{"id":112221,"date":"2026-10-02T21:48:53","date_gmt":"2026-10-02T13:48:53","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/112221.html"},"modified":"2026-10-02T21:48:53","modified_gmt":"2026-10-02T13:48:53","slug":"%ef%bc%88%e4%b8%89%ef%bc%89%e6%95%b0%e6%8d%ae%e7%bb%93%e6%9e%84%e4%b8%8e%e7%ae%97%e6%b3%95-%e7%bb%8f%e5%85%b8%e7%ae%97%e6%b3%95","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/112221.html","title":{"rendered":"\uff08\u4e09\uff09\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u2014\u2014\u7ecf\u5178\u7b97\u6cd5"},"content":{"rendered":"

🗂\ufe0f \u4e00\u3001 \u54c8\u5e0c\u8868：O(1) \u795e\u8bdd\u7684\u7f14\u9020\u8005\u4e0e\u51b2\u7a81\u89e3\u51b3<\/h3>\n

1. \u54c8\u5e0c\u8868\u7684\u539f\u7406\u662f\u4ec0\u4e48？<\/h4>\n

\u54c8\u5e0c\u8868（Hash Table）\u662f\u4e00\u79cd\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u628a\u952e（key）\u6620\u5c04\u5230\u6570\u7ec4\u4e0b\u6807，\u4ece\u800c\u5728\u5e73\u5747 O(1) \u65f6\u95f4\u5185\u5b8c\u6210\u67e5\u627e\u3001\u63d2\u5165\u3001\u5220\u9664\u7684\u6570\u636e\u7ed3\u6784\u3002
\n\u6838\u5fc3\u539f\u7406\u56db\u6b65\u8d70：<\/p>\n

  • \n

    \u51c6\u5907\u4e00\u4e2a\u56fa\u5b9a\u5927\u5c0f\u7684\u6570\u7ec4\u4f5c\u4e3a\u201c\u6876\u6570\u7ec4\u201d\u3002<\/p>\n<\/li>\n

  • \n

    \u7528\u54c8\u5e0c\u51fd\u6570 hash(key) \u628a\u952e\u6620\u5c04\u6210\u4e00\u4e2a\u6574\u6570\u3002<\/p>\n<\/li>\n

  • \n

    \u7528\u8fd9\u4e2a\u6574\u6570\u5bf9\u6570\u7ec4\u957f\u5ea6\u53d6\u6a21（\u6216\u4f4d\u8fd0\u7b97）\u5f97\u5230\u6876\u4e0b\u6807\u3002<\/p>\n<\/li>\n

  • \n

    \u628a\u952e\u503c\u5bf9\u653e\u5230\u8fd9\u4e2a\u6876\u91cc\u3002<\/p>\n<\/li>\n

    2. \u54c8\u5e0c\u51b2\u7a81\u600e\u4e48\u89e3\u51b3？<\/h4>\n

    \u4e0d\u540c\u7684 key \u7ecf\u8fc7\u54c8\u5e0c\u51fd\u6570\u540e\u53ef\u80fd\u6620\u5c04\u5230\u540c\u4e00\u4e2a\u4e0b\u6807，\u8fd9\u5c31\u662f\u54c8\u5e0c\u51b2\u7a81，\u65e0\u6cd5\u5b8c\u5168\u907f\u514d\u3002\u4e3b\u8981\u6709\u4e24\u5927\u89e3\u51b3\u65b9\u6848：<\/p>\n

      \n
    • \n

      \u94fe\u5730\u5740\u6cd5（\u62c9\u94fe\u6cd5）：\u6bcf\u4e2a\u6876\u91cc\u6302\u4e00\u4e2a\u94fe\u8868（\u6216\u7ea2\u9ed1\u6811），\u51b2\u7a81\u7684\u5143\u7d20\u90fd\u6302\u5728\u5bf9\u5e94\u6876\u7684\u94fe\u8868\u4e0a\u3002Java HashMap \u548c Redis \u7684 dict \u90fd\u7528\u8fd9\u79cd\u65b9\u5f0f\u3002\u6ce8\u610f\u70b9：JDK 1.8 \u4e4b\u540e，\u5f53\u67d0\u4e2a\u6876\u7684\u94fe\u8868\u957f\u5ea6 \u2265 8 \u4e14\u6570\u7ec4\u957f\u5ea6 \u2265 64 \u65f6，\u94fe\u8868\u4f1a\u8f6c\u6210\u7ea2\u9ed1\u6811，\u8fdb\u4e00\u6b65\u4f18\u5316\u6700\u574f\u60c5\u51b5\u4e0b\u7684\u67e5\u627e\u3002<\/p>\n<\/li>\n

    • \n

      \u5f00\u653e\u5bfb\u5740\u6cd5：\u51b2\u7a81\u65f6\u6309\u4e00\u5b9a\u89c4\u5219\u5728\u6570\u7ec4\u91cc\u5bfb\u627e\u4e0b\u4e00\u4e2a\u7a7a\u4f4d\u3002\u5e38\u89c1\u63a2\u6d4b\u65b9\u5f0f\u6709\u7ebf\u6027\u63a2\u6d4b（i, i+1, i+2…）\u3001\u4e8c\u6b21\u63a2\u6d4b（i, i+1\u00b2, i+2\u00b2…）\u3001\u53cc\u91cd\u54c8\u5e0c\u3002Java \u7684 ThreadLocalMap \u5c31\u662f\u7528\u7ebf\u6027\u63a2\u6d4b\u3002<\/p>\n<\/li>\n<\/ul>\n

      3. \u8d1f\u8f7d\u56e0\u5b50（Load Factor）<\/h4>\n
        \n
      • \n

        \u516c\u5f0f：\u5143\u7d20\u6570\u91cf \/ \u6876\u6570\u91cf\u3002<\/p>\n<\/li>\n

      • \n

        \u8d1f\u8f7d\u56e0\u5b50\u8d8a\u5927\u51b2\u7a81\u8d8a\u591a\u3002Java HashMap \u9ed8\u8ba4 0.75，\u8d85\u8fc7\u5c31\u89e6\u53d1\u6269\u5bb9（rehash），\u4ee5\u7ef4\u6301\u5e73\u5747 O(1) \u7684\u67e5\u627e\u6027\u80fd\u3002<\/p>\n<\/li>\n<\/ul>\n

        🧩 \u4e8c\u3001 \u5f52\u5e76\u6392\u5e8f：\u5206\u6cbb\u601d\u60f3\u7684\u5b8c\u7f8e\u4f53\u73b0<\/h3>\n

        1. \u539f\u7406\u4e0e\u5b9e\u73b0<\/h4>\n

        \u5f52\u5e76\u6392\u5e8f\u662f\u4e00\u79cd\u5178\u578b\u7684\u5206\u6cbb\u7b97\u6cd5，\u601d\u60f3\u975e\u5e38\u6e05\u6670，\u5c31\u662f\u201c\u5206\u3001\u6cbb\u3001\u5408\u201d\u4e09\u6b65：<\/p>\n

      • \n

        \u5206：\u628a\u6570\u7ec4\u4ece\u4e2d\u95f4\u4e00\u5206\u4e3a\u4e8c，\u5206\u5230\u5e95\u3002<\/p>\n<\/li>\n

      • \n

        \u6cbb：\u9012\u5f52\u5730\u5bf9\u4e24\u534a\u5206\u522b\u6392\u5e8f\u3002<\/p>\n<\/li>\n

      • \n

        \u5408：\u628a\u4e24\u4e2a\u5df2\u6392\u5e8f\u7684\u5b50\u6570\u7ec4\u5408\u5e76\u6210\u4e00\u4e2a\u6709\u5e8f\u6570\u7ec4\u3002<\/p>\n<\/li>\n

        2. \u590d\u6742\u5ea6\u4e0e\u7279\u6027<\/h4>\n
          \n
        • \n

          \u65f6\u95f4\u590d\u6742\u5ea6：\u6700\u597d\u3001\u6700\u574f\u3001\u5e73\u5747\u90fd\u662f O(n log n)（\u5171 log n \u5c42\u9012\u5f52，\u6bcf\u5c42\u5408\u5e76\u603b\u5171\u5904\u7406 n \u4e2a\u5143\u7d20）\u3002<\/p>\n<\/li>\n

        • \n

          \u7a7a\u95f4\u590d\u6742\u5ea6：O(n)（\u9700\u8981\u989d\u5916\u7684\u4e34\u65f6\u6570\u7ec4\u5b58\u653e\u5408\u5e76\u7ed3\u679c）\u3002<\/p>\n<\/li>\n

        • \n

          \u7a33\u5b9a\u6027：\u7a33\u5b9a\u6392\u5e8f，\u5408\u5e76\u65f6\u76f8\u540c\u5143\u7d20\u6309\u539f\u59cb\u987a\u5e8f\u653e\u5165\u3002<\/p>\n<\/li>\n<\/ul>\n

          3. \u5bf9\u6bd4\u5feb\u6392\u4e0e\u5e94\u7528\u573a\u666f<\/h4>\n
            \n
          • \n

            \u76f8\u6bd4\u5feb\u6392\u7684\u4f18\u52bf：\u7a33\u5b9a\u3001\u6700\u574f\u60c5\u51b5\u4e5f\u662f O(n log n)（\u5feb\u6392\u6700\u574f\u662f O(n\u00b2)）\u3002<\/p>\n<\/li>\n

          • \n

            \u76f8\u6bd4\u5feb\u6392\u7684\u52a3\u52bf：\u9700\u8981 O(n) \u989d\u5916\u7a7a\u95f4，\u4e0d\u662f\u539f\u5730\u6392\u5e8f\u3002<\/p>\n<\/li>\n

          • \n

            \u5178\u578b\u5e94\u7528\u573a\u666f：<\/p>\n

              \n
            • \n

              \u5916\u90e8\u6392\u5e8f（\u5185\u5b58\u653e\u4e0d\u4e0b\u7684\u6d77\u91cf\u6570\u636e）\u3002<\/p>\n<\/li>\n

            • \n

              \u94fe\u8868\u6392\u5e8f（\u5bf9\u94fe\u8868\u975e\u5e38\u53cb\u597d，\u7a7a\u95f4\u53ef\u4ee5\u505a\u5230 O(1)）\u3002<\/p>\n<\/li>\n

            • \n

              \u9700\u8981\u7a33\u5b9a\u6027\u7684\u4e1a\u52a1\u573a\u666f\u3002<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n

              #include <iostream>
              \n#include <vector><\/p>\n

              \/\/ \u5408\u5e76\u4e24\u4e2a\u6709\u5e8f\u5b50\u6570\u7ec4
              \nvoid merge(vector<int>& arr, int left, int mid, int right) {
              \n vector<int> temp(right – left + 1);
              \n int i = left, j = mid + 1, k = 0;<\/p>\n

              while (i <= mid && j <= right) {
              \n \/\/ \u6ce8\u610f：\u8fd9\u91cc\u7528 <= \u4fdd\u8bc1\u7a33\u5b9a\u6027
              \n if (arr[i] <= arr[j]) {
              \n temp[k++] = arr[i++];
              \n } else {
              \n temp[k++] = arr[j++];
              \n }
              \n }
              \n while (i <= mid) temp[k++] = arr[i++];
              \n while (j <= right) temp[k++] = arr[j++];<\/p>\n

              \/\/ \u5c06\u4e34\u65f6\u6570\u7ec4\u62f7\u8d1d\u56de\u539f\u6570\u7ec4
              \n for (int p = 0; p < k; ++p) {
              \n arr[left + p] = temp[p];
              \n }
              \n}<\/p>\n

              void mergeSort(vector<int>& arr, int left, int right) {
              \n if (left >= right) return;
              \n int mid = left + (right – left) \/ 2;
              \n mergeSort(arr, left, mid); \/\/ \u9012\u5f52\u6392\u5e8f\u5de6\u534a
              \n mergeSort(arr, mid + 1, right); \/\/ \u9012\u5f52\u6392\u5e8f\u53f3\u534a
              \n merge(arr, left, mid, right); \/\/ \u5408\u5e76
              \n}<\/p>\n

              \u00a0\u4e09\u3001 \u4e8c\u5206\u67e5\u627e：\u601d\u8def\u5f88\u7b80\u5355，\u7ec6\u8282\u662f\u9b54\u9b3c<\/h3>\n

              1. \u6838\u5fc3\u601d\u60f3\u4e0e\u6807\u51c6\u5b9e\u73b0<\/h4>\n

              \u4e8c\u5206\u67e5\u627e\u662f\u5bf9\u6709\u5e8f\u6570\u7ec4\u8fdb\u884c\u67e5\u627e\u7684\u7ecf\u5178\u7b97\u6cd5，\u65f6\u95f4\u590d\u6742\u5ea6 O(log n)\u3002\u6838\u5fc3\u601d\u60f3\u662f\u6bcf\u6b21\u6bd4\u8f83\u4e2d\u95f4\u5143\u7d20，\u6839\u636e\u5927\u5c0f\u5173\u7cfb\u628a\u67e5\u627e\u8303\u56f4\u7f29\u5c0f\u4e00\u534a\u3002<\/p>\n

              2. \u5bb9\u6613\u8e29\u7684\u5751<\/h4>\n
            • \n

              \u6574\u6570\u6ea2\u51fa：\u4e0d\u8981\u5199 (left + right) \/ 2，\u5f53 left \u548c right \u90fd\u63a5\u8fd1 Integer.MAX_VALUE \u65f6\u4f1a\u6ea2\u51fa，\u5fc5\u987b\u5199\u6210 left + (right – left) \/ 2\u3002<\/p>\n<\/li>\n

            • \n

              \u8fb9\u754c\u533a\u95f4\u7684\u5199\u6cd5\u8981\u7edf\u4e00：<\/p>\n

                \n
              • \n

                \u95ed\u533a\u95f4 <\/span>[left, right]<\/span>：\u5faa\u73af\u6761\u4ef6 <\/span>left <= right<\/span>，\u66f4\u65b0\u65f6 <\/span>left = mid + 1<\/span> \u6216 <\/span>right = mid – 1<\/span>\u3002<\/span><\/p>\n<\/li>\n

              • \n

                \u5de6\u95ed\u53f3\u5f00 <\/span>[left, right)<\/span>：\u5faa\u73af\u6761\u4ef6 <\/span>left < right<\/span>，\u66f4\u65b0\u65f6 <\/span>left = mid + 1<\/span>\u6216<\/span>right = mid<\/span>（<\/span>\u4e0d\u51cf 1<\/span>）\u3002<\/span><\/p>\n<\/li>\n

              • \n

                \u26a0\ufe0f\u4e24\u79cd\u5199\u6cd5\u4e0d\u80fd\u6df7\u7528，\u5426\u5219\u8981\u4e48\u6b7b\u5faa\u73af\u8981\u4e48\u6f0f\u89e3\u3002<\/p>\n<\/li>\n<\/ul>\n<\/li>\n

              • \n

                \u6b7b\u5faa\u73af：\u5982\u679c\u5faa\u73af\u91cc\u67d0\u4e2a\u5206\u652f\u5fd8\u8bb0\u8ba9 left \u6216 right \u53d8\u5316，\u4f1a\u5bfc\u81f4\u6b7b\u5faa\u73af\u3002<\/p>\n<\/li>\n

              • \n

                \u53d8\u4f53\u9898：\u67e5\u627e\u201c\u7b2c\u4e00\u4e2a\u7b49\u4e8e target\u201d\u3001\u201c\u6700\u540e\u4e00\u4e2a\u7b49\u4e8e target\u201d\u3001\u201c\u7b2c\u4e00\u4e2a\u5927\u4e8e\u7b49\u4e8e target\u201d\u7b49\u53d8\u4f53，\u9700\u8981\u5c0f\u5fc3 < \u548c <= \u7684\u53d6\u820d\u3002<\/p>\n<\/li>\n

                #include <vector><\/p>\n

                int binarySearch(const vector<int>& arr, int target) {
                \n int left = 0, right = arr.size() – 1;<\/p>\n

                \/\/ \u95ed\u533a\u95f4 [left, right]
                \n while (left <= right) {
                \n \/\/ \u9632\u6ea2\u51fa\u5199\u6cd5，\u9762\u8bd5\u5fc5\u8003！
                \n int mid = left + (right – left) \/ 2; <\/p>\n

                if (arr[mid] == target) {
                \n return mid;
                \n } else if (arr[mid] < target) {
                \n left = mid + 1; \/\/ \u66f4\u65b0\u5de6\u8fb9\u754c
                \n } else {
                \n right = mid – 1; \/\/ \u66f4\u65b0\u53f3\u8fb9\u754c
                \n }
                \n }
                \n return -1; \/\/ \u672a\u627e\u5230
                \n}<\/p>\n

                \u56db\u3001 DFS \u4e0e BFS：\u56fe\/\u6811\u904d\u5386\u7684\u53cc\u5b50\u661f<\/h3>\n

                1. \u533a\u522b\u4e0e\u5b9e\u73b0<\/h4>\n
                  \n
                • \n

                  DFS（\u6df1\u5ea6\u4f18\u5148）：\u4e00\u6761\u8def\u8d70\u5230\u9ed1，\u8d70\u4e0d\u901a\u518d\u56de\u6eaf\u3002\u901a\u5e38\u7528\u9012\u5f52\u6216\u6808\u5b9e\u73b0\u3002<\/p>\n<\/li>\n

                • \n

                  BFS（\u5e7f\u5ea6\u4f18\u5148）：\u4e00\u5c42\u4e00\u5c42\u5411\u5916\u6269\u5c55\u3002\u901a\u5e38\u7528\u961f\u5217\u5b9e\u73b0\u3002<\/p>\n<\/li>\n<\/ul>\n

                  2. \u5bf9\u6bd4\u8868\u683c<\/h4>\n\n\u7ef4\u5ea6DFSBFS<\/tr>\n\n\n\n\n\n
                  \u5b9e\u73b0<\/td>\n\u9012\u5f52 \/ \u6808<\/td>\n\u961f\u5217<\/td>\n<\/tr>\n
                  \u7a7a\u95f4\u590d\u6742\u5ea6<\/td>\nO(h)，h \u662f\u9012\u5f52\u6808\u6df1\u5ea6（\u6811\u7684\u9ad8\u5ea6）<\/td>\nO(w)，w \u662f\u6811\u6700\u5bbd\u4e00\u5c42\u7684\u8282\u70b9\u6570<\/td>\n<\/tr>\n
                  \u80fd\u5426\u6c42\u6700\u77ed\u8def\u5f84<\/td>\n\u4e0d\u80fd\u76f4\u63a5\u6c42<\/td>\n\u53ef\u4ee5（\u8fb9\u6743\u76f8\u7b49\u7684\u56fe，BFS \u7b2c\u4e00\u6b21\u5230\u8fbe\u5373\u6700\u77ed\u8def\u5f84）<\/td>\n<\/tr>\n
                  \u5178\u578b\u5e94\u7528<\/td>\n\u5168\u6392\u5217\u3001\u5b50\u96c6\u3001\u62d3\u6251\u6392\u5e8f\u3001\u68c0\u6d4b\u73af\u3001\u8fde\u901a\u5206\u91cf\u3001\u56de\u6eaf<\/td>\n\u5c42\u5e8f\u904d\u5386\u3001\u6c42\u6700\u77ed\u8def\u5f84\u3001\u201c\u6700\u5c11\u51e0\u6b65\u201d\u7c7b\u95ee\u9898<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                  3. \u7ecf\u5178\u9898\u4e3e\u4f8b<\/h4>\n
                    \n
                  • \n

                    \u4e8c\u53c9\u6811\u524d\/\u4e2d\/\u540e\u5e8f\u904d\u5386 \u2192 DFS<\/p>\n<\/li>\n

                  • \n

                    \u4e8c\u53c9\u6811\u5c42\u5e8f\u904d\u5386 \u2192 BFS<\/p>\n<\/li>\n

                  • \n

                    LeetCode 200 \u5c9b\u5c7f\u6570\u91cf \u2192 DFS\/BFS \u90fd\u884c<\/p>\n<\/li>\n

                  • \n

                    LeetCode 994 \u8150\u70c2\u7684\u6a58\u5b50（\u6c42\u6700\u77ed\u611f\u67d3\u65f6\u95f4） \u2192 BFS<\/p>\n<\/li>\n

                  • \n

                    \u56de\u6eaf\u7c7b\u9898\u76ee（\u5168\u6392\u5217\u3001N \u7687\u540e\u3001\u5b50\u96c6\u3001\u7ec4\u5408） \u2192 DFS<\/p>\n<\/li>\n<\/ul>\n

                    💡 \u4e00\u53e5\u8bdd\u533a\u522b：\u8981\u627e\u6700\u77ed\u8def\u5f84\u3001\u6700\u5c11\u6b65\u6570\u7528 BFS；\u8981\u7a77\u4e3e\u6240\u6709\u65b9\u6848\u6216\u6df1\u6316\u67d0\u6761\u8def\u5f84\u7528 DFS\u3002<\/p>\n

                    \u4e94\u3001 \u94fe\u8868\u624b\u6495：\u53cd\u8f6c\u94fe\u8868\u4e0e\u73af\u5f62\u94fe\u8868<\/h3>\n

                    1. \u5982\u4f55\u53cd\u8f6c\u4e00\u4e2a\u94fe\u8868？<\/h4>\n

                    \u8fd9\u662f\u94fe\u8868\u624b\u6495\u9898\u91cc\u6700\u7ecf\u5178\u7684\u4e00\u9053，\u8981\u4f1a\u4e24\u79cd\u89e3\u6cd5\u3002<\/p>\n

                      \n
                    • \n

                      \u89e3\u6cd5\u4e00：\u8fed\u4ee3（\u63a8\u8350，O(1) \u7a7a\u95f4）
                      \n\u7528\u4e09\u4e2a\u6307\u9488 prev\u3001curr\u3001next \u4f9d\u6b21\u8c03\u8f6c\u6bcf\u4e2a\u8282\u70b9\u7684\u6307\u5411\u3002
                      \n\u65f6\u95f4\u590d\u6742\u5ea6 O(n)，\u7a7a\u95f4\u590d\u6742\u5ea6 O(1)\u3002<\/p>\n<\/li>\n<\/ul>\n

                      struct ListNode {
                      \n int val;
                      \n ListNode *next;
                      \n ListNode(int x) : val(x), next(nullptr) {}
                      \n};<\/p>\n

                      ListNode* reverseList(ListNode* head) {
                      \n ListNode* prev = nullptr;
                      \n ListNode* curr = head;
                      \n while (curr != nullptr) {
                      \n ListNode* nextTemp = curr->next; \/\/ \u6682\u5b58\u4e0b\u4e00\u4e2a\u8282\u70b9
                      \n curr->next = prev; \/\/ \u53cd\u8f6c\u5f53\u524d\u8282\u70b9\u6307\u9488
                      \n prev = curr; \/\/ prev \u524d\u8fdb
                      \n curr = nextTemp; \/\/ curr \u524d\u8fdb
                      \n }
                      \n return prev; \/\/ \u6700\u540e prev \u5c31\u662f\u65b0\u7684\u5934\u8282\u70b9
                      \n}<\/p>\n

                        \n
                      • \n

                        \u89e3\u6cd5\u4e8c：\u9012\u5f52
                        \n\u9012\u5f52\u5230\u6700\u540e\u4e00\u4e2a\u8282\u70b9\u4f5c\u4e3a\u65b0\u5934\u8282\u70b9，\u7136\u540e\u5728\u56de\u6eaf\u7684\u8fc7\u7a0b\u4e2d\u6539\u53d8\u6307\u9488\u6307\u5411\u3002
                        \n\u65f6\u95f4\u590d\u6742\u5ea6 O(n)，\u4f46\u9012\u5f52\u6808\u5360\u7528 O(n) \u7a7a\u95f4\u3002
                        \n\u26a0\ufe0f\u4e00\u5b9a\u8981\u8003\u8651\u7a7a\u94fe\u8868（head == null）\u548c\u53ea\u6709\u4e00\u4e2a\u8282\u70b9\u7684\u8fb9\u754c\u60c5\u51b5，\u4e0a\u9762\u4e24\u79cd\u5199\u6cd5\u90fd\u5df2\u7ecf\u5904\u7406\u4e86\u3002<\/p>\n<\/li>\n<\/ul>\n

                        ListNode* reverseListRecursive(ListNode* head) {
                        \n \/\/ \u8fb9\u754c\u60c5\u51b5：\u7a7a\u94fe\u8868\u6216\u53ea\u6709\u4e00\u4e2a\u8282\u70b9
                        \n if (head == nullptr || head->next == nullptr) return head;<\/p>\n

                        ListNode* newHead = reverseListRecursive(head->next);
                        \n head->next->next = head; \/\/ \u8ba9\u4e0b\u4e00\u4e2a\u8282\u70b9\u7684 next \u6307\u5411\u5f53\u524d\u8282\u70b9
                        \n head->next = nullptr; \/\/ \u5f53\u524d\u8282\u70b9\u7684 next \u8bbe\u4e3a null，\u907f\u514d\u6210\u73af
                        \n return newHead;
                        \n}<\/p>\n

                        2. \u5982\u4f55\u5224\u65ad\u4e00\u4e2a\u94fe\u8868\u662f\u5426\u6709\u73af？\u5982\u4f55\u627e\u5230\u73af\u7684\u5165\u53e3\u8282\u70b9？<\/h4>\n

                        \u8fd9\u662f\u5feb\u6162\u6307\u9488（Floyd \u5224\u5708\u7b97\u6cd5）\u7684\u7ecf\u5178\u5e94\u7528\u9898，\u4e24\u95ee\u90fd\u6709\u6807\u51c6\u7b54\u6848\u3002<\/p>\n

                          \n
                        • \n

                          \u7b2c\u4e00\u95ee：\u5224\u65ad\u6709\u6ca1\u6709\u73af
                          \n\u7528\u4e24\u4e2a\u6307\u9488 slow \u548c fast，slow \u6bcf\u6b21\u8d70 1 \u6b65，fast \u6bcf\u6b21\u8d70 2 \u6b65\u3002
                          \n\u5982\u679c fast \u8d70\u5230\u4e86 null，\u8bf4\u660e\u6ca1\u6709\u73af；\u5982\u679c fast \u8ffd\u4e0a\u4e86 slow（\u76f8\u9047），\u8bf4\u660e\u6709\u73af\u3002<\/p>\n<\/li>\n

                        • \n

                          \u7b2c\u4e8c\u95ee：\u627e\u73af\u7684\u5165\u53e3\u8282\u70b9
                          \n\u5173\u952e\u7ed3\u8bba（Floyd \u7b97\u6cd5\u7684\u6570\u5b66\u63a8\u5bfc）：\u5728 slow \u548c fast \u76f8\u9047\u540e，\u8ba9\u5176\u4e2d\u4e00\u4e2a\u6307\u9488\u56de\u5230\u94fe\u8868\u5934，\u4e24\u4e2a\u6307\u9488\u90fd\u6bcf\u6b21\u8d70\u4e00\u6b65，\u518d\u6b21\u76f8\u9047\u7684\u5730\u65b9\u5c31\u662f\u73af\u7684\u5165\u53e3\u3002
                          \n\u6570\u5b66\u539f\u7406：\u8bbe\u5934\u5230\u5165\u53e3\u8ddd\u79bb\u4e3a a，\u5165\u53e3\u5230\u76f8\u9047\u70b9\u4e3a b，\u76f8\u9047\u70b9\u5230\u5165\u53e3（\u6cbf\u73af\u8d70）\u4e3a c\u3002\u76f8\u9047\u65f6 slow = a + b，fast = a + b + k(b + c)（\u591a\u8dd1\u4e86 k \u5708）\u3002\u7531 fast = 2 * slow \u63a8\u51fa a = (k – 1)(b + c) + c，\u542b\u4e49\u5c31\u662f：\u4ece\u5934\u8d70 a \u6b65，\u7b49\u4ef7\u4e8e\u4ece\u76f8\u9047\u70b9\u8d70 c \u6b65\u518d\u8d70\u82e5\u5e72\u6574\u5708，\u4e24\u8005\u4e00\u5b9a\u4f1a\u540c\u65f6\u5230\u8fbe\u5165\u53e3\u3002<\/p>\n<\/li>\n<\/ul>\n

                          ListNode* detectCycle(ListNode* head) {
                          \n ListNode* slow = head;
                          \n ListNode* fast = head;<\/p>\n

                          while (fast != nullptr && fast->next != nullptr) {
                          \n slow = slow->next;
                          \n fast = fast->next->next;<\/p>\n

                          if (slow == fast) { \/\/ \u76f8\u9047，\u5f00\u59cb\u627e\u5165\u53e3
                          \n ListNode* p = head;
                          \n while (p != slow) {
                          \n p = p->next;
                          \n slow = slow->next;
                          \n }
                          \n return p; \/\/ \u518d\u6b21\u76f8\u9047\u7684\u5730\u65b9\u5c31\u662f\u73af\u7684\u5165\u53e3
                          \n }
                          \n }
                          \n return nullptr; \/\/ \u65e0\u73af
                          \n}<\/p>\n

                          \u00a0\u516d\u3001 \u4e8c\u53c9\u6811\u904d\u5386：\u56db\u79cd\u65b9\u5f0f\u4e00\u7f51\u6253\u5c3d<\/h3>\n

                          \u4e8c\u53c9\u6811\u904d\u5386\u5206\u4e24\u5927\u7c7b：DFS \u4e09\u79cd \u548c BFS \u4e00\u79cd，\u4e00\u5171\u56db\u79cd，\u90fd\u662f\u8d85\u9ad8\u9891\u8003\u70b9\u3002<\/p>\n

                          1. DFS \u4e09\u79cd（\u6839\u636e\u6839\u8282\u70b9\u88ab\u8bbf\u95ee\u7684\u987a\u5e8f\u533a\u5206）<\/h4>\n
                            \n
                          • \n

                            \u524d\u5e8f\u904d\u5386：\u6839 \u2192 \u5de6 \u2192 \u53f3<\/p>\n<\/li>\n

                          • \n

                            \u4e2d\u5e8f\u904d\u5386：\u5de6 \u2192 \u6839 \u2192 \u53f3（\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u4e2d\u5e8f\u904d\u5386\u7ed3\u679c\u662f\u5347\u5e8f\u7684）<\/p>\n<\/li>\n

                          • \n

                            \u540e\u5e8f\u904d\u5386：\u5de6 \u2192 \u53f3 \u2192 \u6839<\/p>\n<\/li>\n<\/ul>\n

                            \u9012\u5f52\u5b9e\u73b0（\u4ee5\u4e2d\u5e8f\u4e3a\u4f8b）：\u4ee3\u7801\u975e\u5e38\u7b80\u6d01，\u7262\u8bb0\u201c\u5de6\u3001\u6839\u3001\u53f3\u201d\u7684\u987a\u5e8f\u9012\u5f52\u8c03\u7528\u5373\u53ef\u3002\u8fed\u4ee3\u5b9e\u73b0\u9700\u8981\u663e\u5f0f\u7528\u6808\u3002<\/p>\n

                            #include <stack><\/p>\n

                            vector<int> inorderTraversal(TreeNode* root) {
                            \n vector<int> result;
                            \n stack<TreeNode*> stk;
                            \n TreeNode* curr = root;<\/p>\n

                            while (curr != nullptr || !stk.empty()) {
                            \n \/\/ \u4e00\u8def\u5411\u5de6，\u628a\u5de6\u8282\u70b9\u5168\u90e8\u538b\u6808
                            \n while (curr != nullptr) {
                            \n stk.push(curr);
                            \n curr = curr->left;
                            \n }
                            \n \/\/ \u5f39\u51fa\u6808\u9876（\u6700\u5de6\u8282\u70b9）\u5e76\u8bbf\u95ee
                            \n curr = stk.top();
                            \n stk.pop();
                            \n result.push_back(curr->val);
                            \n \/\/ \u8f6c\u5411\u53f3\u5b50\u6811
                            \n curr = curr->right;
                            \n }
                            \n return result;
                            \n}<\/p>\n

                            2. BFS \u4e00\u79cd：\u5c42\u5e8f\u904d\u5386（Level Order）<\/h4>\n

                            \u7528\u961f\u5217\u5b9e\u73b0，\u4e00\u5c42\u4e00\u5c42\u8bbf\u95ee\u3002
                            \n\u6838\u5fc3\u6280\u5de7\u662f\u5728\u6bcf\u4e00\u5c42\u5f00\u59cb\u524d，\u8bb0\u5f55\u5f53\u524d\u961f\u5217\u7684\u5927\u5c0f size，\u7136\u540e\u7528\u4e00\u4e2a\u5faa\u73af\u628a\u5f53\u524d\u5c42\u7684\u8282\u70b9\u5168\u90e8\u5904\u7406\u5b8c，\u5e76\u5c06\u5176\u5b50\u8282\u70b9\u52a0\u5165\u961f\u5217\u3002\u8fd9\u6837\u5c31\u80fd\u5b8c\u7f8e\u5206\u5c42\u8f93\u51fa\u3002<\/p>\n

                            #include <queue><\/p>\n

                            vector<vector<int>> levelOrder(TreeNode* root) {
                            \n vector<vector<int>> result;
                            \n if (root == nullptr) return result;<\/p>\n

                            queue<TreeNode*> q;
                            \n q.push(root);<\/p>\n

                            while (!q.empty()) {
                            \n int size = q.size(); \/\/ \u8bb0\u5f55\u5f53\u524d\u5c42\u7684\u8282\u70b9\u6570
                            \n vector<int> level;<\/p>\n

                            for (int i = 0; i < size; ++i) {
                            \n TreeNode* node = q.front();
                            \n q.pop();
                            \n level.push_back(node->val);<\/p>\n

                            if (node->left) q.push(node->left);
                            \n if (node->right) q.push(node->right);
                            \n }
                            \n result.push_back(level);
                            \n }
                            \n return result;
                            \n}<\/p>\n

                            \u00a0\u4e03\u3001\u4ec0\u4e48\u662f\u52a8\u6001\u89c4\u5212（DP）？<\/h3>\n

                            \u52a8\u6001\u89c4\u5212（Dynamic Programming）\u662f\u4e00\u79cd\u628a\u5927\u95ee\u9898\u62c6\u89e3\u4e3a\u91cd\u53e0\u5b50\u95ee\u9898\u3001\u901a\u8fc7\u8bb0\u5f55\u5b50\u95ee\u9898\u7684\u89e3\u6765\u907f\u514d\u91cd\u590d\u8ba1\u7b97\u7684\u7b97\u6cd5\u601d\u60f3\u3002\u5b83\u672c\u8d28\u4e0a\u662f\u201c\u806a\u660e\u7684\u66b4\u529b\u679a\u4e3e\u201d\u3002<\/p>\n

                            \u4e00\u4e2a\u95ee\u9898\u662f\u80fd\u5426\u7528DP，\u901a\u5e38\u770b\u4e24\u4e2a\u7279\u5f81：<\/p>\n

                          • \n

                            \u6700\u4f18\u5b50\u7ed3\u6784：\u539f\u95ee\u9898\u7684\u6700\u4f18\u89e3\u53ef\u4ee5\u7531\u5b50\u95ee\u9898\u7684\u6700\u4f18\u89e3\u7ec4\u5408\u800c\u6765\u3002<\/p>\n<\/li>\n

                          • \n

                            \u91cd\u53e0\u5b50\u95ee\u9898：\u9012\u5f52\u6c42\u89e3\u65f6\u4f1a\u53cd\u590d\u8ba1\u7b97\u76f8\u540c\u7684\u5b50\u95ee\u9898<\/span>（\u8fd9\u6b63\u662fDP\u6bd4\u66b4\u529b\u9012\u5f52\u5feb\u7684\u539f\u56e0）\u3002<\/p>\n<\/li>\n

                            💡 \u638c\u63e1DP\u7684\u89e3\u9898\u5957\u8def（\u6838\u5fc3\u56db\u6b65\u66f2）：<\/p>\n

                              \n
                            • \n

                              \u5b9a\u4e49\u72b6\u6001：dp[i] \u6216 dp[i][j] \u5230\u5e95\u8868\u793a\u4ec0\u4e48？<\/p>\n<\/li>\n

                            • \n

                              \u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b：dp[i] \u5982\u4f55\u4ece\u4e4b\u524d\u7684\u72b6\u6001\u63a8\u5bfc\u51fa\u6765？<\/p>\n<\/li>\n

                            • \n

                              \u521d\u59cb\u5316：\u8fb9\u754c\u72b6\u6001\u5982\u4f55\u8d4b\u503c？<\/p>\n<\/li>\n

                            • \n

                              \u904d\u5386\u987a\u5e8f：\u662f\u4ece\u524d\u5f80\u540e\u3001\u4ece\u540e\u5f80\u524d，\u8fd8\u662f\u6309\u67d0\u79cd\u7ef4\u5ea6？<\/p>\n<\/li>\n<\/ul>\n

                              🔥 \u8fdb\u9636\u6280\u5de7（\u5fc5\u5907）：
                              \n\u5148\u60f3\u66b4\u529b\u9012\u5f52 -> \u53d1\u73b0\u91cd\u53e0\u5b50\u95ee\u9898 -> \u7528\u6570\u7ec4\/\u54c8\u5e0c\u8868\u505a\u8bb0\u5fc6\u5316\u641c\u7d22 -> \u6539\u5199\u6210\u81ea\u5e95\u5411\u4e0a\u7684\u8fed\u4ee3DP -> \u6700\u540e\u7528\u6eda\u52a8\u6570\u7ec4\u4f18\u5316\u7a7a\u95f4\u3002\u638c\u63e1\u8fd9\u5957\u6d41\u7a0b，80%\u7684DP\u9898\u90fd\u80fd\u5957！<\/p>\n

                              💻 \u5b9e\u6218：\u6590\u6ce2\u90a3\u5951（\u6eda\u52a8\u6570\u7ec4\u7a7a\u95f4\u4f18\u5316）<\/h5>\n

                              \u4ee5\u6700\u57fa\u7840\u7684\u722c\u697c\u68af\u4e3a\u4f8b，\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b dp[i] = dp[i-1] + dp[i-2]\u3002\u4e3a\u4e86\u8282\u7701\u7a7a\u95f4，\u6211\u4eec\u4e0d\u7528\u5b58\u6574\u4e2a\u6570\u7ec4，\u53ea\u7528\u4e24\u4e2a\u53d8\u91cf\u3002<\/p>\n

                              int climbStairs(int n) {
                              \n if (n <= 2) return n;
                              \n int prev2 = 1; \/\/ dp[1]
                              \n int prev1 = 2; \/\/ dp[2]
                              \n int current = 0;
                              \n for (int i = 3; i <= n; ++i) {
                              \n current = prev1 + prev2;
                              \n prev2 = prev1;
                              \n prev1 = current;
                              \n }
                              \n return current;
                              \n}<\/p>\n

                              \u5178\u578b\u9898\u76ee\u56fe\u8c31： 0-1\u80cc\u5305\u3001\u5b8c\u5168\u80cc\u5305（\u96f6\u94b1\u5151\u6362）\u3001\u6700\u957f\u516c\u5171\u5b50\u5e8f\u5217（LCS）\u3001\u6700\u957f\u9012\u589e\u5b50\u5e8f\u5217（LIS）\u3001\u6700\u957f\u56de\u6587\u5b50\u4e32\u3001\u7f16\u8f91\u8ddd\u79bb\u3001\u6253\u5bb6\u52ab\u820d<\/p>\n

                              \u8d2a\u5fc3\u7b97\u6cd5\u548c\u52a8\u6001\u89c4\u5212\u6709\u4ec0\u4e48\u533a\u522b？<\/h4>\n

                              \u4e24\u8005\u90fd\u7528\u4e8e\u6c42\u6700\u4f18\u89e3\u95ee\u9898，\u4f46\u6838\u5fc3\u533a\u522b\u5728\u4e8e\u505a\u51b3\u7b56\u65f6\u662f\u5426\u56de\u5934\u770b\u3002<\/p>\n\n\u7ef4\u5ea6\u8d2a\u5fc3\u7b97\u6cd5\u52a8\u6001\u89c4\u5212<\/tr>\n\n\n\n\n\n\n
                              \u51b3\u7b56\u65b9\u5f0f<\/td>\n\u6bcf\u4e00\u6b65\u90fd\u505a\u5f53\u524d\u770b\u8d77\u6765\u6700\u4f18\u7684\u9009\u62e9<\/td>\n\u679a\u4e3e\u6240\u6709\u53ef\u80fd\u7684\u5b50\u95ee\u9898\u89e3\u518d\u505a\u9009\u62e9<\/td>\n<\/tr>\n
                              \u662f\u5426\u6709\u540e\u6548\u6027<\/td>\n\u5047\u8bbe\u5f53\u524d\u9009\u62e9\u4e0d\u5f71\u54cd\u672a\u6765<\/td>\n\u57fa\u4e8e\u4e4b\u524d\u6240\u6709\u72b6\u6001\u63a8\u5bfc<\/td>\n<\/tr>\n
                              \u6b63\u786e\u6027<\/td>\n\u53ea\u5728\u201c\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u201d\u6210\u7acb\u65f6\u624d\u80fd\u5f97\u5230\u6700\u4f18\u89e3<\/td>\n\u53ea\u8981\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u5b9a\u4e49\u5bf9\u4e86，\u4e00\u5b9a\u80fd\u5f97\u5230\u6700\u4f18\u89e3<\/td>\n<\/tr>\n
                              \u65f6\u95f4\u590d\u6742\u5ea6<\/td>\n\u901a\u5e38\u66f4\u5feb，O(n) \u6216 O(n log n)<\/td>\n\u901a\u5e38 O(n\u00b2) \u6216\u66f4\u9ad8<\/td>\n<\/tr>\n
                              \u7a7a\u95f4\u590d\u6742\u5ea6<\/td>\n\u901a\u5e38 O(1)<\/td>\n\u901a\u5e38 O(n) \u6216 O(n\u00b2)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                              \u5178\u578b\u8d2a\u5fc3\u9898：<\/p>\n

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                              • \n

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                              • \n

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                              • \n

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                              • \n

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