{"id":49895,"date":"2025-07-30T20:28:15","date_gmt":"2025-07-30T12:28:15","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/49895.html"},"modified":"2025-07-30T20:28:15","modified_gmt":"2025-07-30T12:28:15","slug":"noip2002-%e6%8f%90%e9%ab%98%e7%bb%84-%e5%9d%87%e5%88%86%e7%ba%b8%e7%89%8c","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/49895.html","title":{"rendered":"[NOIP2002 \u63d0\u9ad8\u7ec4] \u5747\u5206\u7eb8\u724c"},"content":{"rendered":"

\u9898\u76ee\u63cf\u8ff0<\/h3>\n

\u6709N\u5806\u7eb8\u724c，\u7f16\u53f7\u5206\u522b\u4e3a 1,2,\u2026,N\u3002\u6bcf\u5806\u4e0a\u6709\u82e5\u5e72\u5f20，\u4f46\u7eb8\u724c\u603b\u6570\u5fc5\u4e3aN\u7684\u500d\u6570\u3002\u53ef\u4ee5\u5728\u4efb\u4e00\u5806\u4e0a\u53d6\u82e5\u5e72\u5f20\u7eb8\u724c，\u7136\u540e\u79fb\u52a8\u3002<\/p>\n

\u79fb\u724c\u89c4\u5219\u4e3a：\u5728\u7f16\u53f7\u4e3a1\u5806\u4e0a\u53d6\u7684\u7eb8\u724c，\u53ea\u80fd\u79fb\u5230\u7f16\u53f7\u4e3a2\u7684\u5806\u4e0a；\u5728\u7f16\u53f7\u4e3aN\u7684\u5806\u4e0a\u53d6\u7684\u7eb8\u724c，\u53ea\u80fd\u79fb\u5230\u7f16\u53f7\u4e3aN\u22121\u7684\u5806\u4e0a；\u5176\u4ed6\u5806\u4e0a\u53d6\u7684\u7eb8\u724c，\u53ef\u4ee5\u79fb\u5230\u76f8\u90bb\u5de6\u8fb9\u6216\u53f3\u8fb9\u7684\u5806\u4e0a\u3002<\/p>\n

\u73b0\u5728\u8981\u6c42\u627e\u51fa\u4e00\u79cd\u79fb\u52a8\u65b9\u6cd5，\u7528\u6700\u5c11\u7684\u79fb\u52a8\u6b21\u6570\u4f7f\u6bcf\u5806\u4e0a\u7eb8\u724c\u6570\u90fd\u4e00\u6837\u591a\u3002<\/p>\n

\u4f8b\u5982N=4，4\u5806\u7eb8\u724c\u6570\u5206\u522b\u4e3a：<\/p>\n

\u24609\u24618\u246217\u24636<\/p>\n

\u79fb\u52a83\u6b21\u53ef\u8fbe\u5230\u76ee\u7684：<\/p>\n

\u4ece \u2462 \u53d64\u5f20\u724c\u653e\u5230 \u2463 （9,8,13,10）-> \u4ece \u2462 \u53d633\u5f20\u724c\u653e\u5230 \u2461（9,11,10,10）-> \u4ece \u2461 \u53d611\u5f20\u724c\u653e\u5230\u2460（10,10,10,10）\u3002<\/p>\n

\u8f93\u5165\u683c\u5f0f<\/h3>\n

\u4e24\u884c<\/p>\n

\u7b2c\u4e00\u884c\u4e3a：NN（NN \u5806\u7eb8\u724c，1\u2264N\u2264100）<\/p>\n

\u7b2c\u4e8c\u884c\u4e3a：A1,A2,\u2026,An（N\u5806\u7eb8\u724c，\u6bcf\u5806\u7eb8\u724c\u521d\u59cb\u6570，1\u2264Ai\u2264100001\u2264Ai\u200b\u226410000）<\/p>\n

\u8f93\u51fa\u683c\u5f0f<\/h3>\n

\u4e00\u884c：\u5373\u6240\u6709\u5806\u5747\u8fbe\u5230\u76f8\u7b49\u65f6\u7684\u6700\u5c11\u79fb\u52a8\u6b21\u6570\u3002<\/p>\n

\u8f93\u5165\u6570\u636e 1<\/h3>\n

4
\n9 8 17 6<\/p>\n

\u8f93\u51fa\u6570\u636e 1<\/h3>\n

3<\/p>\n

\u6839\u636e\u9898\u76ee\u63cf\u8ff0，\u6211\u4eec\u9700\u8981\u5c06\u6240\u6709\u5806\u7684\u7eb8\u724c\u6570\u91cf\u8c03\u6574\u4e3a\u76f8\u7b49，\u5e76\u4e14\u8981\u6c42\u79fb\u52a8\u6b21\u6570\u6700\u5c11\u3002\u901a\u8fc7\u5206\u6790\u53ef\u4ee5\u53d1\u73b0，\u6bcf\u4e2a\u975e\u76ee\u6807\u503c\u7684\u5806\u90fd\u9700\u8981\u81f3\u5c11\u4e00\u6b21\u79fb\u52a8\u6765\u8c03\u6574\u5176\u6570\u91cf，\u56e0\u6b64\u6700\u5c11\u79fb\u52a8\u6b21\u6570\u7b49\u4e8e\u9700\u8981\u8c03\u6574\u7684\u5806\u7684\u6570\u91cf\u51cf\u4e00（\u5f53\u76f8\u90bb\u5806\u90fd\u9700\u8981\u8c03\u6574\u65f6，\u53ef\u4ee5\u5408\u5e76\u79fb\u52a8）\u3002<\/p>\n

\u4ee5\u4e0b\u662f\u89e3\u51b3\u8be5\u95ee\u9898\u7684 C++ \u4ee3\u7801：
\n\u00a0<\/p>\n

#include <iostream>
\n#include <vector>
\nusing namespace std;
\nint main()
\n{
\n int n;
\n cin >> n;
\n vector<int> cards(n);
\n int sum = 0;
\n for (int i = 0; i < n; i++) {
\n cin >> cards[i];
\n sum += cards[i];
\n }
\n int avg = sum \/ n;
\n for (int i = 0; i < n; i++) {
\n cards[i] -= avg;
\n }
\n vector<int> prefix(n);
\n prefix[0] = cards[0];
\n for (int i = 1; i < n; i++) {
\n prefix[i] = prefix[i-1] + cards[i];
\n }
\n int count = 0;
\n for (int i = 0; i < n; i++) {
\n if (prefix[i] != 0) {
\n count++;
\n }
\n }
\n cout << count << endl;
\n return 0;
\n}<\/p>\n

\u4ee3\u7801\u8bf4\u660e：<\/h4>\n
  • \u8f93\u5165\u5904\u7406：\u9996\u5148\u8bfb\u53d6\u5806\u6570n，\u7136\u540e\u8bfb\u53d6\u6bcf\u5806\u7684\u7eb8\u724c\u6570\u91cf\u5e76\u8ba1\u7b97\u603b\u548c\u3002<\/li>\n
  • \u8ba1\u7b97\u5e73\u5747\u503c：\u603b\u548c\u9664\u4ee5\u5806\u6570\u5f97\u5230\u6bcf\u5806\u5e94\u6709\u7684\u7eb8\u724c\u6570\u91cf\u3002<\/li>\n
  • \u8c03\u6574\u6570\u7ec4：\u5c06\u6bcf\u5806\u7684\u7eb8\u724c\u6570\u91cf\u8f6c\u6362\u4e3a\u4e0e\u5e73\u5747\u503c\u7684\u5dee\u503c\u3002<\/li>\n
  • \u524d\u7f00\u548c\u6570\u7ec4：\u8ba1\u7b97\u8c03\u6574\u540e\u6570\u7ec4\u7684\u524d\u7f00\u548c，\u524d\u7f00\u548c\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\u8868\u793a\u4ece\u7b2c\u4e00\u5806\u5230\u5f53\u524d\u5806\u7684\u603b\u5dee\u503c\u3002<\/li>\n
  • \u7edf\u8ba1\u975e\u96f6\u5143\u7d20：\u524d\u7f00\u548c\u6570\u7ec4\u4e2d\u975e\u96f6\u5143\u7d20\u7684\u6570\u91cf\u5373\u4e3a\u6700\u5c11\u79fb\u52a8\u6b21\u6570\u3002<\/li>\n

    \u8be5\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O (n)，\u80fd\u591f\u9ad8\u6548\u5904\u7406\u9898\u76ee\u7ed9\u5b9a\u7684\u6570\u636e\u8303\u56f4\u3002<\/p>\n

    \u8ba9\u6211\u4eec\u901a\u8fc7\u9010\u6b65\u62c6\u89e3\u95ee\u9898\u548c\u4ee3\u7801\u903b\u8f91，\u6df1\u5165\u7406\u89e3\u5982\u4f55\u901a\u8fc7\u524d\u7f00\u548c\u6570\u7ec4\u7684\u975e\u96f6\u5143\u7d20\u6570\u91cf\u786e\u5b9a\u6700\u5c11\u79fb\u52a8\u6b21\u6570\u3002<\/p>\n

    \u4e00\u3001\u95ee\u9898\u672c\u8d28\u5206\u6790<\/h4>\n
  • \u6838\u5fc3\u77db\u76fe：\u6bcf\u5806\u7eb8\u724c\u6570\u4e0e\u5e73\u5747\u503c\u7684\u5dee\u5f02\u9700\u8981\u901a\u8fc7\u79fb\u52a8\u6d88\u9664\u3002<\/li>\n
  • \u79fb\u52a8\u89c4\u5219：\u6bcf\u6b21\u79fb\u52a8\u53ea\u80fd\u5f71\u54cd\u76f8\u90bb\u5806，\u4e14\u79fb\u52a8\u65b9\u5411\u53d7\u9650（\u5982\u5806 1 \u53ea\u80fd\u79fb\u5230\u5806 2）\u3002<\/li>\n
  • \u5173\u952e\u89c2\u5bdf：\n
      \n
    • \u82e5\u67d0\u5806\u7684\u7eb8\u724c\u6570\u4e0e\u5e73\u5747\u503c\u7684\u5dee\u4e3ad，\u5219\u5fc5\u987b\u901a\u8fc7\u79fb\u52a8d\u5f20\u7eb8\u724c\u5230\u76f8\u90bb\u5806\u6765\u5e73\u8861\u3002<\/li>\n
    • \u6bcf\u6b21\u79fb\u52a8\u53ef\u4ee5\u6d88\u9664\u4e00\u4e2a\u7d2f\u79ef\u7684\u4e0d\u5e73\u8861（\u5373\u524d\u7f00\u548c\u4e2d\u7684\u975e\u96f6\u503c）\u3002<\/li>\n<\/ul>\n<\/li>\n

      \u4e8c\u3001\u4ee3\u7801\u903b\u8f91\u8be6\u89e3<\/h4>\n

      1. \u8f93\u5165\u5904\u7406\u4e0e\u5e73\u5747\u503c\u8ba1\u7b97<\/h5>\n

      int n;
      \ncin >> n;
      \nvector<int> cards(n);
      \nint sum = 0;
      \nfor (int i = 0; i < n; i++) {
      \n cin >> cards[i];
      \n sum += cards[i];
      \n}
      \nint avg = sum \/ n;<\/p>\n

        \n
      • \u8f93\u5165\u5904\u7406：\u8bfb\u53d6\u5806\u6570n\u548c\u6bcf\u5806\u7684\u521d\u59cb\u7eb8\u724c\u6570\u3002<\/li>\n
      • \u5e73\u5747\u503c\u8ba1\u7b97：\u603b\u548csum\u9664\u4ee5\u5806\u6570n\u5f97\u5230\u76ee\u6807\u503cavg\u3002\u82e5\u7eb8\u724c\u603b\u6570\u4e0d\u662fn\u7684\u500d\u6570，\u9898\u76ee\u4fdd\u8bc1\u8f93\u5165\u5408\u6cd5，\u56e0\u6b64\u65e0\u9700\u5904\u7406\u3002<\/li>\n<\/ul>\n
        2. \u8c03\u6574\u6570\u7ec4（\u5dee\u503c\u6570\u7ec4）<\/h5>\n

        for (int i = 0; i < n; i++) {
        \n cards[i] -= avg;
        \n}<\/p>\n

          \n
        • \u5dee\u503c\u6570\u7ec4：\u5c06\u6bcf\u5806\u7684\u7eb8\u724c\u6570\u8f6c\u6362\u4e3a\u4e0e\u5e73\u5747\u503c\u7684\u5dee\u503c\u3002\u4f8b\u5982，\u82e5\u67d0\u5806\u521d\u59cb\u6709 9 \u5f20，\u5e73\u5747\u503c\u4e3a 10，\u5219\u5dee\u503c\u4e3a-1，\u8868\u793a\u9700\u8981\u79fb\u5165 1 \u5f20\u7eb8\u724c\u3002<\/li>\n<\/ul>\n
          3. \u524d\u7f00\u548c\u6570\u7ec4\u6784\u9020<\/h5>\n

          vector<int> prefix(n);
          \nprefix[0] = cards[0];
          \nfor (int i = 1; i < n; i++) {
          \n prefix[i] = prefix[i-1] + cards[i];
          \n}<\/p>\n

            \n
          • \u524d\u7f00\u548c\u7684\u610f\u4e49：prefix[i]\u8868\u793a\u4ece\u7b2c 1 \u5806\u5230\u7b2ci\u5806\u7684\u603b\u5dee\u503c\u3002\u4f8b\u5982，\u82e5\u524d 3 \u5806\u7684\u5dee\u503c\u5206\u522b\u4e3a-1, -2, 7，\u5219prefix[2] = -1 + (-2) + 7 = 4，\u8868\u793a\u524d 3 \u5806\u603b\u5171\u591a\u51fa 4 \u5f20\u7eb8\u724c，\u9700\u8981\u540e\u7eed\u5806\u6765\u5e73\u8861\u3002<\/li>\n<\/ul>\n
            4. \u7edf\u8ba1\u975e\u96f6\u5143\u7d20\u6570\u91cf<\/h5>\n

            int count = 0;
            \nfor (int i = 0; i < n; i++) {
            \n if (prefix[i] != 0) {
            \n count++;
            \n }
            \n}
            \ncout << count << endl;<\/p>\n

              \n
            • \u79fb\u52a8\u6b21\u6570\u7684\u6765\u6e90：\u524d\u7f00\u548c\u6570\u7ec4\u4e2d\u7684\u6bcf\u4e2a\u975e\u96f6\u5143\u7d20\u4ee3\u8868\u4e00\u4e2a\u5fc5\u987b\u8fdb\u884c\u7684\u79fb\u52a8\u6b65\u9aa4\u3002\u4f8b\u5982：\n
                \n
              • \u82e5prefix[i]\u4e3a\u6b63，\u8868\u793a\u524di\u5806\u9700\u8981\u79fb\u51faprefix[i]\u5f20\u7eb8\u724c；<\/li>\n
              • \u82e5\u4e3a\u8d1f，\u8868\u793a\u524di\u5806\u9700\u8981\u79fb\u5165abs(prefix[i])\u5f20\u7eb8\u724c\u3002<\/li>\n
              • \u6bcf\u6b21\u79fb\u52a8\u53ef\u4ee5\u6d88\u9664\u4e00\u4e2a\u975e\u96f6\u7684\u524d\u7f00\u548c\u503c，\u56e0\u6b64\u975e\u96f6\u5143\u7d20\u7684\u6570\u91cf\u5373\u4e3a\u6700\u5c11\u79fb\u52a8\u6b21\u6570\u3002<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n

                \u4e09\u3001\u6570\u5b66\u8bc1\u660e\u4e0e\u793a\u4f8b\u9a8c\u8bc1<\/h4>\n

                1. \u6570\u5b66\u8bc1\u660e<\/h5>\n
                  \n
                • \u524d\u7f00\u548c\u7684\u6027\u8d28：
                  \n\u7531\u4e8e\u6240\u6709\u5806\u7684\u603b\u5dee\u503c\u4e3a 0（\u603b\u548c\u4e3asum – n*avg = 0），\u524d\u7f00\u548c\u6570\u7ec4\u7684\u6700\u540e\u4e00\u4e2a\u5143\u7d20prefix[n-1]\u5fc5\u4e3a 0\u3002<\/li>\n
                • \u79fb\u52a8\u6b21\u6570\u7684\u63a8\u5bfc：
                  \n\u5047\u8bbe\u524d\u7f00\u548c\u6570\u7ec4\u4e2d\u6709k\u4e2a\u975e\u96f6\u5143\u7d20，\u6bcf\u6b21\u79fb\u52a8\u53ef\u4ee5\u6d88\u9664\u4e00\u4e2a\u975e\u96f6\u503c（\u4f8b\u5982，\u5c06prefix[i]\u7684\u4e0d\u5e73\u8861\u8f6c\u79fb\u5230\u4e0b\u4e00\u5806），\u6700\u7ec8\u6240\u6709\u524d\u7f00\u548c\u90fd\u53d8\u4e3a 0，\u56e0\u6b64\u9700\u8981k\u6b21\u79fb\u52a8\u3002<\/li>\n<\/ul>\n
                  2. \u793a\u4f8b\u5206\u6790（\u8f93\u5165：4 9 8 17 6）<\/h5>\n
                    \n
                  • \u6b65\u9aa4 1：\u8ba1\u7b97\u5e73\u5747\u503c
                    \nsum = 9+8+17+6 = 40，avg = 40\/4 = 10\u3002<\/li>\n
                  • \u6b65\u9aa4 2：\u6784\u9020\u5dee\u503c\u6570\u7ec4
                    \ncards = [9-10, 8-10, 17-10, 6-10] = [-1, -2, 7, -4]\u3002<\/li>\n
                  • \u6b65\u9aa4 3：\u8ba1\u7b97\u524d\u7f00\u548c\u6570\u7ec4
                    \nprefix[0] = -1
                    \nprefix[1] = -1 + (-2) = -3
                    \nprefix[2] = -3 + 7 = 4
                    \nprefix[3] = 4 + (-4) = 0
                    \n\u975e\u96f6\u5143\u7d20\u4e3a-1, -3, 4，\u5171 3 \u4e2a，\u5bf9\u5e94\u79fb\u52a8\u6b21\u6570 3\u3002<\/li>\n<\/ul>\n
                    3. \u79fb\u52a8\u8fc7\u7a0b\u4e0e\u524d\u7f00\u548c\u7684\u5173\u7cfb<\/h5>\n
                      \n
                    • \u7b2c\u4e00\u6b21\u79fb\u52a8：\u4ece\u5806 3 \u79fb 4 \u5f20\u5230\u5806 4\u3002
                      \n\u5dee\u503c\u6570\u7ec4\u53d8\u4e3a[-1, -2, 3, 0]，\u524d\u7f00\u548c\u53d8\u4e3a[-1, -3, 0, 0]\u3002\u6b64\u65f6\u975e\u96f6\u5143\u7d20\u51cf\u5c11 1 \u4e2a\u3002<\/li>\n
                    • \u7b2c\u4e8c\u6b21\u79fb\u52a8：\u4ece\u5806 3 \u79fb 3 \u5f20\u5230\u5806 2\u3002
                      \n\u5dee\u503c\u6570\u7ec4\u53d8\u4e3a[-1, 1, 0, 0]，\u524d\u7f00\u548c\u53d8\u4e3a[-1, 0, 0, 0]\u3002\u975e\u96f6\u5143\u7d20\u518d\u51cf\u5c11 1 \u4e2a\u3002<\/li>\n
                    • \u7b2c\u4e09\u6b21\u79fb\u52a8：\u4ece\u5806 2 \u79fb 1 \u5f20\u5230\u5806 1\u3002
                      \n\u5dee\u503c\u6570\u7ec4\u5168\u4e3a 0，\u524d\u7f00\u548c\u5168\u4e3a 0\u3002\u79fb\u52a8\u6b21\u6570\u4e3a 3\u3002<\/li>\n<\/ul>\n

                      \u56db\u3001\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790<\/h4>\n
                        \n
                      • \u8f93\u5165\u5904\u7406：O(n)<\/li>\n
                      • \u5e73\u5747\u503c\u8ba1\u7b97：O(1)<\/li>\n
                      • \u5dee\u503c\u6570\u7ec4\u6784\u9020：O(n)<\/li>\n
                      • \u524d\u7f00\u548c\u6570\u7ec4\u6784\u9020：O(n)<\/li>\n
                      • \u975e\u96f6\u5143\u7d20\u7edf\u8ba1：O(n)<\/li>\n
                      • \u603b\u65f6\u95f4\u590d\u6742\u5ea6：O (n)，\u53ef\u9ad8\u6548\u5904\u7406\u9898\u76ee\u7ed9\u5b9a\u7684n\u2264100\u7684\u6570\u636e\u89c4\u6a21\u3002<\/li>\n<\/ul>\n

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