p>\n \u6211\u4eec\u77e5\u9053，C++\u7684std::sort\u51fd\u6570\u9ed8\u8ba4\u4f7f\u7528\u201c<"\u8fd0\u7b97\u7b26\u6765\u6bd4\u8f83\u4e24\u4e2a\u5143\u7d20\u7684\u5927\u5c0f，\u5bf9\u4e8e\u81ea\u5b9a\u4e49\u7ed3\u6784\u4f53Pos\u800c\u8a00，\u5f53\u6211\u4eec\u91cd\u8f7d\u4e86\u8fd0\u7b97\u7b26"<"\u540e，sort\u5c31\u4f1a\u8c03\u7528\u8fd9\u4e2a\u91cd\u8f7d\u7248\u672c\u6765\u51b3\u5b9a\u5143\u7d20\u7684\u6392\u5217\u987a\u5e8f\u3002<\/p>\n \u5bf9\u4e8ePos\u800c\u8a00，\u6211\u4eec\u5173\u6ce8\u7684\u662f\u4ed6\u7684X\u503c，\u56e0\u4e3a\u6211\u4eec\u8981\u4fdd\u8bc1\u6240\u9009\u8282\u70b9\u5728\u57fa\u51c6\u8282\u70b9\u53f3\u4fa7\u3002\u4f46\u60c5\u51b5\u5b9e\u9645\u4e0a\u8fd8\u6709\u53e6\u4e00\u79cd\u60c5\u51b5，\u4e5f\u662f\u552f\u4e00\u7684\u4e00\u79cd\u60c5\u51b5：\u6240\u9009\u4e24\u70b9X\u503c\u76f8\u540c\u3002\u8fd9\u663e\u7136\u4e5f\u9700\u8981\u4e00\u79cd\u6392\u5e8f\u65b9\u5f0f，\u56e0\u4e3a\u4ed6\u662f\u771f\u5b9e\u5b58\u5728\u7684\u60c5\u51b5，\u6211\u4eec\u65e0\u6cd5\u5ffd\u7565\u4ed6\u3002\u6240\u4ee5，\u6211\u4eec\u4e5f\u8981\u4e3a\u4ed6\u6765\u6d89\u53ca\u4e00\u4e2a\u6bd4\u8f83\u65b9\u5f0f，\u7b80\u5355\u8d77\u89c1，\u4ecd\u65e7\u6bd4\u8f83Ya<Yb\u5373\u53ef\u3002\u6211\u4eec\u5728\u7ed3\u6784\u4f53\u4e2d\u8865\u5145\u4e00\u6761\u8fd0\u7b97\u7b26\u91cd\u8f7d\u51fd\u6570，\u4ee5\u4fbfsort\u51fd\u6570\u53ef\u4ee5\u8c03\u7528\u4ed6\u3002<\/p>\n struct Pos { \u8fd9\u91cc\u9700\u8981\u6ce8\u610f\u7684\u662f，\u4e09\u76ee\u8fd0\u7b97\u7b26\u201c\u00a0 ？：\u201d\u5f62\u6210\u7684\u662f\u4e00\u4e2a\u6574\u4f53，\u5e76\u4e0d\u662f\u76f4\u63a5return x；\u800c\u662f\u901a\u8fc7\u5927\u5c0f\u7684\u6bd4\u8f83\u8fd4\u56de\u4e00\u4e2abool\u503c\u3002<\/p>\n \u7136\u540e\u8fdb\u884c\u6392\u5e8f<\/p>\n sort(pos.begin(), pos.end()); <\/p>\n \u73b0\u5728，\u6570\u7ec4pos\u4ee5X\u503c\u4ece\u5c0f\u5230\u5927\u7684\u5f62\u5f0f\u6392\u5217\u597d\u4e86\u3002<\/p>\n<\/p>\n \u5b8c\u6210\u4e86\u7b2c\u4e00\u4e2a\u6761\u4ef6\u7684\u5224\u5b9a\u540e，\u5c31\u9700\u8981\u8fdb\u884c\u7b2c\u4e8c\u4e2a\u6761\u4ef6\u7684\u5224\u65ad，\u6d89\u53ca\u5230\u8282\u70b9\u7684\u7279\u5f81\u503cYb-Xb<\/p>\n \u5bf9\u4e8e\u8fd9\u6837\u7684\u8ba1\u7b97，\u6211\u4eec\u53ef\u4ee5\u590d\u7528\u7ed3\u6784\u4f53Pos，\u8ba1\u7b97\u6bcf\u4e00\u4e2a\u8282\u70b9\u7684\u7279\u5f81\u503c，\u5bf9\u6b64，\u6211\u4eec\u540c\u6837\u9700\u8981\u4e00\u4e2a\u5bb9\u5668\u6765\u5b58\u50a8\u8fd9\u4e9b\u8ba1\u7b97\u7684\u503c\u3002\u7531\u4e8e\u8fd9\u662f\u4e00\u4e2a\u6574\u6570\u5b9a\u503c，\u6211\u4eec\u628a\u7c7b\u578b\u8bbe\u4e3aint<\/p>\n vector<int> a(n); <\/p>\n \u7136\u540e\u904d\u5386vector<Pos> pos\u4e2d\u7684\u6240\u6709\u5143\u7d20，\u5c06\u8ba1\u7b97\u503c\u8bb0\u5f55\u5230vector<int> a(n)\u4e2d<\/p>\n for (int i = 0; i < n; i++) { \u6709\u65f6\u4f1a\u51fa\u73b0\u4e00\u4e9b\u8282\u70b9Y-X\u76f8\u540c\u7684\u60c5\u51b5，\u8fd9\u610f\u5473\u7740\u8fd9\u4e9b\u8282\u70b9\u90fd\u5904\u4e8e\u57fa\u51c6\u8282\u70b9\u7684\u540c\u4e00\u4e2a\u533a\u57df\u5185，\u6bd4\u5982\u90fd\u5904\u4e8e\u57fa\u51c6\u5143\u7d20\u7684P1\u533a\u57df\u5185\u3002<\/p>\n \u8fd9\u4e2a\u73b0\u8c61\u9664\u4e86\u544a\u8bc9\u6211\u4eec\u4ed6\u4eec\u5904\u4e8e\u540c\u4e00\u4e2a\u533a\u57df，\u8fd8\u544a\u8bc9\u6211\u4eec\u4ed6\u4f1a\u8ba9\u7a0b\u5e8f\u68c0\u7d22\u65f6\u95f4\u53d8\u957f，\u8fd9\u662f\u6211\u4eec\u4e0d\u613f\u770b\u5230\u7684，\u6bd4\u8f83\u597d\u7684\u529e\u6cd5\u662f\u8ba9\u6bcf\u4e00\u4e2a\u7279\u5f81\u503cY-X\u53d8\u5f97\u72ec\u7279，\u6210\u4e3a\u7d22\u5f15\u4e00\u6837\u7684\u5b58\u5728，\u4f46\u6211\u4eec\u53c8\u4e0d\u5e0c\u671b\u6539\u53d8\u6570\u7ec4a，\u56e0\u4e3a\u6570\u7ec4a\u7684\u7d22\u5f15\u4ee3\u8868\u4ed6\u662f\u4e0d\u540c\u8282\u70b9\u4ea7\u751f\u7684\u7279\u5f81\u503c\u3002<\/p>\n \u6240\u4ee5\u6211\u4eec\u53e6\u8bbe\u4e00\u4e2a\u6570\u7ec4b，\u4e13\u95e8\u7528\u4e8e\u5b58\u653e\u6ca1\u6709\u91cd\u590d\u7684\u7279\u5f81\u503cY-X，\u8ba9\u5176\u6210\u4e3a\u5206\u754c\u7ebf\u4e00\u6837\u7684\u5b58\u5728，\u901a\u8fc7\u68c0\u7d22b\u7684\u7d22\u5f15，\u6211\u4eec\u5c31\u80fd\u77e5\u9053\u5750\u6807\u5904\u4e8e\u54ea\u4e00\u4e2a\u533a\u57df\u3002\u7531\u4e8e\u6211\u4eec\u4e0d\u77e5\u9053\u91cd\u590d\u503c\u7684\u4e2a\u6570，\u5148\u6682\u4e14\u8bbe\u5bb9\u5668\u5927\u5c0f\u4e3an<\/p>\n vector<int> b(n)； <\/p>\n \u5f53\u7136，\u5408\u7406\u8d77\u89c1，\u6211\u4eec\u5e94\u8be5\u4e0ea\u4e00\u540c\u5904\u7406，\u4e5f\u5c31\u662f\u5728\u4e0a\u4e00\u6761\u7a0b\u5e8f\u7684for\u5faa\u73af\u52a0\u4e0a\u6570\u7ec4b\u7684\u5904\u7406<\/p>\n for (int i = 0; i < n; i++) { \u7136\u540e，\u5bf9\u6570\u7ec4b\u8fdb\u884c\u4e0a\u8ff0\u64cd\u4f5c，\u8ba9\u6bcf\u4e2a\u7279\u5f81\u503c\u72ec\u7acb，\u5e76\u4f7f\u4e4b\u53ef\u4ee5\u6210\u4e3a\u88ab\u5408\u7406\u68c0\u7d22\u7684\u5b58\u5728\u3002<\/p>\n \u4e3a\u4e86\u53ef\u4ee5\u8f7b\u6613\u68c0\u7d22\u4ed6，\u6211\u4eec\u5148\u628a\u6570\u7ec4b\u8fdb\u884c\u6392\u5e8f，\u8fd9\u6837，\u6211\u4eec\u5c31\u53ef\u4ee5\u901a\u8fc7\u4e0b\u6807\u7684\u5f62\u5f0f\u8fdb\u884c\u533a\u57df\u5212\u5206\u3002\u540c\u65f6，\u8fd9\u4e5f\u662f\u4e3a\u4e86\u9002\u914d\u53bb\u91cd\u65f6erase\u51fd\u6570\u7684\u4f7f\u7528，\u56e0\u4e3aerase\u51fd\u6570\u53ea\u80fd\u5220\u53bb\u7279\u5b9a\u533a\u57df\u7684\u6570\u5b57\u3002<\/p>\n sort(b.begin(), b.end()); <\/p>\n \u7136\u540e\u8fdb\u884c\u53bb\u91cd\u3002<\/p>\n b.erase(unique(b.begin(), b.end()), b.end()); <\/p>\n \u9996\u5148\u662f\u5229\u7528unique\u51fd\u6570\u628a\u6570\u7ec4b\u6240\u6709\u91cd\u590d\u7684\u5143\u7d20\u79fb\u5230\u961f\u5217\u672b\u5c3e，\u7136\u540e\u8fd4\u56de\u4e00\u4e2a\u6307\u5411\u7b2c\u4e00\u4e2a\u91cd\u590d\u5143\u7d20\u7684\u8fed\u4ee3\u5668\u3002\u4e4b\u540e，erase\u51fd\u6570\u5927\u624b\u53d1\u529b，\u62b9\u9664\u4e24\u4e2a\u8fed\u4ee3\u5668\u4e4b\u95f4\u7684\u91cd\u590d\u5143\u7d20\u3002\u73b0\u5728，\u6570\u7ec4b\u6709\u4e86\u4e00\u4e2a\u597d\u542c\u7684\u540d\u5b57\u2014\u2014\u3010\u6620\u5c04\u8868\u3011\u3002\u5f53\u7136，\u4f60\u559c\u6b22\u7ba1\u4ed6\u53eb，\u3010\u5b57\u5178\u3011，\u4e5f\u662f\u53ef\u4ee5\u88ab\u63a5\u53d7\u7684\u3002<\/p>\n \u8bf4\u4e86\u8fd9\u4e48\u591a，\u4e0d\u8981\u5fd8\u8bb0\u521d\u5fc3，\u6211\u4eec\u662f\u8981\u89e3\u91capos\u8fd9\u4e2a\u6570\u7ec4tree\u7684\u7d22\u5f15\u662f\u600e\u4e48\u6765\u7684<\/p>\n \u4e3a\u4e86\u628a\u7d22\u5f15\u540d\u79f0\u548c\u6570\u7ec4\u540dpos\u533a\u5206，\u6211\u4eec\u628a\u8ba1\u7b97\u7528\u7684\u7d22\u5f15\u540d\u79f0\u53ebposs<\/p>\n int poss = lower_bound(b.begin(), b.end(), a[i]) – b.begin() + 1; <\/p>\n \u901a\u8fc7lower_bound\u51fd\u6570，\u4ed6\u4f1a\u8fd4\u56de\u4e00\u4e2a\u5728\u6570\u7ec4b\u4e2d\u4e0e\u6570\u7ec4a\u6570\u503c\u76f8\u7b49\u65f6\u7684\u8fed\u4ee3\u5668，\u5f53\u7136，\u6211\u4eec\u9700\u8981\u628a\u4ed6\u5904\u7406\u4e3a\u6570\u7ec4\u7d22\u5f15，\u8fed\u4ee3\u5668\u5f53\u7136\u4e0d\u80fd\u6210\u4e3a\u7d22\u5f15\u3002\u4f46\u6570\u7ec4b\u662f\u4e00\u4e2avector\u6570\u7ec4，\u4ed6\u7684\u5185\u5bb9\u662f\u8fde\u7eed\u5b58\u50a8\u7684，\u90a3\u4e48\u8fed\u4ee3\u5668\u4e4b\u95f4\u76f8\u51cf，\u5f97\u5230\u4e00\u4e2a\u5dee\u503c，\u8fd9\u5c31\u662f\u6570\u7ec4b\u6b64\u65f6\u7684\u7d22\u5f15\u4e86\u3002\u4f46\u8fd9\u8fd8\u4e0d\u591f，\u56e0\u4e3a\u4e8c\u53c9\u7d22\u5f15\u6811\u7684\u6807\u53f7\u662f\u4ece\u30101\u3011\u5f00\u59cb\u7684，\u800c\u6570\u7ec4b\u662f\u4ece\u30100\u3011\u5f00\u59cb\u7684，\u4e0d\u8fdb\u884c\u6807\u53f7\u5bf9\u9f50\u7684\u8bdd，\u4f1a\u51fa\u73b0<\/p>\n lowbit（0）\u65e0\u610f\u4e49\u7684\u95ee\u9898，\u6240\u4ee5，\u6211\u4eec\u9700\u8981\u4e3a\u6570\u7ec4b\u7684\u7d22\u5f15<\/p>\n \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0lower_bound(b.begin(), b.end(), a[i]) – b.begin()\u00a0<\/p>\n \u8fdb\u884c\u52a01\u7684\u5904\u7406，\u4ee5\u4fbf\u5bf9\u9f50\u4e8c\u53c9\u7d22\u5f15\u6811\u7684\u6807\u53f7\u3002<\/p>\n \u8fd9\u5c31\u662fpos\u7684\u6765\u6e90\u4e86，\u5982\u4f55\u4f7f\u7528\u5b83，\u6211\u4eec\u540e\u9762\u4f1a\u7ee7\u7eed\u8bb2\u89e3<\/p>\n<\/p>\n \u63a5\u4e0b\u6765\u8bb2\u89e3\u7684\u662f\u6570\u7ec4tree\u4e2d\u5b58\u50a8\u7684val，\u8fd9\u53c8\u662f\u600e\u4e48\u6765\u7684<\/p>\n \u6587\u7ae0\u6709\u4e9b\u592a\u957f\u4e86，\u6211\u4eec\u91cd\u6e29\u4e00\u4e0b\u4e0a\u9762\u7684\u5185\u5bb9<\/p>\n \u5728\u5b8c\u6210\u533a\u57df\u7684\u5224\u522b\u4e4b\u540e，\u6211\u4eec\u5c31\u9700\u8981\u8fdb\u884c\u6700\u77ed\u8fb9\u7684\u5224\u65ad\u3002\u6211\u4eec\u77e5\u9053，\u5bf9\u4e8e\u5904\u4e8eP1\u533a\u57df\u7684\u70b9\u800c\u8a00，\u66fc\u54c8\u987f\u8ddd\u79bb\u53ef\u4ee5\u88ab\u8868\u793a\u4e3aD=（Xz-Xa）+（Yz-Ya）=（Xz+Yz）-（Xa+Ya），\u6709\u610f\u601d\u7684\u662f，Xa+Ya\u5728\u6b64\u5904\u5b9e\u9645\u4e0a\u662f\u4e2a\u5b9a\u503c，\u5728\u5904\u7406\u7684\u8fc7\u7a0b\u4e2d，\u4f1a\u6539\u53d8\u7684\u53ea\u6709Xz+Yz\u3002\u4e5f\u5c31\u662f\u8bf4：（Xz+Yz）\u8d8a\u5927，\u66fc\u54c8\u987f\u8ddd\u79bb\u5c31\u8d8a\u5927<\/p>\n pos\u8fd9\u4e2a\u7d22\u5f15，\u4e8b\u5b9e\u4e0a\u5c31\u5b8c\u6210\u4e86\u533a\u5757\u7684\u5212\u5206，\u6240\u4ee5，tree\u5b58\u50a8\u7684\u503c\u5e94\u8be5\u662f\u5904\u4e8eP1\u533a\u5757\u7684\u8282\u70b9\u7684\u66fc\u54c8\u987f\u8ddd\u79bb\u7684\u9b54\u6539\u503c（Xz+Yz），\u5bf9\u4e8e\u8fd9\u4e2a\u503c\u7684\u4f7f\u7528，\u6211\u4eec\u540e\u9762\u4f1a\u8fdb\u884c\u8bb2\u89e3<\/p>\n \u2461\u7528\u4e8e\u8bb0\u5f55\u7d22\u5f15\u7684\u6570\u7ec4：\u901a\u8fc7\u76f8\u540c\u7684\u7f16\u53f7\u6765\u8bb0\u5f55\u7d22\u5f15\u3002\u4e8b\u5b9e\u4e0a，\u8fd9\u4e2a\u6570\u7ec4\u6709\u4e24\u4e2a\u7d22\u5f15，\u6211\u4eec\u9700\u8981\u5206\u6e05\u695a\u4e00\u4e9b\u3002\u6211\u4eec\u8bbe\u8fd9\u4e2a\u6570\u7ec4\u4e3aidRec\u3002\u5bf9\u4e8e\u8fd9\u4e2aidRec，\u8fdb\u884c\u67e5\u627e\u65f6，\u6709idRec[pos] = id\u3002\u5176\u4e2dpos\u7684\u6765\u6e90，\u5728\u2460\u4e2d\u5df2\u7ecf\u89e3\u91ca\u8fc7\u3002\u73b0\u5728，\u6211\u4eec\u6765\u89e3\u91ca\u201cid\u201d\u5b58\u50a8\u4e86\u4ec0\u4e48\u503c\u3002<\/p>\n id，\u5c31\u662f\u5f53\u524d\u6570\u7ec4tree\u6240\u4f7f\u7528\u7684poss\u7d22\u5f15，\u5728\u6570\u7ec4pos\u4e2d\u7684\u7f16\u53f7<\/p>\n \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0pos[i]\u2014\u2014>a[i]\u2014\u2014>b[?]\u2014\u2014>poss\u2014\u2014>tree[poss]<\/p>\n \u4e5f\u5c31\u662f\u8bf4，id=i，\u4fdd\u7559\u8fd9\u4e2a\u4fe1\u606f，\u662f\u4e3a\u4e86\u4e4b\u540e\u5f97\u80dc\u6700\u5c0f\u8fb9\u65f6，\u8ba1\u7b97\u66fc\u54c8\u987f\u8ddd\u79bb\u7684\u4f9d\u636e<\/p>\n \u2462\u6784\u9020BIT\u5b9e\u4f8b\u82b1\u8d39\u7684\u6570\u7ec4\u5927\u5c0f：\u6211\u4eec\u628a\u8fd9\u4e2a\u4f20\u5165\u7684\u4e1c\u897f\u53eb\u505an\u3002<\/p>\n \u5bf9\u4e8en，\u5e94\u8be5\u4e0e\u6570\u7ec4tree\u542b\u6709\u7684\u5143\u7d20\u4e2a\u6570\u4e00\u81f4，\u8fd9\u6837，\u624d\u80fd\u7cbe\u786e\u6784\u5efa\u6570\u7ec4tree\u7684\u5927\u5c0f，\u907f\u514d\u4e0d\u5fc5\u8981\u7684\u6d6a\u8d39，\u4e5f\u662f\u4e3a\u4e86\u9632\u6b62\u5b58\u50a8\u7a7a\u95f4\u4e0d\u8db3\u800c\u8d8a\u754c\u3002\u6240\u4ee5\u6709<\/p>\n int num = b.size(); <\/p>\n \u8fd9\u91cc\u7684num\u4e0en\u4e0d\u662f\u4e00\u4e2a\u4e1c\u897f，\u7531\u4e8e\u6784\u5efa\u7684\u6570\u7ec4\u5f80\u5f80\u4ece0\u5f00\u59cb，\u6240\u4ee5\u5b9e\u9645\u4e0a\u6570\u7ec4tree\u548c\u6570\u7ec4idRec\u90fd\u8981\u6bd4\u5b9e\u9645\u5b58\u50a8\u5143\u7d20\u603b\u91cf\u5927\u4e00\u4f4d，\u4ee5\u7b26\u5408\u4e8c\u53c9\u7d22\u5f15\u6811\u7684\u5e94\u7528\u89c4\u5219，[0]\u53f7\u6570\u7ec4\u4e0d\u5b58\u50a8\u4e1c\u897f，\u53ea\u662f\u7528\u6765\u52a0\u585e\u3002\u4e5f\u5c31\u662f\u8bf4，n=num+1<\/p>\n \u5bf9\u4e8enum\u800c\u8a00，\u8fd9\u4e2a\u503c\u8d8a\u5927，\u8bf4\u660eP1\u533a\u57df\u5212\u5206\u7684\u8d8a\u7cbe\u7ec6，\u4f46\u540c\u65f6\u4e5f\u610f\u5473\u7740\u641c\u7d22\u6548\u7387\u7684\u964d\u4f4e\u3002<\/p>\n<\/p>\n \u62e5\u6709\u6240\u6709\u6784\u5efa\u4e8c\u53c9\u7d22\u5f15\u6811\u7684\u8981\u7d20\u4e4b\u540e，\u6211\u4eec\u5c31\u5f00\u59cb\u6784\u5efa\u4e8c\u53c9\u7d22\u5f15\u6811<\/p>\n BIT(int _n) { \u6211\u4eec\u5728\u521d\u59cb\u5316\u65f6，\u5e94\u8be5\u628a\u6570\u7ec4tree\u7684\u6bcf\u4e00\u4e2a\u5143\u7d20\u8bbe\u7f6e\u4e3a\u6700\u5927，\u7ed9\u4e88\u4e00\u4e2a\u521d\u59cb\u7684\u6bd4\u8f83\u503c，\u56e0\u4e3a\u6211\u4eec\u7684tree\u603b\u662f\u5b58\u50a8\u6700\u5c0f\u7684（Xz+Yz），\u6240\u4ee5\u521d\u59cb\u503c，\u6211\u4eec\u8981\u8bbe\u4e3a\u65e0\u7a77\u5927，\u867d\u7136\u65e0\u6cd5\u5728\u8ba1\u7b97\u673a\u5b9e\u73b0\u65e0\u7a77\u5927，\u4e0d\u8fc7\u6211\u4eec\u53ef\u4ee5\u7528int\u6574\u5f62\u53ef\u4ee5\u8868\u8fbe\u7684\u6700\u5927\u503c\u6765\u8868\u793a\u65e0\u7a77\u3002<\/p>\n \u5bf9\u4e8e\u6570\u7ec4idRec\u800c\u8a00，\u4e3a\u4ec0\u4e48\u8981\u628a\u521d\u59cb\u5143\u7d20\u8bbe\u7f6e\u4e3a-1，\u8fd9\u5e76\u6ca1\u6709\u4ec0\u4e48\u7279\u6b8a\u7684，\u53ea\u662f\u56e0\u4e3a\u6b63\u6570\u548c\u96f6\u90fd\u9700\u8981\u88ab\u4f7f\u7528\u4e3a\u7f16\u53f7\u7d22\u5f15，\u6240\u4ee5\u6211\u4eec\u53ea\u80fd\u5728\u8d1f\u6570\u4e2d\u968f\u673a\u9009\u62e9\u4e00\u4e2a\u6570，\u8868\u793a\u6b64\u65f6\u5c1a\u672a\u5b58\u5728\u8be5\u533a\u57df\u7684tree\u88ab\u5173\u8054\u3002<\/p>\n \u00a0\u6b64\u65f6\u4e8c\u53c9\u7d22\u5f15\u6811\u521d\u59cb\u5316\u5b8c\u6210，\u63a5\u4e0b\u6765，\u6211\u4eec\u6765\u5b9e\u73b0\u4e8c\u53c9\u7d22\u5f15\u6811\u7684\u529f\u80fd\u3002<\/p>\n \u54c8\u54c8，\u9996\u5148，\u662f\u4e8c\u53c9\u7d22\u5f15\u6811\u6700\u5173\u952e\u7684\u641c\u67e5\u7ec4\u4ef6，lowbit（）<\/p>\n int lowbit(int k) { return k & (-k); } <\/p>\n \u4f4d\u4e0e\u8fd0\u7b97\u7b26&\u9996\u5148\u4f1a\u628a\u5341\u8fdb\u5236\u7684k\u8f6c\u5316\u4e3a\u4e8c\u8fdb\u5236，\u7136\u540e-k\u53d8\u6210k\u53d6\u8865\u7801，\u4e5f\u5c31\u662fk\u6309\u4f4d\u53d6\u53cd，\u7136\u540e\u52a0\u4e00\u3002\u63a5\u7740&\u8fd0\u7b97\u7b26\u5c31\u4f1a\u6bd4\u5bf9k\u4e0e-k\u4e4b\u95f4\u7684\u6570\u5b57，\u82e5\u540c\u4e00\u4f4d\u6b21\u90fd\u662f1，\u5219\u53d61，\u5426\u5219\u53d60\u3002\u8fd9\u6837，\u6211\u4eec\u5c31\u53ef\u4ee5\u5f97\u5230\u4e8c\u8fdb\u5236\u7684k\u6700\u53f3\u4fa7\u76841\u4e86\u3002<\/p>\n \u8fd9\u6837\u8bbe\u8ba1\u7684\u76ee\u7684\u662f\u4ec0\u4e48？<\/p>\n \u6765\u770b\u770b\u6811\u72b6\u6570\u7ec4\u7684\u5f62\u72b6，\u8fd9\u91cc，\u6211\u4eec\u501f\u7528\u4e00\u4e0bB\u7ad9up\u4e3b\u9e64\u7fd4\u4e07\u91cc\u7684\u56fe，\u4ee5\u4e0b\u662f\u94fe\u63a5<\/p>\n https:\/\/www.bilibili.com\/video\/BV1pE41197Qj\/?spm_id_from=333.1007.top_right_bar_window_history.content.click&vd_source=f8b81fa5a70ab21de75192e5c7ec720c<\/span> \u5bf9\u4e8e\u6700\u5e95\u4e0b\u7684\u767d\u8272\u6570\u7ec4\u800c\u8a00(\u6ce8\u610f\u8fd9\u91cc\u7684\u6570\u7ec4a\u4e0d\u662f\u6211\u4eec\u7f16\u5199\u7684，\u53ea\u662f\u56fe\u7247\u7684\u4e00\u4e2a\u8303\u4f8b），\u4ed6\u8bb0\u5f55\u7684\u662f\u6211\u4eec\u5904\u7406\u8fc7\u540e\u7684\u6570\u7ec4b[\u00a0 ]，\u4e5f\u5c31\u662f\u5728P1\u533a\u57df\u67d0\u4e2a\u8303\u56f4\u5185\u7684\u6700\u5c0f\u503c\u3002\u73b0\u5728，\u6211\u4eec\u5e0c\u671b\u5728\u4e4b\u540e\u53ef\u4ee5\u66f4\u52a0\u5feb\u901f\u7684\u68c0\u7d22\u7531\u6570\u7ec4b\u751f\u6210\u7684poss\u5bf9\u5e94\u7684\u6700\u5c0f\u503c，\u6240\u4ee5，\u6211\u4eec\u6784\u5efa\u4e86\u53e6\u5916\u4e00\u4e2a\u6570\u7ec4 t [ ]\u6765\u5b58\u653e\u67d0\u4e00\u533a\u95f4（\u8fd9\u91cc\u7684\u533a\u95f4\u6307\u7684\u662f\u7531poss\u751f\u6210\u7684\u7d22\u5f15\u533a\u95f4，\u56e0\u4e3aposs\u7684\u751f\u6210\u662f\u6709\u5e8f\u7684，\u6240\u4ee5\u533a\u95f4\u4e5f\u662f\u6709\u5e8f\u7684）\u5185\u7684\u6700\u5c0f\u503c，\u5728\u6211\u4eec\u7684\u4ee3\u7801\u91cc，\u6570\u7ec4tree[\u00a0 ]\u7b49\u6548\u4e8e\u4e0a\u9762\u7684\u6570\u7ec4 t [\u00a0 ]\u3002<\/p>\n \u6211\u4eec\u6765\u4e3e\u4f8b\u7406\u89e3\u4ed6\u5427：t[2]\u5b58\u50a8\u7684\u662fa[1]~a[2]\u91cc\u6bd4\u8f83\u5c0f\u7684\u90a3\u4e2a\u503c，t[4]\u5b58\u50a8\u7684\u662fa[1]~a[4]\u4e2d\u6700\u5c0f\u7684\u90a3\u4e2a\u503c，t[8]\u5b58\u50a8\u7684\u662fa[1]~a[8]\u91cc\u6700\u5c0f\u7684\u90a3\u4e2a\u503c\u3002<\/p>\n \u8fd9\u79cd\u53ef\u4ee5\u901a\u8fc7\u7236\u533a\u95f4\u5feb\u901f\u67e5\u627e\u6211\u4eec\u8bbe\u5b9a\u7684\u89c4\u5219（\u672c\u9898\u662f\u6700\u5c0f\u503c，\u5f53\u7136，\u4f60\u53ef\u4ee5\u81ea\u5df1DIY\u5b58\u50a8\u5404\u79cd\u6570\u636e\u7684\u6811\u72b6\u6570\u7ec4）\u7684\u6811\u72b6\u7ed3\u6784，\u5c31\u662f\u6211\u4eec\u7684\u6811\u72b6\u6570\u7ec4\u4e86\u3002<\/p>\n \u4e5f\u5c31\u662f\u8bf4，\u53ea\u8981\u67e5\u8be2\u5bf9\u5e94\u7684\u7236\u8282\u70b9，\u5c31\u53ef\u4ee5\u83b7\u6089\u5176\u7ba1\u7406\u7684\u5b50\u8282\u70b9\u7684\u6838\u5fc3\u4fe1\u606f\u3002\u5728\u67e5\u8be2\u4e4b\u524d，\u6811\u72b6\u6570\u7ec4\u662f\u5df2\u7ecf\u7ecf\u8fc7\u4e00\u6b21\u6bd4\u5bf9，\u624d\u88ab\u88ab\u6784\u5efa\u51fa\u6765\u7684，\u6811\u72b6\u6570\u7ec4\u5b9e\u8d28\u4e0a\u53ea\u662f\u4e3a\u4e86\u65b9\u4fbf\u540e\u7eed\u67e5\u8be2\u7684\u7ed3\u6784\u3002<\/p>\n \u6811\u72b6\u6570\u7ec4\u7684\u6838\u5fc3\u601d\u60f3\u662f\u7528\u7236\u8282\u70b9\u7ba1\u7406\u82e5\u5e72\u5b50\u8282\u70b9\u7684\u4fe1\u606f，\u800c\u00a0lowbit\u00a0\u5c31\u662f\u7528\u6765\u786e\u5b9a\u7236\u8282\u70b9\u4e0e\u5b50\u8282\u70b9\u7684\u7ba1\u8f96\u5173\u7cfb\u7684\u5173\u952e\u3002<\/p>\n \u2460lowbit\u53ef\u4ee5\u5e2e\u52a9\u6211\u4eec\u83b7\u5f97\u6570\u7ec4\u00a0 t [\u00a0 ]\u76ee\u524d\u7ba1\u7406\u7684\u5143\u7d20\u4e2a\u6570，\u4ee5\u53ca\u8ba1\u7b97\u88ab\u7ba1\u7406\u5143\u7d20\u7684\u7d22\u5f15<\/p>\n \u4f8b\u5982：<\/p>\n \u2461lowbit\u53ef\u4ee5\u4f5c\u4e3a\u6b65\u957f\u8fdb\u884c\u8fc1\u8d8a，\u627e\u5230\u7236\u8282\u70b9，\u6216\u5b50\u8282\u70b9<\/p>\n \u73b0\u5728，\u7406\u89e3\u4e86\u57fa\u672c\u539f\u7406\u540e，\u6211\u4eec\u6765\u6784\u5efaupdate\u7ec4\u4ef6，\u4e0equery\u7ec4\u4ef6\u6df7\u5408\u4f7f\u7528\u3002<\/p>\n update\u7528\u4e8e\u4e3a\u6811\u72b6\u6570\u7ec4tree\u63d2\u5165\u6570\u636e，\u4e5f\u5c31\u662f\u4e0a\u9762\u6811\u72b6\u6570\u7ec4\u7684\u56fe\u7247\u4e2d\u7684\u6570\u7ec4\u00a0t [\u00a0 ]\u3002<\/p>\n \u2460tree\u662f\u6570\u7ec4b\u7684\u6620\u5c04，\u4ee5\u00a0int poss = lower_bound(b.begin(), b.end(), a[i]) – b.begin() + 1;\u7684\u65b9\u5f0f<\/p>\n \u2461\u4f46\u6211\u4eec\u4ecd\u9700\u4ece\u6570\u7ec4a\u4e0b\u624b，\u56e0\u4e3a\u6570\u7ec4a\u7684\u7d22\u5f15\u4e0e\u6570\u7ec4pos\u7684\u7d22\u5f15\u662f\u4e00\u4e00\u5bf9\u5e94\u7684，\u6570\u7ec4tree\u53ea\u662f\u5b58\u50a8\u4e2d\u95f4\u91cf\u7684\u5de5\u5177，\u6211\u4eec\u4ecd\u7136\u5f97\u901a\u8fc7\u51c6\u786e\u7684\u8282\u70b9pos[i]\u6765\u8ba1\u7b97\u66fc\u54c8\u987f\u8ddd\u79bb\u3002<\/p>\n \u2462tree\u5b58\u50a8\u7684\u503cval\u662f\u7279\u5f81\u503cX+Y，\u4e5f\u5c31\u662f\u6211\u4eec\u9700\u8981\u4f20\u5165pos\u8282\u70b9\u672c\u8eab\u7684x，y\u4fe1\u606f<\/p>\n \u8fd9\u6837\u4e00\u6765，update\u5c31\u9700\u8981\u4f20\u5165\u4e09\u4e2a\u6570\u636e：poss\u3001i\u3001\u5bf9\u5e94\u7684\u771f\u5b9e\u8282\u70b9pos[i].x\u4e0epos[i].y\u7684\u4fe1\u606f<\/p>\n \u4ed6\u7684\u539f\u578b\u8bbe\u7f6e\u4e3a<\/p>\n void update(int poss, int val, int id) <\/p>\n \u53cd\u6b63\u8981\u8bb0\u5f55\u7279\u5f81\u503c，\u4e00\u4e0d\u505a\u4e8c\u4e0d\u4f11，\u76f4\u63a5\u5728\u4f20\u5165\u53c2\u6570\u7684\u65f6\u5019\u5c31\u8ba1\u7b97<\/p>\n update(poss, pos[i].x + pos[i].y, i) <\/p>\n \u7136\u540e\u5f00\u59cb\u6784\u5efa\u5185\u90e8\u903b\u8f91<\/p>\n \u7531\u4e8e\u4e00\u5f00\u59cb，tree\u6240\u6709\u5143\u7d20\u88ab\u521d\u59cb\u5316\u4e3a\u65e0\u7a77\u5927，\u6240\u4ee5，\u5f53\u6211\u4eec\u6309pos[i]\u63d2\u5165\u65f6，\u4ed6\u4e00\u5b9a\u662f\u6574\u4e2a\u6570\u7ec4tree\u4e2d\u6700\u5c0f\u7684\u503c，\u5219\u6570\u7ec4\u5fc5\u7136\u4f1a\u66f4\u65b0\u3002<\/p>\n \u5728\u5f80\u540e\u7684\u64cd\u4f5c\u4e2d，\u4e5f\u662f\u9700\u8981\u6bd4\u5bf9\u6570\u503c，\u5982\u679c\u65b0\u63d2\u5165\u7684\u6570\u503c\u6bd4\u539f\u6765\u7684\u5927，\u5c31\u4e0d\u7ba1\u4ed6，\u8df3\u8fc7，\u4e0d\u66f4\u65b0，\u5982\u679c\u63d2\u5165\u7684\u6570\u503c\u66f4\u5c0f，\u5c31\u6309\u6b65\u957f\u5f80\u4e0b\u8d70，\u66f4\u65b0\u6cbf\u9014\u6240\u6709\u8282\u70b9\u7684\u503c，\u76f4\u5230\u8282\u70b9\u5c0f\u4e8e0\u3002<\/p>\n \u5f80\u4e0b\u8d70\u7684\u539f\u56e0\u662f，\u6211\u4eec\u5e0c\u671b\u53ef\u4ee5\u8fd9\u4e48\u8bbe\u8ba1\u3002\u56e0\u4e3a\u8fd9\u6837\u53ef\u4ee5\u8282\u7701\u66f4\u65b0\u8282\u70b9\u7684\u65f6\u95f4\u3002\u53ef\u4ee5\u5f97\u77e5，\u6570\u7ec4tree\u6700\u540e\u4e00\u4e2a\u5143\u7d20，\u5fc5\u7136\u662f\u6574\u4e2a\u6570\u7ec4\u6700\u5c0f\u7684\u5143\u7d20，\u8fd9\u4e00\u70b9，\u53ef\u4ee5\u7531\u4e0a\u9762\u7684\u6811\u72b6\u6570\u7ec4\u56fe\u5f88\u8f7b\u6613\u7684\u83b7\u77e5\u3002<\/p>\n \u6240\u4ee5\u5047\u8bbe\u6211\u4eec\u4eceX\u503c\u5927\u5230\u5c0f，\u4e5f\u5c31\u662f\u4ece\u6700\u672b\u5c3e\u7684\u7f16\u53f7\u5f00\u59cb\u66f4\u65b0，\u4ed6\u5c31\u4f1a\u4ece\u7236\u8282\u70b9\u5f00\u59cb\u66f4\u65b0，\u4e0d\u9700\u8981\u904d\u5386\u6240\u6709\u5b50\u8282\u70b9\u3002\u5728\u540e\u7eed\u7684\u6d41\u7a0b\u4e2d\u4e5f\u662f\u4e00\u6837\u7684\u9053\u7406\u3002\u8fd9\u6837\u4e00\u6765，\u904d\u5386\u4fee\u6539\u503c\u7684\u6b21\u6570\u660e\u663e\u6bd4\u4eceX\u503c\u4ece\u5c0f\u5230\u5927\u7684\u6b21\u6570\u8981\u5c0f，\u8282\u7701\u4e86\u4e0d\u5c11\u65f6\u95f4\u3002<\/p>\n \u90a3\u4e48，\u6211\u4eec\u7684\u6b65\u957f\u5c31\u8bbe\u7f6e\u4e3a\u81ea\u51cf，\u800c\u4e0d\u662f\u6211\u4eec\u5e38\u7528\u7684\u81ea\u589e\u3002<\/p>\n \u7136\u540e\u8bb0\u5f55\u6211\u4eec\u5f53\u524d\u6700\u5c0f\u503c\u7684\u8282\u70b9poss\u8eab\u4efd\u8bc1\u7f16\u53f7i,\u5b58\u50a8\u5230idRec，\u63a5\u7740poss\u5411\u4e0b\u904d\u5386，\u76f4\u81f3poss<=0,\u9000\u51fa\u5faa\u73af，\u56e0\u4e3a\u6211\u4eec\u7684\u7f16\u53f7\u662f\u4ece1\u5f00\u59cb\u7684\u3002<\/p>\n void update(int poss, int val, int id) { \u7531\u4e8e\u6700\u9876\u5c42\u7684\u6570\u7ec4\u4e00\u5b9a\u662f\u6700\u5c0f\u503c，\u6211\u4eec\u4e5f\u53ea\u5173\u5fc3\u6700\u5c0f\u503c，\u5f80\u4e0b\u662f\u5982\u4f55\u5b9e\u73b0\u7684，\u5176\u5b9e\u6211\u4eec\u5e76\u4e0d\u9700\u8981\u5bf9\u5176\u8fc7\u591a\u5173\u6ce8<\/p>\n \u73b0\u5728，\u6211\u4eec\u9700\u8981\u53e6\u4e00\u4e2a\u7ec4\u4ef6query\u7528\u4e8e\u67e5\u8be2\u7279\u5b9a\u533a\u57df\u6700\u5c0f\u7684\u6570\u503c\u3002\u56e0\u4e3aupdate\u53ea\u63d0\u4f9b\u63d2\u5165\u529f\u80fd，\u6ca1\u6709\u63d0\u4f9b\u67e5\u8be2\u529f\u80fd\u3002<\/p>\n<\/p>\n \u5bf9\u4e8e\u67e5\u8be2\u529f\u80fd，\u6211\u4eec\u9700\u8981\u4f20\u5165\u5f53\u524dposs\u503c，\u627e\u5230\u5f53\u524d\u533a\u57df\u6700\u5c0f\u7684\u8fb9\u957f\u3002\u503c\u5f97\u4e00\u63d0\u7684\u662f，\u6811\u72b6\u6570\u7ec4\u4e2d\u7684\u5143\u7d20\u5e76\u4e0d\u4f1a\u6700\u7ec8\u90fd\u94fe\u63a5\u5230\u540c\u4e00\u4e2a\u7236\u7ed3\u70b9\u4e0a，\u4e00\u4e2a\u6811\u72b6\u6570\u7ec4\u6700\u7ec8\u53ef\u4ee5\u6709\u591a\u4e2a\u7236\u8282\u70b9\u3002<\/p>\n \u5e78\u8fd0\u7684\u662f，\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7poss\u8fd9\u4e2a\u7d22\u5f15\u53ef\u4ee5\u901a\u8fc7\u81ea\u589e\u4f5c\u4e3a\u6865\u6881，\u6765\u8de8\u8fdb\u4e0d\u540c\u7236\u8282\u70b9\u6240\u5c5e\u7684\u6811，\u8fdb\u884c\u6700\u5c0f\u503c\u67e5\u8be2\u3002\u8fd9\u65f6\u5019，\u5c31\u7b26\u5408\u4e8c\u53c9\u7d22\u5f15\u6811\u7684\u903b\u8f91，\u81ea\u4e0b\u5411\u4e0a，\u81ea\u5de6\u5f80\u53f3\u7684\u641c\u7d22\u4e0d\u540c\u7684\u6811，\u5229\u7528lowbit\u6b65\u957f\u8de8\u8d8a\u7684\u529f\u80fd，\u6211\u4eec\u4e5f\u80fd\u5f88\u5feb\u7684\u68c0\u7d22\u5230\u6811\u72b6\u6570\u7ec4\u4e2d\u7684\u6700\u5c0f\u503c\u3002<\/p>\n \u65e2\u7136\u662f\u68c0\u7d22\u6700\u5c0f\u503c，\u90a3\u4e48\u6211\u4eec\u5c31\u5e94\u8be5\u5b58\u5728\u8fd4\u56de\u6700\u5c0f\u503c\u7684\u529f\u80fd，\u4e0d\u8fc7\u7531\u4e8e\u6211\u4eec\u5b58\u50a8\u7684\u662f\u7279\u5f81\u503cval，\u5230\u65f6\u6211\u4eec\u8981\u5224\u65ad\u662f\u5426\u6210\u73af\u8fd8\u8981\u7528\u5230\u6bcf\u4e2a\u8282\u70b9\u81ea\u8eab\u7684\u72ec\u7acb\u6027，\u5426\u5219，\u6211\u4eec\u5c31\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528tree\u5b58\u50a8\u7684\u503c\u6765\u8ba1\u7b97\u66fc\u54c8\u987f\u8ddd\u79bb\u3002\u6240\u4ee5\u6b64\u5904，\u6211\u4eec\u8fd4\u56de\u539f\u59cb\u8282\u70b9\u7684\u7d22\u5f15\u3002<\/p>\n \u8fd9\u91cc\u6d89\u53ca\u5230\u4e00\u4e2a\u95ee\u9898，\u5c31\u662f\u6700\u5c0f\u503c\u80fd\u5426\u662f\u7531\u4e0d\u540c\u7684\u70b9\u8ba1\u7b97\u51fa\u76f8\u540c\u7684\u7b54\u6848？\u7b54\u6848\u662f\u4e0d\u53ef\u80fd\u3002\u56e0\u4e3a\u8fdb\u5165\u6811\u7684\u8282\u70b9\u90fd\u662f\u7ecf\u8fc7Y-X\u7684\u7b5b\u9009\u7684，\u4e0d\u53ef\u80fd\u5728Y-X\u6bd4\u57fa\u51c6\u8282\u70b9\u5c0f\u7684\u60c5\u51b5\u4e0b，Y+X\u7684\u503c\u4e0e\u57fa\u51c6\u8282\u70b9\u8ba1\u7b97\u7684\u76f8\u540c\u3002<\/p>\n \u7136\u540e\u6211\u4eec\u5c31\u9700\u8981\u8003\u8651\u68c0\u7d22\u4e0d\u5230\u503c\u7684\u60c5\u51b5\u4e86\u3002\u5bf9\u4e8e\u68c0\u7d22\u4e0d\u5230\u503c\u7684\u60c5\u51b5，\u6211\u4eec\u4e5f\u5e94\u6709\u8fd4\u56de\u503c，\u4e3a\u4e86\u4e0e\u8282\u70b9\u7f16\u53f7\u6709\u6240\u533a\u5206，\u6211\u4eec\u628a\u65e0\u8fd4\u56de\u503c\u8bbe\u4e3a-1\u3002<\/p>\n \u4e3a\u4e86\u53ef\u4ee5\u627e\u5230\u6700\u5c0f\u503c，\u4e14\u5b58\u5728\u6bd4\u5bf9\u7684\u5bf9\u8c61，\u6211\u4eec\u8bbe\u7528\u4e8e\u6bd4\u8f83\u7684\u503cminval\u521d\u59cb\u65f6\u662f\u65e0\u7a77\u5927\u3002<\/p>\n int query(int pos) { \u73b0\u5728，\u6211\u4eec\u5df2\u7ecf\u5b8c\u6210\u4e86\u5bf9\u4e8c\u53c9\u7d22\u5f15\u6811\u7684\u5168\u90e8\u6784\u5efa，\u6765\u4e00\u5f20\u5168\u5bb6\u798f\u5427\u3002<\/p>\n class BIT { BIT(int _n) { int lowbit(int k) { return k & (-k); }<\/p>\n void update(int pos, int val, int id) { int query(int pos) { \u4e8c\u53c9\u7d22\u5f15\u6811\u53ea\u662f\u5b8c\u6210\u4e86\u5bf9\u533a\u57df\u6700\u5c0f\u8fb9\u7684\u9009\u62e9，\u6211\u4eec\u9700\u8981\u4e00\u4e2a\u53e6\u5916\u4e00\u4e2a\u7ec4\u4ef6，\u6765\u5229\u7528\u4e8c\u53c9\u7d22\u5f15\u6811\u6784\u5efa\u8fb9\u3002<\/p>\n \u662f\u7684，\u5230\u8fd9\u91cc\u7b14\u8005\u5df2\u7ecf\u5199\u4e8612000\u5b57\u7684\u601d\u8def\u4e86\u00a0，\u624d\u521a\u5230\u8fb9\u7684\u6784\u5efa\u3002\u54c8\u54c8，\u4e0d\u8fc7\u5be5\u5be5\u4e00\u767e\u591a\u884c\u7684\u4ee3\u7801，\u5c45\u7136\u6709\u8fd9\u4e48\u591a\u95e8\u9053\u3002\u5e9e\u6742\u7684\u601d\u8def\u88ab\u6d53\u7f29\u6210\u7cbe\u7b80\u7684\u4ee3\u7801，\u771f\u662f\u8ba9\u4eba\u7740\u8ff7\u3002<\/p>\n \u4e3a\u4e86\u7f16\u5199\u6784\u9020\u8fb9\u7684\u7ec4\u4ef6，\u6211\u4eec\u9700\u8981\u53e6\u8bbe\u4e00\u4e2a\u51fd\u6570，\u5c31\u7ba1\u4ed6\u53ebbuild\u5427\u3002<\/p>\n \u5728\u8fdb\u884cbuild\u4e4b\u524d，\u6211\u4eec\u9700\u8981\u77e5\u9053\u4fdd\u7559\u7684\u8fb9\u7684\u4fe1\u606f：<\/p>\n \u7b2c\u4e00：\u6784\u5efa\u4ed6\u6700\u4e3b\u8981\u7684\u76ee\u7684\u662f\u77e5\u9053\u8fb9\u7684\u957f\u5ea6<\/p>\n \u7b2c\u4e8c：\u6211\u4eec\u4e0d\u5e0c\u671b\u5b83\u6210\u73af，\u6240\u4ee5\u6211\u4eec\u5e0c\u671b\u8bb0\u5f55\u8fb9\u7684\u540c\u65f6，\u53ef\u4ee5\u540c\u6b65\u8bb0\u5f55\u6784\u6210\u5b83\u7684\u4e24\u4e2a\u8282\u70b9\u7684\u7f16\u53f7<\/p>\n \u7b2c\u4e09：\u6211\u4eec\u77e5\u9053，\u540e\u9762\u4f1a\u6709\u5bb9\u5668\u5b58\u50a8\u8fb9\u7684\u4fe1\u606f，\u800c\u4e14\u9700\u8981\u9009\u53d6\u6700\u5c0f\u503c\u3002\u4f46\u5b58\u50a8\u4e86\u8fd9\u4e48\u591a\u4fe1\u606f\u7684\u4e2a\u4f53，\u9700\u8981\u4e00\u4e2a\u4e3b\u5bfc\u5143\u7d20\u6765\u4f9b\u7ed9\u6bd4\u8f83，\u8fd9\u4e2a\u4e3b\u5bfc\u5143\u7d20\u5c31\u662f\u66fc\u54c8\u987f\u957f\u5ea6\u3002<\/p>\n \u57fa\u4e8e\u4e0a\u8ff0\u8981\u6c42，\u6784\u5efa\u7ed3\u6784\u4f53\u662f\u4e00\u4e2a\u4e0d\u9519\u7684\u9009\u62e9，\u63a5\u7740，\u4e3a\u4e86\u6ee1\u8db3\u6bd4\u8f83\u7684\u9700\u6c42，\u6211\u4eec\u9700\u8981\u91cd\u8f7d\u8fd0\u7b97\u7b26\u6765\u4f9b\u7ed9\u540e\u9762\u7684\u6392\u5e8f\u9700\u8981\u3002<\/p>\n struct Edge { \u8fd9\u91cc\u9700\u8981\u6ce8\u610f\u7684\u662f，\u91cd\u8f7d\u8fd0\u7b97\u7b26\u524d\u540e\u90fd\u6709\u4e00\u4e2aconst\u3002<\/p>\n \u524d\u9762\u7684const，\u662f\u4e3a\u4e86\u4fdd\u8bc1\u4f20\u5165\u7684\u53c2\u6570\u4e0d\u88ab\u4fee\u6539<\/p>\n \u540e\u9762\u7684const，\u662f\u4e3a\u4e86\u4fdd\u8bc1this\u6307\u9488\u6240\u6307\u7684\u5bf9\u8c61\u53c2\u6570\u4e0d\u88ab\u4fee\u6539<\/p>\n \u65e2\u7136\u5b8c\u6210\u4e86\u5bf9\u8fb9\u7684\u6784\u5efa，\u6211\u4eec\u5c31\u9700\u8981\u6709\u4e00\u4e2a\u5bb9\u5668\u6765\u5b58\u50a8\u751f\u6210\u7684\u8fb9<\/p>\n vector<Edge> edges; <\/p>\n \u5bf9\u4e8ebuild，\u65e2\u7136\u5b83\u662f\u6784\u5efa\u8fb9\u7684\u51fd\u6570，\u90a3\u4ed6\u5c31\u5e94\u8be5\u5305\u62ec\u6211\u4eec\u6784\u5efa\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u6240\u6709\u64cd\u4f5c，\u56e0\u4e3a\u8fd9\u672c\u5c31\u662f\u4e3a\u4ed6\u670d\u52a1\u7684\u3002<\/p>\n \u6211\u4eec\u6574\u7406\u4e00\u4e0b\u601d\u8def\u5427：<\/p>\n \u2460\u5bf9\u6240\u6709\u8282\u70b9pos\u6309\u7167X\u503c\u7531\u5c0f\u5230\u5927\u6392\u5e8f<\/p>\n \u2461\u8bbe\u7acb\u4e24\u4e2a\u6570\u7ec4a\u4e0eb<\/p>\n \u2462\u5bf9b\u8fdb\u884c\u53bb\u91cd<\/p>\n \u2463\u4ee5b\u7684\u5927\u5c0f\u4e3aBIT\u6784\u5efa\u5b9e\u4f8b<\/p>\n \u2464\u8fdb\u5165\u5faa\u73af，\u7531\u7f16\u53f7\u4ece\u5927\u5230\u5c0f\u4e3aBIT\u66f4\u65b0\u6700\u5c0f\u503c<\/p>\n \u2465\u4f7f\u7528query\u641c\u67e5\u6700\u5c0f\u503c<\/p>\n \u2466\u82e5\u662f\u641c\u67e5\u5230\u6700\u5c0f\u503c\u5c31\u8ba1\u7b97\u66fc\u54c8\u987f\u8ddd\u79bb，\u5c06\u5176\u585e\u5165\u6570\u7ec4edges\u4e2d，\u4f9b\u7ed9Kruskal\u7b97\u6cd5\u5907\u9009<\/p>\n \u2467\u65e0\u8bba\u662f\u5426\u5b58\u5728\u6700\u5c0f\u8fb9，\u90fd\u4ee5\u5f53\u524d\u5faa\u73af\u7684\u7f16\u53f7\u4f7f\u7528update\u585e\u5165\u65b0\u8282\u70b9\u8fdbBIT<\/p>\n \u2468\u8fdb\u5165\u4e0b\u4e00\u8f6e\u5faa\u73af，\u91cd\u590d\u2464~\u2467，\u76f4\u81f3\u6240\u6709\u5143\u7d20\u88ab\u5904\u7406\u5b8c\u6210\u3002<\/p>\n \u8fd9\u91cc\u6211\u4eec\u603b\u662f\u53ea\u8003\u8651\u533a\u57dfP1，\u6240\u4ee5\u8fd9\u4e2a\u51fd\u6570\u5e94\u8be5\u8fd8\u88ab\u522b\u7684\u51fd\u6570\u6240\u4f7f\u7528，\u8fd9\u91cc\u6211\u4eec\u53ea\u4f20\u5165\u53c2\u6570n\u6765\u8ba9\u4e0b\u8ff0\u64cd\u4f5c\u6b63\u786e\u6784\u9020\u6570\u7ec4\u5373\u53ef，\u5176\u4ed6\u53c2\u6570\u7531\u8c03\u7528build\u7684\u51fd\u6570\u6765\u63d0\u4f9b\u3002<\/p>\n \u4e3a\u4ec0\u4e48\u8981\u8fd9\u4e48\u505a？<\/p>\n \u56e0\u4e3a\u5728build\u51fd\u6570\u4e2d\u65e0\u8bba\u4ec0\u4e48\u60c5\u51b5，\u90fd\u53ea\u6709n，\u4ee5\u53ca\u6bcf\u4e2a\u8282\u70b9\u7684pos[i].x\u4e0epos[i].y\u662f\u4e0d\u53d8\u7684，\u4e14n\u662f\u4e00\u4e2a\u53c2\u6570，\u4e0d\u4f1a\u5f71\u54cd\u7a0b\u5e8f\u8fd0\u884c\u901f\u5ea6\u3002\u4f46\u82e5\u662f\u6bcf\u6b21n\u90fd\u4f20\u5165pos\u7684\u8bdd，\u9ebb\u70e6\u5c31\u5927\u4e86\u3002\u56e0\u4e3apos\u5b9e\u9645\u4e0a\u662f\u6211\u4eec\u5bf9\u4f20\u5165\u7684vector<vector<int>>& points\u7684\u4e00\u79cd\u6539\u9020，\u5982\u679c\u6bcf\u6b21\u4f7f\u7528pos，\u90fd\u8981\u4f20\u5165\u4e00\u6b21points\u6570\u7ec4\u8fdb\u884c\u8d4b\u503c\u7684\u8bdd，\u7a0b\u5e8f\u5c31\u4f1a\u53d8\u7684\u5947\u6162\u65e0\u6bd4\u3002\u6240\u4ee5\u6211\u4eec\u5e0c\u671b\u53ea\u5bf9pos\u8d4b\u503c\u4e00\u6b21\u5c31\u591f\u4e86，\u540e\u9762\u5c31\u53ef\u4ee5\u5c3d\u60c5\u7684\u4f7f\u7528\u5b83，\u800c\u4e0d\u662f\u6bcf\u8c03\u7528\u4e00\u6b21build\u5c31\u6784\u9020\u4e00\u6b21pos\u3002<\/p>\n void build(int n) { \u00a0\u9700\u8981\u6ce8\u610f\u7684\u662f，\u66fc\u54c8\u987f\u8ddd\u79bb\u662f\u4e00\u4e2a\u7edd\u5bf9\u503c，\u6240\u4ee5\u9700\u8981\u7528\u5230abs（）\u51fd\u6570\u53d6\u7edd\u5bf9\u503c<\/p>\n<\/p>\n \u63a5\u4e0b\u6765，\u5c31\u53ea\u5269\u4e0b\u6700\u540e\u4e00\u4e2a\u7ec4\u4ef6\u4e86<\/p>\n \u7531\u4e8e\u6211\u4eec\u4ee5\u4e0a\u64cd\u4f5c，\u90fd\u9ed8\u8ba4\u662f\u5728P1\u533a\u57df\u8fdb\u884c\u7684，\u800c\u6211\u4eec\u9700\u8981\u751f\u6210P1，P2，P3，P4\u56db\u4e2a\u533a\u57df\u7684\u8fb9，\u663e\u7136，\u6211\u4eec\u9700\u8981\u4e00\u4e2a\u51fd\u6570\u6765\u4e3a\u6211\u4eec\u8fbe\u6210\u8fd9\u4e2a\u76ee\u7684，\u4f7f\u4e4b\u590d\u7528\u4e0a\u9762\u7684build\u51fd\u6570\u6765\u5b8c\u6210\u56db\u4e2a\u533a\u57df\u6240\u6709\u8fb9\u7684\u751f\u6210，\u5c31\u53eb\u5b83MST\u5427<\/p>\n \u6211\u4eec\u4ee5\u4e0a\u64cd\u4f5c，\u90fd\u662f\u57fa\u4e8e\u9898\u76ee\u4f20\u5165\u7684\u6570\u7ec4\u548c\u8282\u70b9\u4e2a\u6570\u5b9e\u73b0\u7684\u3002\u4e3a\u4e86\u8fbe\u6210\u8fb9\u751f\u6210\u7684\u76ee\u7684，\u8fd9\u4e9b\u4fe1\u606f\u4e5f\u5fc5\u4e0d\u53ef\u5c11，\u6240\u4ee5\u9664\u4e86\u4f20\u5165\u7684points\u6570\u7ec4\u4e4b\u5916，\u6211\u4eec\u9700\u8981\u83b7\u53d6\u6570\u7ec4\u4e2a\u6570\u7684\u53c2\u6570<\/p>\n void MST(vector<vector<int>>& points, int n) <\/p>\n \u4ed6\u6ca1\u6709\u8fd4\u56de\u503c，\u56e0\u4e3a\u4ed6\u53ea\u662f\u8c03\u7528build\u51fd\u6570\u751f\u6210\u8fb9\u800c\u5df2<\/p>\n \u9996\u5148，\u6211\u4eec\u9700\u8981\u4e3abuild\u51fd\u6570\u63d0\u4f9b\u5b83\u9700\u8981\u7684pos\u8282\u70b9\u4fe1\u606f<\/p>\n pos.resize(n); \u6784\u9020\u5bf9pos\u6570\u7ec4\u8fdb\u884c\u91cd\u6784，\u4f7f\u4e4b\u5927\u5c0f\u4e3an，\u7136\u540e\u5229\u7528\u4f20\u5165\u7684\u4e8c\u7ef4\u6570\u7ec4\u8d4b\u503c，\u5c06\u5176\u8868\u793a\u4e3ax，y，\u5e76\u8d4b\u4e88\u8fd9\u4e9b\u8282\u70b9\u72ec\u7279\u7f16\u53f7，\u4ee5\u4fbf\u6211\u4eec\u4f7f\u7528\u5e76\u67e5\u96c6\u65f6\u53ef\u4ee5\u63a2\u67e5\u662f\u5426\u6210\u73af，\u4ee5\u53ca\u4f7f\u7528BIT\u65f6\u6709\u68c0\u7d22\u7684\u76ee\u6807\u3002<\/p>\n \u7136\u540e\u4f9d\u6b21\u6784\u5efaP1，P2，P3，P4\u533a\u57df\u7684\u8fb9<\/p>\n P1：\u76f4\u63a5\u8c03\u7528build\u51fd\u6570\u5373\u53ef，\u56e0\u4e3a\u6211\u4eec\u9ed8\u8ba4\u5c31\u662f\u8bbe\u5b9aP1\u4e3a\u793a\u4f8b\u533a\u57df<\/p>\n build(n); <\/p>\n P2：\u65e2\u7136build\u7684\u9ed8\u8ba4\u5904\u7406\u65b9\u5f0f\u662fP1，\u90a3\u6211\u4eec\u5c31\u628aP2\u4e5f\u5316\u6210P1\u533a\u57df\u7684\u6a21\u6837\u5c31\u597d\u4e86，\u6211\u4eec\u53ea\u9700\u8981\u5bf9\u5750\u6807\u8fdb\u884c\u4e00\u4e9b\u53d8\u5316，\u6bd4\u5982\u8bf4：\u7ffb\u8f6c<\/p>\n \u73b0\u5728，X\u4e0eY\u90fd\u662f\u6b63\u6570，\u4e14P1\u533a\u57df\u662f\u9ecf\u5728Y\u8f74\u4e0a\u7684，\u800cP2\u662f\u9ecf\u5728X\u8f74\u4e0a\u7684，\u4e14\u6761\u4ef6\u4e5f\u662fX\u4e0eY\u90fd\u662f\u6b63\u6570\u3002<\/p>\n \u806a\u660e\u7684\u4f60\u5df2\u7ecf\u60f3\u5230\u4e86，\u6211\u53ea\u9700\u8981\u4ea4\u6362X\u4e0eY\u7684\u5730\u4f4d，\u5c31\u53ef\u4ee5\u4ee4P2\u533a\u57df\u50cfP1\u533a\u57df\u4e00\u6837\u5904\u7406\u4e86<\/p>\n for (int i = 0; i < n; i++) { P3：\u6211\u4eec\u77e5\u9053，P3\u4e0eP2\u5728\u539f\u6765\u7684\u5750\u6807\u8f74\u4e2d\u662f\u4ee5X\u8f74\u5bf9\u79f0\u7684，\u73b0\u5728X\u4e0eY\u6570\u503c\u4ea4\u6362<\/p>\n \u672c\u6765\u5e94\u8be5\u662f\u65cb\u8f6cX\u8f74\u7684\u64cd\u4f5c，\u73b0\u5728\u5e94\u8be5\u53d8\u6210\u65cb\u8f6cY\u8f74，\u6b64\u65f6Y\u8f74\u7531pos[i].x\u8868\u793a，\u65cb\u8f6c\u5b83，\u5373\u8ba9\u5176\u53d8\u4e3a\u8d1f\u7684<\/p>\n for (int i = 0; i < n; i++) { P4：\u7531\u4e8e\u73b0\u5728P3\u7b49\u6548\u4e8eP1，\u90a3\u4e48P4\u5c31\u7b49\u6548\u4e3aP2.\u60f3\u60f3\u4ee5P1\u4e3a\u57fa\u51c6\u65f6，\u5bf9P2\u600e\u4e48\u64cd\u4f5c\u7684\u5427：\u4ea4\u6362X\u548cY\u7684\u503c\u3002\u90a3\u4e48\u73b0\u5728\u6211\u4eec\u4e5f\u8981\u5bf9P4\u8fdb\u884c\u540c\u7b49\u64cd\u4f5c，\u8ba9\u4ed6\u53d8\u6210P1\u7684\u5f62\u72b6\u3002<\/p>\n build(n); \u4ee5\u4e0b\u662f\u5b8c\u6574\u7684MST\u51fd\u6570\u5b9e\u73b0<\/p>\n void MST(vector<vector<int>>& points, int n) { \u6765\u6784\u5efa\u9898\u76ee\u7684\u8c03\u7528\u51fd\u6570\u5427，\u4ed6\u5df2\u7ecf\u7ed9\u4e86\u57fa\u672c\u683c\u5f0f\u4e86<\/p>\n int minCostConnectPoints(vector<vector<int>>& points) <\/p>\n \u73b0\u5728，\u6211\u4eec\u9700\u8981\u6309\u6d41\u7a0b\u8c03\u7528\u6211\u4eec\u4e0a\u9762\u6240\u6709\u7684\u7ec4\u4ef6<\/p>\n \u2460\u83b7\u53d6\u5143\u7d20\u4e2a\u6570n，\u8fd9\u5e94\u8be5\u5728\u8c03\u7528\u51fd\u6570\u4e4b\u5916\u8fdb\u884c，\u56e0\u4e3a\u6240\u6709\u51fd\u6570\u90fd\u7528\u5230\u4e86\u5b83，\u4ed6\u5fc5\u987b\u662f\u4e2a\u5168\u5c40\u53d8\u91cf<\/p>\n \u2461\u751f\u6210\u6240\u6709\u53ef\u7528\u7684\u8fb9，\u8c03\u7528MST\u51fd\u6570<\/p>\n \u2462\u521d\u59cb\u5316\u5e76\u67e5\u96c6，\u8c03\u7528DisjointSetUnion\u7c7b\u7684\u6784\u9020\u51fd\u6570<\/p>\n \u2463\u6211\u4eec\u9700\u8981\u4e00\u4e2a\u8fd4\u56de\u503c，\u521d\u59cb\u6210\u672cret\u4e3a0，\u7136\u540e\u4e00\u4e2a\u7528\u4e8e\u8ba1\u6570\u7684\u5b9e\u4f8bnum，\u7528\u4e8e\u8bb0\u5f55\u6b64\u65f6\u63d2\u5165\u7684\u8282\u70b9\u4e2a\u6570，\u65b9\u4fbf\u540e\u7eed\u5224\u65ad\u662f\u5426\u6240\u6709\u8282\u70b9\u5df2\u7ecf\u63d2\u5165，\u7136\u540e\u53ef\u4ee5\u7ec8\u6b62\u7a0b\u5e8f\u3002\u6ce8\u610f，num\u88ab\u521d\u59cb\u5316\u4e3a1，\u56e0\u4e3a\u6bcf\u9009\u7528\u4e00\u6761\u8fb9，\u610f\u5473\u7740\u7eb3\u5165\u4e00\u4e2a\u8282\u70b9\u3002\u867d\u7136\u6709\u65f6\u5019\u9009\u53d6\u7684\u8fb9\u94fe\u63a5\u7684\u8282\u70b9\u5e76\u4e0d\u5904\u4e8e\u6211\u4eec\u4e4b\u524d\u7684\u7236\u8282\u70b9\u91cc\u9762，\u5f62\u6210\u4e24\u68f5\u6811，\u4f46\u5b83\u6700\u540e\u603b\u662f\u4f1a\u6210\u4e3a\u4e00\u4e2a\u6574\u4f53\u7684，\u6240\u4ee5\u521d\u59cb\u4e3a1\u662f\u5f88\u5408\u7406\u7684\u3002<\/p>\n \u2464\u628aMST\u51fd\u6570\u6240\u6709\u751f\u6210\u7684\u8fb9\u4ece\u5c0f\u5230\u5927\u6392\u5e8f<\/p>\n \u2465\u904d\u5386\u6574\u4e2aedges，\u5373\u5b58\u50a8\u6700\u5c0f\u8fb9\u7684\u6570\u7ec4<\/p>\n \u2466\u5229\u7528\u5e76\u67e5\u96c6\u5224\u65ad\u5176\u662f\u5426\u6210\u73af，\u6210\u73af，\u5e76\u67e5\u96c6\u8fd4\u56defalse，\u8fdb\u884c\u4e0b\u4e00\u6b21\u5faa\u73af<\/p>\n \u2467\u4e0d\u6210\u73af，\u7d2f\u52a0\u5f53\u524d\u8fb9\u7684\u957f\u5ea6ret，num\u8ba1\u6570\u52a0\u4e00<\/p>\n \u2468\u5224\u65adnum\u662f\u5426\u7b49\u4e8en，\u5373\u662f\u5426\u6240\u6709\u70b9\u90fd\u5df2\u7ecf\u52a0\u5165，\u82e5\u662f，\u7ec8\u6b62\u7a0b\u5e8f，\u8fd4\u56de\u6210\u672cret<\/p>\n \u2469\u82e5\u6240\u6709\u8282\u70b9\u4ecd\u672a\u52a0\u5165\u6811\u4e2d，\u5faa\u73af\u2465~\u2468，\u76f4\u81f3\u7a0b\u5e8f\u7ec8\u6b62，\u8fd4\u56de\u6210\u672cret<\/p>\n \u8fd9\u5c31\u662f\u6240\u6709\u4e86<\/p>\n \u5199\u7684\u6700\u957f\u7684\u4e00\u6b21，\u524d\u9762\u6240\u6709\u6587\u7ae0\u7684\u5b57\u6570\u52a0\u8d77\u6765\u7ffb\u4e00\u500d\u90fd\u6ca1\u6709\u8fd9\u4e00\u7bc7\u957f\u3002<\/p>\n \u6700\u540e，\u9644\u4e0a\u529b\u6263\u5b98\u65b9\u7684\u5b8c\u6574\u9898\u89e3<\/p>\n \u4f5c\u8005：\u529b\u6263\u5b98\u65b9\u9898\u89e3<\/p>\n \u94fe\u63a5：https:\/\/leetcode.cn\/problems\/min-cost-to-connect-all-points\/solutions\/565801\/lian-jie-suo-you-dian-de-zui-xiao-fei-yo-kcx7\/<\/p>\n \u6765\u6e90：\u529b\u6263（LeetCode）<\/p>\n \u8457\u4f5c\u6743\u5f52\u4f5c\u8005\u6240\u6709\u3002\u5546\u4e1a\u8f6c\u8f7d\u8bf7\u8054\u7cfb\u4f5c\u8005\u83b7\u5f97\u6388\u6743，\u975e\u5546\u4e1a\u8f6c\u8f7d\u8bf7\u6ce8\u660e\u51fa\u5904\u3002<\/p>\n class DisjointSetUnion { public: int find(int x) { int unionSet(int x, int y) { class BIT { BIT(int _n) { int lowbit(int k) { void update(int pos, int val, int id) { int query(int pos) { struct Edge { struct Pos { class Solution { void build(int n) { void solve(vector<vector<int>>& points, int n) { int minCostConnectPoints(vector<vector<int>>& points) { DisjointSetUnion dsu(n); \u611f\u8c22\u5404\u4f4d\u9605\u8bfb\u5230\u6b64\u5904，\u6211\u4eec\u4e0d\u65e5\u76f8\u89c1\u3002<\/p>\n \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/p>\n","protected":false},"excerpt":{"rendered":" \u6587\u7ae0\u6d4f\u89c8\u9605\u8bfb861\u6b21\uff0c\u70b9\u8d5e15\u6b21\uff0c\u6536\u85cf18\u6b21\u3002\u672c\u6587\u63a2\u8ba8\u4e86\u4f7f\u7528Kruskal\u7b97\u6cd5\u89e3\u51b3LeetCode 1584\u9898\uff08\u8fde\u63a5\u6240\u6709\u70b9\u7684\u6700\u5c0f\u8d39\u7528\uff09\u7684\u4f18\u5316\u65b9\u6cd5\u3002\u9488\u5bf9\u7a20\u5bc6\u56fe\u7684\u7279\u6027\uff0c\u6587\u7ae0\u63d0\u51fa\u901a\u8fc7\u51e0\u4f55\u5173\u7cfb\uff0845\u5ea6\u89d2\u5206\u5272\uff09\u5c06\u7a20\u5bc6\u56fe\u8f6c\u5316\u4e3a\u7a00\u758f\u56fe\uff0c\u4ece\u800c\u63d0\u5347Kruskal\u7b97\u6cd5\u6548\u7387\u3002\u8be6\u7ec6\u4ecb\u7ecd\u4e86\u5e76\u67e5\u96c6\u548c\u4e8c\u53c9\u7d22\u5f15\u6811\uff08BIT\uff09\u7684\u5b9e\u73b0\uff0c\u7528\u4e8e\u5224\u65ad\u6210\u73af\u548c\u5feb\u901f\u68c0\u7d22\u533a\u57df\u6700\u5c0f\u8fb9\u3002\u6700\u7ec8\u901a\u8fc7\u5750\u6807\u53d8\u6362\u5904\u7406\u56db\u4e2a\u8c61\u9650\uff0c\u751f\u6210\u5019\u9009\u8fb9\u96c6\u5408\uff0c\u5e76\u4f7f\u7528Kruskal\u7b97\u6cd5\u6784\u5efa\u6700\u5c0f\u751f\u6210\u6811\u3002\u8be5\u65b9\u6848\u5c06\u539f\u59cbO(n\u00b2)\u7684\u8fb9\u6570\u4f18\u5316\u4e3aO(n)\u7ea7\u522b\uff0c\u663e\u8457\u63d0\u9ad8\u4e86\u7b97\u6cd5\u6027\u80fd\u3002<\/p>\n","protected":false},"author":2,"featured_media":58075,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[55,2692,1813,427,925],"topic":[],"class_list":["post-58080","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-server","tag-c","tag-leetcode","tag-1813","tag-427","tag-925"],"yoast_head":"\n![](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250815180033-689f75c1e06e1.png\")
https:\/\/www.bilibili.com\/video\/BV1pE41197Qj\/?spm_id_from=333.1387.homepage.video_card.click&vd_source=f8b81fa5a70ab21de75192e5c7ec720c<\/span><\/span><\/p>\n
\n int id, x, y;
\n}； <\/p>\n
\n int id, x, y;
\n bool operator<(const Pos& a) const { return x == a.x ? y < a.y : x < a.x; }
\n}; <\/p>\n
\n a[i] = pos[i].y – pos[i].x;
\n} <\/p>\n
\n a[i] = pos[i].y – pos[i].x;
\n b[i] = pos[i].y – pos[i].x;
\n } <\/p>\n
\n n = _n;
\n tree.resize(n, INT_MAX);
\n idRec.resize(n, -1);
\n } <\/p>\nhttps:\/\/www.bilibili.com\/video\/BV1pE41197Qj\/?spm_id_from=333.1007.top_right_bar_window_history.content.click&vd_source=f8b81fa5a70ab21de75192e5c7ec720c<\/span><\/span>![\"\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2025\/08\/20250815180034-689f75c272a51.png\")
<\/p>\n\n
\n
\u793a\u4f8b：\u4fee\u6539\u00a0a[3]（\u4e8c\u8fdb\u5236\u00a00011）<\/h5>\n<\/li>\n
\n
\n while (pos > 0) {
\n if (tree[poss] > val) {
\n tree[poss] = val;
\n idRec[poss] = id;
\n }
\n poss -= lowbit(poss);
\n }<\/p>\n
\n int minval = INT_MAX;
\n int j = -1;
\n while (pos < n) {
\n if (minval > tree[pos]) {
\n minval = tree[pos];
\n j = idRec[pos];
\n }
\n pos += lowbit(pos);
\n }
\n return j;
\n } <\/p>\n
\npublic:
\n vector<int> tree, idRec;
\n int n;<\/p>\n
\n n = _n;
\n tree.resize(n, INT_MAX);
\n idRec.resize(n, -1);
\n }<\/p>\n
\n while (pos > 0) {
\n if (tree[pos] > val) {
\n tree[pos] = val;
\n idRec[pos] = id;
\n }
\n pos -= lowbit(pos);
\n }
\n }<\/p>\n
\n int minval = INT_MAX;
\n int j = -1;
\n while (pos < n) {
\n if (minval > tree[pos]) {
\n minval = tree[pos];
\n j = idRec[pos];
\n }
\n pos += lowbit(pos);
\n }
\n return j;
\n }
\n}; <\/p>\n
\n int len, i, j;
\n Edge(int len, int i, int j) : len(len), i(i), j(j) {
\n }
\n bool operator<(const Edge& a) const {
\n return len < a.len;
\n }
\n}; <\/p>\n
\n sort(pos.begin(), pos.end());
\n vector<int> a(n), b(n);
\n for (int i = 0; i < n; i++) {
\n a[i] = pos[i].y – pos[i].x;
\n b[i] = pos[i].y – pos[i].x;
\n }
\n sort(b.begin(), b.end());
\n b.erase(unique(b.begin(), b.end()), b.end());
\n int num = b.size();
\n BIT bit(num + 1);
\n for (int i = n – 1; i >= 0; i–) {
\n int poss = lower_bound(b.begin(), b.end(), a[i]) – b.begin() + 1;
\n int j = bit.query(poss);
\n if (j != -1) {
\n int dis = abs(pos[i].x – pos[j].x) + abs(pos[i].y – pos[j].y);
\n edges.emplace_back(dis, pos[i].id, pos[j].id);
\n }
\n bit.update(poss, pos[i].x + pos[i].y, i);
\n }
\n }<\/p>\n
\n for (int i = 0; i < n; i++) {
\n pos[i].x = points[i][0];
\n pos[i].y = points[i][1];
\n pos[i].id = i;
\n } <\/p>\n
\n swap(pos[i].x, pos[i].y);
\n }
\n build(n); <\/p>\n
\n pos[i].x = -pos[i].x;
\n }
\n build(n); <\/p>\n
\n for (int i = 0; i < n; i++) {
\n swap(pos[i].x, pos[i].y);
\n }
\n build(n); <\/p>\n
\n pos.resize(n);
\n for (int i = 0; i < n; i++) {
\n pos[i].x = points[i][0];
\n pos[i].y = points[i][1];
\n pos[i].id = i;
\n }
\n build(n);
\n for (int i = 0; i < n; i++) {
\n swap(pos[i].x, pos[i].y);
\n }
\n build(n);
\n for (int i = 0; i < n; i++) {
\n pos[i].x = -pos[i].x;
\n }
\n build(n);
\n for (int i = 0; i < n; i++) {
\n swap(pos[i].x, pos[i].y);
\n }
\n build(n);
\n } <\/p>\n\u6211\u5df2\u7ecf\u4ec0\u4e48\u90fd\u4e0d\u7f3a\u4e86<\/span><\/h2>\n
\nprivate:
\n vector<int> f, rank;
\n int n;<\/p>\n
\n DisjointSetUnion(int _n) {
\n n = _n;
\n rank.resize(n, 1);
\n f.resize(n);
\n for (int i = 0; i < n; i++) {
\n f[i] = i;
\n }
\n }<\/p>\n
\n return f[x] == x ? x : f[x] = find(f[x]);
\n }<\/p>\n
\n int fx = find(x), fy = find(y);
\n if (fx == fy) {
\n return false;
\n }
\n if (rank[fx] < rank[fy]) {
\n swap(fx, fy);
\n }
\n rank[fx] += rank[fy];
\n f[fy] = fx;
\n return true;
\n }
\n};<\/p>\n
\npublic:
\n vector<int> tree, idRec;
\n int n;<\/p>\n
\n n = _n;
\n tree.resize(n, INT_MAX);
\n idRec.resize(n, -1);
\n }<\/p>\n
\n return k & (-k);
\n }<\/p>\n
\n while (pos > 0) {
\n if (tree[pos] > val) {
\n tree[pos] = val;
\n idRec[pos] = id;
\n }
\n pos -= lowbit(pos);
\n }
\n }<\/p>\n
\n int minval = INT_MAX;
\n int j = -1;
\n while (pos < n) {
\n if (minval > tree[pos]) {
\n minval = tree[pos];
\n j = idRec[pos];
\n }
\n pos += lowbit(pos);
\n }
\n return j;
\n }
\n};<\/p>\n
\n int len, x, y;
\n Edge(int len, int x, int y) : len(len), x(x), y(y) {
\n }
\n bool operator<(const Edge& a) const {
\n return len < a.len;
\n }
\n};<\/p>\n
\n int id, x, y;
\n bool operator<(const Pos& a) const {
\n return x == a.x ? y < a.y : x < a.x;
\n }
\n};<\/p>\n
\npublic:
\n vector<Edge> edges;
\n vector<Pos> pos;<\/p>\n
\n sort(pos.begin(), pos.end());
\n vector<int> a(n), b(n);
\n for (int i = 0; i < n; i++) {
\n a[i] = pos[i].y – pos[i].x;
\n b[i] = pos[i].y – pos[i].x;
\n }
\n sort(b.begin(), b.end());
\n b.erase(unique(b.begin(), b.end()), b.end());
\n int num = b.size();
\n BIT bit(num + 1);
\n for (int i = n – 1; i >= 0; i–) {
\n int poss = lower_bound(b.begin(), b.end(), a[i]) – b.begin() + 1;
\n int j = bit.query(poss);
\n if (j != -1) {
\n int dis = abs(pos[i].x – pos[j].x) + abs(pos[i].y – pos[j].y);
\n edges.emplace_back(dis, pos[i].id, pos[j].id);
\n }
\n bit.update(poss, pos[i].x + pos[i].y, i);
\n }
\n }<\/p>\n
\n pos.resize(n);
\n for (int i = 0; i < n; i++) {
\n pos[i].x = points[i][0];
\n pos[i].y = points[i][1];
\n pos[i].id = i;
\n }
\n build(n);
\n for (int i = 0; i < n; i++) {
\n swap(pos[i].x, pos[i].y);
\n }
\n build(n);
\n for (int i = 0; i < n; i++) {
\n pos[i].x = -pos[i].x;
\n }
\n build(n);
\n for (int i = 0; i < n; i++) {
\n swap(pos[i].x, pos[i].y);
\n }
\n build(n);
\n }<\/p>\n
\n int n = points.size();
\n solve(points, n);<\/p>\n
\n sort(edges.begin(), edges.end());
\n int ret = 0, num = 1;
\n for (auto& [len, x, y] : edges) {
\n if (dsu.unionSet(x, y)) {
\n ret += len;
\n num++;
\n if (num == n) {
\n break;
\n }
\n }
\n }
\n return ret;
\n }
\n};<\/p>\n