{"id":885,"date":"2026-03-07T16:50:22","date_gmt":"2026-03-07T08:50:22","guid":{"rendered":"https:\/\/vite66.cn\/?p=885"},"modified":"2026-03-07T18:08:47","modified_gmt":"2026-03-07T10:08:47","slug":"kadane%e7%ae%97%e6%b3%95-%e8%bf%9e%e7%bb%ad%e5%ad%90%e6%ae%b5%e5%92%8c%e9%97%ae%e9%a2%98","status":"publish","type":"post","link":"https:\/\/vite66.cn\/?p=885","title":{"rendered":"Kadane\u7b97\u6cd5-\u8fde\u7eed\u5b50\u6bb5\u548c\u95ee\u9898"},"content":{"rendered":"

Kadane\u7b97\u6cd5\u53ca\u5176\u5728\u4e00\u7ef4\u4e0e\u4e8c\u7ef4\u6700\u5927\u5b50\u6bb5\u548c\u95ee\u9898\u4e2d\u7684\u5e94\u7528<\/h1>\n

\u6458\u8981<\/h2>\n

\u672c\u6587\u7cfb\u7edf\u9610\u8ff0Kadane\u7b97\u6cd5\u7684\u6838\u5fc3\u539f\u7406\u3001\u6570\u5b66\u8bc1\u660e\u53ca\u5176\u5728\u4e00\u7ef4\u6700\u5927\u5b50\u6570\u7ec4\u548c\u3001\u4e8c\u7ef4\u6700\u5927\u5b50\u77e9\u9635\u548c\u95ee\u9898\u4e2d\u7684\u5e94\u7528\u3002\u901a\u8fc7\u4e25\u8c28\u7684\u6570\u5b66\u63a8\u5bfc\u4e0e\u4ee3\u7801\u5b9e\u73b0\uff0c\u5c55\u793a\u8be5\u7b97\u6cd5\u4ece\u4e00\u7ef4\u5230\u4e8c\u7ef4\u7684\u6269\u5c55\u903b\u8f91\uff0c\u5e76\u8865\u5145\u8bc1\u660e\u4e00\u7ef4\u573a\u666f\u4e0b\u524d\u7f00\u548c\u65b9\u6cd5\u4e0eKadane\u7b97\u6cd5\u7684\u6570\u5b66\u7b49\u4ef7\u6027\uff0c\u4e3a\u7b97\u6cd5\u590d\u4e60\u4e0e\u5e94\u7528\u63d0\u4f9b\u53c2\u8003\u3002<\/p>\n

\n

\u5f15\u8a00<\/h2>\n

\u6700\u5927\u5b50\u6bb5\u548c\u95ee\u9898\u662f\u7b97\u6cd5\u8bbe\u8ba1\u4e2d\u7684\u7ecf\u5178\u95ee\u9898\uff0c\u5176\u6838\u5fc3\u662f\u5728\u7ed9\u5b9a\u5e8f\u5217\u4e2d\u5bfb\u627e\u8fde\u7eed\u5b50\u5e8f\u5217\uff0c\u4f7f\u5176\u5143\u7d20\u548c\u6700\u5927\u3002Kadane\u7b97\u6cd5\u4ee5\u5176 $O(n)$ \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u6210\u4e3a\u89e3\u51b3\u4e00\u7ef4\u6700\u5927\u5b50\u6570\u7ec4\u548c\u95ee\u9898\u7684\u6700\u4f18\u89e3\u6cd5\uff1b\u901a\u8fc7\u964d\u7ef4\u601d\u60f3\uff0c\u8be5\u7b97\u6cd5\u53ef\u8fdb\u4e00\u6b65\u6269\u5c55\u81f3\u4e8c\u7ef4\u6700\u5927\u5b50\u77e9\u9635\u548c\u95ee\u9898\u3002<\/p>\n

\n

\u4e00\u7ef4\u6700\u5927\u5b50\u6570\u7ec4\u548c\u95ee\u9898<\/h2>\n

\u95ee\u9898\u5b9a\u4e49<\/h3>\n

\u7ed9\u5b9a\u957f\u5ea6\u4e3a $n$ \u7684\u6574\u6570\u6570\u7ec4 $a[1..n]$ \uff08\u4e0b\u6807\u4ece1\u5f00\u59cb\uff09\uff0c\u5bfb\u627e\u8fde\u7eed\u5b50\u6570\u7ec4 $a[i..j]$ \uff08 $1 \\le i \\le j \\le n$ \uff09\uff0c\u4f7f\u5f97\u5176\u548c $S(i,j) = \\sum_{k=i}^j a[k]$ \u6700\u5927\u3002\u82e5\u6240\u6709\u5b50\u6570\u7ec4\u548c\u5747\u4e3a\u8d1f\uff0c\u5219\u8fd4\u56de0\uff08\u5bf9\u5e94\u201c\u653e\u5f03\u9009\u62e9\u201d\u7684\u573a\u666f\uff09\u3002<\/p>\n

Kadane\u7b97\u6cd5\u63cf\u8ff0<\/h3>\n

\u72b6\u6001\u5b9a\u4e49<\/h4>\n

\u5b9a\u4e49 $dp[j]$ \u4e3a\u4ee5\u7b2c <\/strong> $j$ \u4e2a\u5143\u7d20\u7ed3\u5c3e\u7684\u6240\u6709\u8fde\u7eed\u5b50\u6570\u7ec4\u7684\u6700\u5927\u548c<\/strong>\u3002<\/p>\n

\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b<\/h4>\n

\u5bf9\u4e8e\u7b2c $j$ \u4e2a\u5143\u7d20\uff0c\u4ee5\u5176\u7ed3\u5c3e\u7684\u8fde\u7eed\u5b50\u6570\u7ec4\u6709\u4e24\u79cd\u9009\u62e9\uff1a<\/p>\n

    \n
  1. \n

    \u4ec5\u5305\u542b $a[j]$ \u81ea\u8eab\uff0c\u548c\u4e3a $a[j]$ \uff1b<\/p>\n<\/li>\n

  2. \n

    \u5305\u542b $a[j]$ \u53ca\u4ee5 $j-1$ \u7ed3\u5c3e\u7684\u6700\u5927\u5b50\u6570\u7ec4\uff0c\u548c\u4e3a $dp[j-1] + a[j]$ \u3002<\/p>\n<\/li>\n<\/ol>\n

    \u56e0\u6b64\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e3a\uff1a<\/p>\n

    $dp[j] = \\max\\left(a[j], dp[j-1] + a[j]\\right)$ <\/p>\n

    \u5168\u5c40\u6700\u5927\u503c\u8ba1\u7b97<\/h4>\n

    \u7531\u4e8e\u4efb\u610f\u8fde\u7eed\u5b50\u6570\u7ec4\u5fc5\u7136\u4ee5\u67d0\u4e2a\u4f4d\u7f6e $j$ \u7ed3\u5c3e\uff0c\u5168\u5c40\u6700\u5927\u5b50\u6570\u7ec4\u548c\u4e3a\uff1a<\/p>\n

    $max\\_sum = \\max\\left(dp[1], dp[2], \\dots, dp[n]\\right)$ <\/p>\n

    \u6700\u7ec8\u7ed3\u679c\u4e3a $\\max(max\\_sum, 0)$ \uff0c\u4ee5\u5904\u7406\u5168\u8d1f\u6570\u7ec4\u7684\u60c5\u51b5\u3002<\/p>\n

    \u6570\u5b66\u6b63\u786e\u6027\u8bc1\u660e<\/h3>\n

    \u91c7\u7528\u6570\u5b66\u5f52\u7eb3\u6cd5\u8bc1\u660e\u201c $dp[j]$ \u662f\u4ee5 $j$ \u7ed3\u5c3e\u7684\u6700\u5927\u5b50\u6570\u7ec4\u548c\u201d\uff1a<\/p>\n

      \n
    • \n

      \u57fa\u4f8b\uff08<\/strong> $j=1$ \uff09<\/strong>\uff1a<\/p>\n

      $dp[1] = a[1]$ \uff0c\u663e\u7136\u6210\u7acb\u2014\u2014\u4ee5\u7b2c\u4e00\u4e2a\u5143\u7d20\u7ed3\u5c3e\u7684\u5b50\u6570\u7ec4\u53ea\u80fd\u662f\u5176\u81ea\u8eab\u3002<\/p>\n<\/li>\n

    • \n

      \u5f52\u7eb3\u6b65\u9aa4\uff08\u5047\u8bbe <\/strong> $j=k$ \u65f6\u6210\u7acb\uff0c\u8bc1\u660e <\/strong> $j=k+1$ \u65f6\u4e5f\u6210\u7acb\uff09<\/strong>\uff1a<\/p>\n<\/li>\n<\/ul>\n

      \u5047\u8bbe $dp[k]$ \u662f\u4ee5 $k$ \u7ed3\u5c3e\u7684\u6700\u5927\u5b50\u6570\u7ec4\u548c\u3002\u8003\u8651\u4ee5 $k+1$ \u7ed3\u5c3e\u7684\u5b50\u6570\u7ec4\uff1a<\/p>\n

        \n
      • \n

        \u82e5\u5b50\u6570\u7ec4\u4ec5\u5305\u542b $a[k+1]$ \uff0c\u548c\u4e3a $a[k+1]$ \uff1b<\/p>\n<\/li>\n

      • \n

        \u82e5\u5b50\u6570\u7ec4\u5305\u542b\u524d\u9762\u7684\u5143\u7d20\uff0c\u5219\u5176\u5f62\u5f0f\u4e3a\u201c\u4ee5 $k$ \u7ed3\u5c3e\u7684\u5b50\u6570\u7ec4 + $a[k+1]$ \u201d\u3002\u6839\u636e\u5f52\u7eb3\u5047\u8bbe\uff0c $dp[k]$ \u662f\u4ee5 $k$ \u7ed3\u5c3e\u7684\u6700\u5927\u548c\uff0c\u56e0\u6b64\u201c $dp[k] + a[k+1]$ \u201d\u662f\u8be5\u60c5\u51b5\u4e0b\u7684\u6700\u5927\u503c\u3002<\/p>\n<\/li>\n<\/ul>\n

        \u56e0\u6b64 $dp[k+1] = \\max(a[k+1], dp[k] + a[k+1])$ \u6210\u7acb\uff0c\u5f52\u7eb3\u6b65\u9aa4\u5f97\u8bc1\u3002<\/p>\n

        \u524d\u7f00\u548c\u89c6\u89d2\u4e0eKadane\u7b97\u6cd5\u7684\u672c\u8d28\u8054\u7cfb\u4e0e\u7b49\u4ef7\u6027\u8bc1\u660e<\/h3>\n

        \u672c\u8282\u5c06\u4e25\u683c\u8bc1\u660e\uff1aKadane\u7b97\u6cd5\u5728\u6570\u5b66\u4e0a\u4e0e\u201c\u524d\u7f00\u548c + \u7ef4\u62a4\u6700\u5c0f\u503c\u201d\u7684\u65b9\u6cd5\u5b8c\u5168\u7b49\u4ef7\u3002<\/p>\n

        \u524d\u7f00\u548c\u5b9a\u4e49<\/h4>\n

        \u5b9a\u4e49\u524d\u7f00\u548c\u6570\u7ec4 $s[0..n]$ \uff0c\u5176\u4e2d\uff1a<\/p>\n

          \n
        • \n

          $s[0] = 0$ <\/p>\n<\/li>\n

        • \n

          $s[j] = a[1] + a[2] + \\dots + a[j] = \\sum_{k=1}^j a[k]$ <\/p>\n<\/li>\n<\/ul>\n

          \u6839\u636e\u5b9a\u4e49\uff0c\u4efb\u610f\u5b50\u6570\u7ec4 $a[i..j]$ \u7684\u548c\u53ef\u8868\u793a\u4e3a\uff1a<\/p>\n

          $S(i,j) = s[j] - s[i-1]$ <\/p>\n

          \u524d\u7f00\u548c\u89c6\u89d2\u4e0b\u7684\u76ee\u6807<\/h4>\n

          \u5bf9\u4e8e\u56fa\u5b9a\u7684\u53f3\u7aef\u70b9 $j$ \uff0c\u6211\u4eec\u7684\u76ee\u6807\u662f\u627e\u5230\u5de6\u7aef\u70b9 $i$ \uff08 $1 \\le i \\le j$ \uff09\uff0c\u4f7f\u5f97 $S(i,j)$ \u6700\u5927\u3002\u7531\u4e8e $s[j]$ \u662f\u5b9a\u503c\uff0c\u6700\u5927\u5316 $s[j] - s[i-1]$ \u7b49\u4ef7\u4e8e\u6700\u5c0f\u5316<\/strong> $s[i-1]$ \u3002<\/p>\n

          \u56e0\u6b64\uff0c\u4ee5 $j$ \u7ed3\u5c3e\u7684\u6700\u5927\u5b50\u6570\u7ec4\u548c\u4e3a\uff1a<\/p>\n

          max\\_ending\\_at\\_j = s[j] - \\min_{0 \\le k \\le j-1} s[k]<\/code><\/pre>\n
          \u7b49\u4ef7\u6027\u63a8\u5bfc<\/h4>\n
          \u6211\u4eec\u5c06\u8bc1\u660e\uff1aKadane\u7b97\u6cd5\u4e2d\u7684  $dp[j]$  \u4e0e\u4e0a\u5f0f\u4e2d\u7684  $max\\_ending\\_at\\_j$  \u5b8c\u5168\u76f8\u7b49\u3002<\/p>\n
          \u8bc1\u660e\uff1a<\/strong><\/p>\n
          \u9996\u5148\uff0c\u6839\u636e\u524d\u7f00\u548c\u5b9a\u4e49\uff0c\u6211\u4eec\u6709\uff1a<\/p>\n
          $a[j] = s[j] - s[j-1]$ <\/p>\n
          \u5c06Kadane\u7684\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u5c55\u5f00\uff1a<\/p>\n
          $dp[j] = \\max\\left(a[j], dp[j-1] + a[j]\\right)$ <\/p>\n
          \u6839\u636e\u5f52\u7eb3\u5047\u8bbe\uff08\u6216\u9012\u63a8\u5173\u7cfb\uff09\uff0c $dp[j-1]$  \u53ef\u8868\u793a\u4e3a\uff1a<\/p>\n
          $dp[j-1] = s[j-1] - \\min_{0 \\le k \\le j-2} s[k]$ <\/p>\n
          \u5c06  $a[j]$  \u548c  $dp[j-1]$  \u4ee3\u5165Kadane\u65b9\u7a0b\u7684\u53f3\u4fa7\u7b2c\u4e8c\u9879\uff1a<\/p>\n
          \\begin{aligned}dp[j-1] + a[j] &= \\left(s[j-1] - \\min_{0 \\le k \\le j-2} s[k]\\right) + \\left(s[j] - s[j-1]\\right) &= s[j] - \\min_{0 \\le k \\le j-2} s[k] \\end{aligned}<\/code><\/pre>\n
          \u73b0\u5728\u770bKadane\u65b9\u7a0b\u7684\u53f3\u4fa7\u7b2c\u4e00\u9879  $a[j]$ \uff0c\u5b83\u53ef\u4ee5\u6539\u5199\u4e3a\uff1a<\/p>\n
          $a[j] = s[j] - s[j-1]$ <\/p>\n
          \u8fd9\u5bf9\u5e94\u4e86\u5728  $k = j-1$  \u65f6\u7684\u53d6\u503c\u3002<\/p>\n
          \u56e0\u6b64\uff0cKadane\u7684\u53d6\u6700\u5927\u503c\u64cd\u4f5c\uff1a<\/p>\n
          $dp[j] = \\max\\left(s[j] - s[j-1], s[j] - \\min_{0 \\le k \\le j-2} s[k]\\right)$ <\/p>\n
          \u53ef\u4ee5\u63d0\u53d6\u516c\u56e0\u5b50  $s[j]$ \uff0c\u8f6c\u5316\u4e3a\uff1a<\/p>\n
          $dp[j] = s[j] - \\min\\left(s[j-1], \\min_{0 \\le k \\le j-2} s[k]\\right)$<\/p>\n
          \u7531\u4e8e<\/p>\n
          \\min\\left(s[j-1], \\min_{0 \\le k \\le j-2} s[k]\\right) = \\min_{0 \\le k \\le j-1} s[k]<\/code><\/pre>\n
          \u6211\u4eec\u6700\u7ec8\u5f97\u5230\uff1a<\/p>\n
          $dp[j] = s[j] - \\min_{0 \\le k \\le j-1} s[k]$ <\/p>\n
          \u8bc1\u6bd5\u3002<\/strong><\/p>\n
          \u7ed3\u8bba<\/h4>\n
          Kadane\u7b97\u6cd5\u4e0e\u524d\u7f00\u548c\u65b9\u6cd5\u662f\u540c\u4e00\u6570\u5b66\u672c\u8d28\u7684\u4e24\u79cd\u4e0d\u540c\u8868\u73b0\u5f62\u5f0f<\/strong>\uff1a<\/p>\n
            \n
          • \n
            \u524d\u7f00\u548c\u89c6\u89d2<\/strong>\u662f\u201c\u5168\u5c40\u89c6\u89d2\u201d\uff0c\u901a\u8fc7\u7ef4\u62a4\u5386\u53f2\u524d\u7f00\u548c\u7684\u6700\u5c0f\u503c\u6765\u8ba1\u7b97\u5f53\u524d\u6700\u4f18\u89e3\uff1b<\/p>\n<\/li>\n
          • \n
            Kadane\u7b97\u6cd5<\/strong>\u662f\u201c\u589e\u91cf\u89c6\u89d2\u201d\u6216\u201c\u8d2a\u5fc3\u89c6\u89d2\u201d\uff0c\u901a\u8fc7\u52a8\u6001\u89c4\u5212\u8f6c\u79fb\u65b9\u7a0b\u9690\u5f0f\u5730\u7ef4\u62a4\u4e86\u8fd9\u4e00\u6700\u5c0f\u503c\u3002<\/p>\n<\/li>\n<\/ul>\n
            \u4e24\u8005\u5728\u65f6\u95f4\u590d\u6742\u5ea6\u4e0a\u5747\u4e3a  $O(n)$ \uff0c\u4f46Kadane\u7b97\u6cd5\u5728\u4ee3\u7801\u5b9e\u73b0\u4e0a\u901a\u5e38\u66f4\u7b80\u6d01\uff0c\u4e14\u65e0\u9700\u663e\u5f0f\u5b58\u50a8\u524d\u7f00\u548c\u6570\u7ec4\u3002<\/p>\n
            \u4ee3\u7801\u5b9e\u73b0<\/h3>\n
            \n#include <iostream>\n#include <vector>\n#include <algorithm>\nusing namespace std;\n\nint main() {\n    int n;\n    cin >> n;\n    vector<int> a(n);\n    for (int i = 0; i < n; ++i) cin >> a[i];\n\n    int current_sum = a[0];\n    int max_sum = a[0];\n    for (int i = 1; i < n; ++i) {\n        current_sum = max(a[i], current_sum + a[i]);\n        max_sum = max(max_sum, current_sum);\n    }\n\n    cout << max(max_sum, 0) << endl;\n    return 0;\n}<\/code><\/pre>\n
            \n
            \u4e8c\u7ef4\u6700\u5927\u5b50\u77e9\u9635\u548c\u95ee\u9898<\/h2>\n
            \u95ee\u9898\u5b9a\u4e49<\/h3>\n
            \u7ed9\u5b9a  $R \\times C$  \u7684\u6574\u6570\u77e9\u9635  $M$ \uff0c\u5bfb\u627e\u884c\u548c\u5217\u5747\u8fde\u7eed\u7684\u975e\u7a7a\u5b50\u77e9\u9635\uff0c\u4f7f\u5176\u5143\u7d20\u548c\u6700\u5927\u3002<\/p>\n
            \u964d\u7ef4\u601d\u8def\uff1a\u4e8c\u7ef4\u8f6c\u4e00\u7ef4<\/h3>\n
            \u6838\u5fc3\u601d\u60f3\u662f\u679a\u4e3e\u6240\u6709\u8fde\u7eed\u884c\u533a\u95f4<\/strong>\uff0c\u5c06\u8be5\u533a\u95f4\u5185\u6bcf\u4e00\u5217\u7684\u5143\u7d20\u6c42\u548c\uff0c\u538b\u7f29\u4e3a\u4e00\u7ef4\u6570\u7ec4\uff0c\u4ece\u800c\u5c06\u4e8c\u7ef4\u95ee\u9898\u8f6c\u5316\u4e3a\u4e00\u7ef4\u6700\u5927\u5b50\u6570\u7ec4\u548c\u95ee\u9898\u3002<\/p>\n
            \u7b97\u6cd5\u63cf\u8ff0<\/h3>\n
              \n
            1. \n
              \u679a\u4e3e\u4e0a\u8fb9\u754c<\/strong> $top$ \uff08 $0 \\le top < R$ \uff09\uff1b<\/p>\n<\/li>\n
            2. \n
              \u521d\u59cb\u5316\u5217\u548c\u6570\u7ec4<\/strong> $col\\_sum$ \uff0c\u957f\u5ea6\u4e3a  $C$ \uff0c\u521d\u59cb\u503c\u4e3a0\uff1b<\/p>\n<\/li>\n
            3. \n
              \u679a\u4e3e\u4e0b\u8fb9\u754c<\/strong> $bottom$ \uff08 $top \\le bottom < R$ \uff09\uff1a
              \n\u5c06\u7b2c  $bottom$  \u884c\u7684\u5143\u7d20\u7d2f\u52a0\u5230 $col\\_sum$  \u4e2d\uff08 $col\\_sum[k] += M[bottom][k]$ \uff09\uff1b<\/p>\n<\/li>\n
            4. \n
              \u5bf9  $col\\_sum$  \u5e94\u7528Kadane\u7b97\u6cd5\uff0c\u6c42\u5f97\u5f53\u524d\u884c\u533a\u95f4\u5185\u7684\u6700\u5927\u5b50\u77e9\u9635\u548c\uff1b<\/p>\n<\/li>\n
            5. \n
              \u66f4\u65b0\u5168\u5c40\u6700\u5927\u503c<\/strong>\uff0c\u904d\u5386\u6240\u6709\u884c\u533a\u95f4\u540e\u5f97\u5230\u6700\u7ec8\u7ed3\u679c\u3002<\/p>\n<\/li>\n<\/ol>\n
              \u6570\u5b66\u6b63\u786e\u6027\u8bc1\u660e<\/h3>\n
              \u7b26\u53f7\u5b9a\u4e49<\/h4>\n
              \u8bbe\u5b50\u77e9\u9635\u7684\u884c\u8303\u56f4\u4e3a  $[top, bottom]$ \uff0c\u5217\u8303\u56f4\u4e3a  $[left, right]$ \uff0c\u5176\u548c\u4e3a\uff1a<\/p>\n
               Sum = \\sum_{i=top}^{bottom} \\sum_{j=left}^{right} M[i][j]<\/code><\/pre>\n
              \u4ea4\u6362\u6c42\u548c\u987a\u5e8f<\/h4>\n
              \u5c06\u201c\u5148\u52a0\u884c\u3001\u518d\u52a0\u5217\u201d\u7684\u987a\u5e8f\u4ea4\u6362\u4e3a\u201c\u5148\u52a0\u5217\u3001\u518d\u52a0\u884c\u201d\uff1a<\/p>\n
              Sum = \\sum_{j=left}^{right} \\left( \\sum_{i=top}^{bottom} M[i][j] \\right)<\/code><\/pre>\n
              \u4ee4\u5217\u548c\u6570\u7ec4  <\/p>\n
              col\\_sum[j] = \\sum_{i=top}^{bottom} M[i][j]<\/code><\/pre>\n
              \uff0c\u4ee3\u5165\u5f97\uff1a<\/p>\n
               Sum = \\sum_{j=left}^{right} col\\_sum[j]<\/code><\/pre>\n
              \u7531\u6b64\u53ef\u8bc1\uff1a\u56fa\u5b9a\u884c\u8303\u56f4 <\/strong> $[top, bottom]$  \u65f6\uff0c\u5b50\u77e9\u9635\u7684\u548c\u7b49\u4ef7\u4e8e <\/strong> $col\\_sum$  \u6570\u7ec4\u7684\u8fde\u7eed\u5b50\u6570\u7ec4\u548c<\/strong>\u3002<\/p>\n
              \u8986\u76d6\u6240\u6709\u5b50\u77e9\u9635<\/h4>\n
              \u7531\u4e8e\u4efb\u610f\u5b50\u77e9\u9635\u90fd\u5bf9\u5e94\u552f\u4e00\u7684\u8fde\u7eed\u884c\u8303\u56f4  $[top, bottom]$ \uff0c\u679a\u4e3e\u6240\u6709\u884c\u533a\u95f4\u53ef\u8986\u76d6\u5168\u90e8\u5b50\u77e9\u9635\u3002\u7ed3\u5408Kadane\u7b97\u6cd5\u7684\u6b63\u786e\u6027\uff0c\u904d\u5386\u6240\u6709\u884c\u533a\u95f4\u540e\u5f97\u5230\u7684\u6700\u5927\u503c\u5373\u4e3a\u5168\u5c40\u6700\u4f18\u89e3\u3002<\/p>\n
              \u4ee3\u7801\u5b9e\u73b0<\/h3>\n
              \n#include <iostream>\n#include <vector>\n#include <algorithm>\nusing namespace std;\n\nint a[55][55];\nvoid solve() {\n    int n, m;\n    cin >> n >> m;\n    for (int i = 0; i < n; ++i) {\n        for (int j = 0; j < m; ++j) {\n            cin >> a[i][j];\n        }\n    }\n    int ans = -100000;\n    for (int i = 0; i < n; ++i) {\n        vector<int> col(m);\n        for (int j = i; j < n; ++j) {\n            int ans2 = -100000;\n            int cursum = 0;\n            for (int k = 0; k < m; ++k) {\n                col[k] += a[j][k];\n                cursum = max(cursum + col[k], col[k]);\n                ans2 = max(ans2, cursum);\n            }\n            ans = max(ans, ans2);\n        }\n    }\n    cout << ans << endl;\n}\n\nint main() {\n    int t = 1;\n    while (t--) solve();\n    return 0;\n}<\/code><\/pre>\n
              \n
              \u590d\u6742\u5ea6\u5206\u6790<\/h2>\n
                \n
              • \n
                \u4e00\u7ef4\u95ee\u9898<\/strong>\uff1a\u65f6\u95f4\u590d\u6742\u5ea6  $O(n)$ \uff0c\u4ec5\u9700\u4e00\u6b21\u904d\u5386\u6570\u7ec4\uff1b\u7a7a\u95f4\u590d\u6742\u5ea6  $O(1)$ \uff08\u4f18\u5316\u540e\u65e0\u9700\u5b58\u50a8\u5b8c\u6574  $dp$  \u6570\u7ec4\u6216\u524d\u7f00\u548c\u6570\u7ec4\uff09\u3002<\/p>\n<\/li>\n
              • \n
                \u4e8c\u7ef4\u95ee\u9898<\/strong>\uff1a\u65f6\u95f4\u590d\u6742\u5ea6  $O(R^2C)$ \uff0c\u5176\u4e2d $R$  \u4e3a\u884c\u6570\u3001 $C$  \u4e3a\u5217\u6570\uff1b\u7a7a\u95f4\u590d\u6742\u5ea6  $O(C)$ \uff0c\u7528\u4e8e\u5b58\u50a8\u5217\u548c\u6570\u7ec4\u3002<\/p>\n<\/li>\n<\/ul>\n
                \n
                \u603b\u7ed3<\/h2>\n
                Kadane\u7b97\u6cd5\u7684\u6838\u5fc3\u5728\u4e8e\u901a\u8fc7\u52a8\u6001\u89c4\u5212\u5b9a\u4e49\u201c\u4ee5\u5f53\u524d\u5143\u7d20\u7ed3\u5c3e\u7684\u6700\u5927\u5b50\u6bb5\u548c\u201d\uff0c\u5c06\u95ee\u9898\u62c6\u89e3\u4e3a\u9012\u63a8\u5b50\u95ee\u9898\u3002\u672c\u6587\u901a\u8fc7\u4e25\u8c28\u7684\u6570\u5b66\u63a8\u5bfc\uff0c\u8865\u5145\u8bc1\u660e\u4e86Kadane\u7b97\u6cd5\u4e0e\u201c\u524d\u7f00\u548c+\u7ef4\u62a4\u6700\u5c0f\u503c\u201d\u65b9\u6cd5\u7684\u5b8c\u5168\u7b49\u4ef7\u6027\uff0c\u63ed\u793a\u4e86\u5176\u589e\u91cf\u89c6\u89d2\u80cc\u540e\u7684\u5168\u5c40\u6570\u5b66\u672c\u8d28\u3002\u901a\u8fc7\u964d\u7ef4\u601d\u60f3\uff0c\u8be5\u7b97\u6cd5\u53ef\u4ece\u4e00\u7ef4\u6269\u5c55\u81f3\u4e8c\u7ef4\uff0c\u89e3\u51b3\u6700\u5927\u5b50\u77e9\u9635\u548c\u95ee\u9898\u3002\u5176\u9ad8\u6548\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e0e\u4e25\u8c28\u7684\u6570\u5b66\u57fa\u7840\uff0c\u4f7f\u5176\u6210\u4e3a\u89e3\u51b3\u8fde\u7eed\u5b50\u6bb5\u548c\u95ee\u9898\u7684\u7ecf\u5178\u7b97\u6cd5\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"
                Kadane\u7b97\u6cd5\u53ca\u5176\u5728\u4e00\u7ef4\u4e0e\u4e8c\u7ef4\u6700\u5927\u5b50\u6bb5\u548c\u95ee\u9898\u4e2d\u7684\u5e94\u7528 \u6458\u8981 \u672c\u6587\u7cfb\u7edf\u9610\u8ff0Kadane\u7b97\u6cd5\u7684\u6838\u5fc3\u539f\u7406\u3001\u6570\u5b66\u8bc1\u660e […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[7],"tags":[],"class_list":["post-885","post","type-post","status-publish","format-standard","hentry","category-science-and-engineering"],"_links":{"self":[{"href":"https:\/\/vite66.cn\/index.php?rest_route=\/wp\/v2\/posts\/885","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/vite66.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/vite66.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/vite66.cn\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/vite66.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=885"}],"version-history":[{"count":22,"href":"https:\/\/vite66.cn\/index.php?rest_route=\/wp\/v2\/posts\/885\/revisions"}],"predecessor-version":[{"id":929,"href":"https:\/\/vite66.cn\/index.php?rest_route=\/wp\/v2\/posts\/885\/revisions\/929"}],"wp:attachment":[{"href":"https:\/\/vite66.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=885"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/vite66.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=885"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/vite66.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=885"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}