{"id":911,"date":"2023-11-15T20:28:03","date_gmt":"2023-11-15T12:28:03","guid":{"rendered":"http:\/\/ggapa.net:81\/?p=911"},"modified":"2023-11-17T11:42:32","modified_gmt":"2023-11-17T03:42:32","slug":"noip2023-%e6%a8%a1%e6%8b%9f%e8%b5%9b2023-11-14","status":"publish","type":"post","link":"http:\/\/ggapa.net:81\/2023\/11\/15\/noip2023-%e6%a8%a1%e6%8b%9f%e8%b5%9b2023-11-14\/","title":{"rendered":"NOIP2023 \u6a21\u62df\u8d5b(2023.11.14)"},"content":{"rendered":"

${\\color{Red} \\mathrm{\u5199\u7684\u5f88\u5783\u573e\uff0c\u5f85\u8865\u5145} } $
\nUpdate on 2023.11.17 \u4fee\u6539\u7b14\u8bef\uff0c\u8865\u5145\u5185\u5bb9\u3002<\/p>\n

T1<\/a><\/h1>\n

\"piYOWCV.png\"<\/div><\/a><\/p>\n
\u9898\u76ee\u5927\u610f<\/h5>\n

\u6c42 $m!$ \u5728\u6a21 $p$ \u4e0b\u7684\u503c\uff0c\u4fdd\u8bc1 $p$ \u662f\u7d20\u6570\u3002<\/p>\n

\u5206\u6790\u4e0e\u89e3\u7b54<\/h5>\n

\u8003\u70b9\uff1a\u5a01\u5c14\u900a\u5b9a\u7406\u3002<\/p>\n

\u7531\u5a01\u5c14\u900a\u5b9a\u7406\u53ef\u77e5\uff0c\u82e5 $p$ \u4e3a\u7d20\u6570\uff0c\u5219\u6709$(p-1)! \\equiv p-1\\pmod{p}$\u3002<\/p>\n

\u53ef\u901a\u8fc7\u4ee5\u4e0a\u5b9a\u7406\u5f97 $(p-1)! \\pmod{p}$ \u7684\u503c\u3002<\/p>\n

\u82e5 $m \\ge p$\uff0c\u7ed3\u679c\u4e3a $0$\u3002<\/p>\n

\u82e5 $m < p$\uff0c\u9898\u76ee\u4e2d\u8ba9\u6211\u4eec\u6c42\u7684\u662f\uff1a<\/p>\n

$$\\prod_{1}^{m} \\pmod{p}$$<\/p>\n

\u4f46\u6211\u4eec\u5df2\u77e5\uff1a<\/p>\n

$$\\prod_{1}^{p - 1} \\pmod{p}$$<\/p>\n

\u6240\u4ee5\u5c31\u9700\u8981\u4e58\u4e0a\u4ee5\u4e0b\u5f0f\u5b50\uff0c\u6765\u5220\u53bb\u591a\u4f59\u7684\u90e8\u5206\u3002<\/p>\n

$$\\prod_{i=m+1}^{p-1} i^{-1} \\pmod{p}$$<\/p>\n

\u6700\u7ec8\u7684\u7b54\u6848\u4e3a\uff1a<\/p>\n

$$(p-1) \\times \\prod_{i=m+1}^{p-1} i^{-1} \\pmod{p}$$<\/p>\n

\u6c42\u9006\u5143\u53ef\u901a\u8fc7\u8d39\u9a6c\u5c0f\u5b9a\u7406\u5b9e\u73b0\u3002<\/p>\n

#include <bits\/stdc++.h>\nusing namespace std;\nlong long qpow(long long a, long long b, long long p) {\n    long long ans = 1;\n    while (b) {\n        if (b & 1)\n            ans = ans * a % p;\n        a = a * a % p;\n        b >>= 1;\n    }\n    return ans;\n}\nint main() {\n    freopen("factorial.in", "r", stdin);\n    freopen("factorial.out", "w", stdout);\n    cin.tie(0)->sync_with_stdio(0);\n    int T;\n    cin >> T;\n    while (T--) {\n        long long m, p;\n        cin >> m >> p;\n        if (m >= p)\n            cout << 0 << '\\n';\n        else {\n            long long ans = p - 1;\n            for (long long i = p - 1; i > m; i--) ans = 1ll * ans * qpow(i, p - 2, p) % p;\n            cout << ans << '\\n';\n        }\n    }\n    return 0;\n}<\/code><\/pre>\n
T2<\/a><\/h1>\n
\"piYOHER.png\"<\/div><\/a><\/p>\n
\u9898\u76ee\u5927\u610f<\/h5>\n
\u5206\u6790\u4e0e\u89e3\u7b54<\/h5>\n
\u8003\u70b9\uff1a\u5206\u5757<\/p>\n
\u8fd9\u9053\u9898\u7684 \u505a\u6cd5\u5f88\u591a\u3002<\/p>\n
\u8003\u8651\u5efa\u7acb\u53cd\u56fe\uff0c\u7528\u7ebf\u6bb5\u6811\u4f18\u5316\u3002<\/p>\n
\u5bf9\u4e8e\u6bcf\u4e00\u4e2a\u70b9\u6784\u5efa\u5168\u5c40\u7ebf\u6bb5\u6811\uff0c\u5728\u8fde\u8fb9\u7684\u65f6\u5019\u7531\u66f4\u5927\u7684\u8282\u70b9\u5411\u66f4\u5c0f\u7684\u8282\u70b9\u8fde\u8fb9\u3002\u4f46\u6bcf\u4e2a\u70b9\u8fde\u51fa\u7684\u8fb9\u662f $O(\\log n)$ \u7684\uff0c\u518d\u8dd1\u4e00\u904d\u6700\u77ed\u8def\uff0c\u6574\u4f53\u65f6\u95f4\u590d\u6742\u5ea6\u53d8\u4e3a $O(n \\log^2 n)$ \u4e0d\u80fd\u901a\u8fc7\u8fd9\u9053\u9898\u76ee\u3002<\/p>\n
\u56e0\u4e3a\u6bcf\u6761\u8fb9\u7684\u8fb9\u6743\u53ef\u4ee5\u5b9a\u4e3a $1$ \uff0c\u6bcf\u4e2a\u70b9\u5728\u7b2c\u4e00\u6b21\u8fbe\u5230\u65f6\u76f4\u63a5\u6254\u8fdb\u961f\u5217\u91cc\uff0c\u6545\u53ef\u4e0d\u7528\u4f18\u5148\u961f\u5217\u5b8c\u6210\u8fd9\u9053\u9898\u3002\u65f6\u95f4\u590d\u6742\u5ea6 $O(n\\log n)$\uff0c\u7406\u8bba\u4e0a\u4e0d\u80fd\u901a\u8fc7\u6b64\u9898\u3002\u5199\u7684\u4f18\u79c0\u53ef\u4ee5\u5f97 86pts\u3002<\/p>\n
\u53d1\u73b0\u4e0d\u80fd\u518d\u901a\u8fc7\u7ebf\u6bb5\u6811\u8fdb\u884c\u4f18\u5316\uff0c\u8003\u8651\u5728\u6784\u5efa\u8fd4\u56fe\u65f6\u4e0d\u8fde\u90a3\u4e48\u591a\u8fb9\u6765\u4f18\u5316\u7a0b\u5e8f\u3002\u53d1\u73b0 $(n-L+R) \\times d^{-1}$ \u5c31\u53ef\u4ee5\u5f97\u5230\u6211\u4eec\u6240\u9700\u8981\u7684\u70b9\u3002\u4f8b\u5982 :
\n$$n = 10,d = 4, \\gcd(n, d) = 2\uff0cL=1,R=3$$
\n\u6ca1\u6709\u5fc5\u8981\u4e00\u6b21\u6027\u628a\u56fe\u5168\u90e8\u6784\u5efa\u51fa\u6765\uff0c\u56e0\u4e3a\u6784\u5efa\u51fa\u6765\u5c31\u5df2\u7ecf\u7206\u4e86\u3002<\/p>\n
<\/p>\n
T3<\/a><\/h1>\n
\"piYOhgU.png\"<\/div><\/a><\/p>\n
\u5206\u6790\u4e0e\u89e3\u7b54<\/h5>\n
\u8003\u70b9\uff1a \u6574\u4f53\u4e8c\u5206<\/p>\n
\u83ab\u961f\u90e8\u5206\u5206\uff0c\u80cc\u5305\u5408\u5e76\u90e8\u5206\u5206<\/p>\n
\u9700\u8981\u8fdb\u884c\u79bb\u7ebf\u64cd\u4f5c\uff0c\u5e76\u4e14\u628a\u6bcf\u4e00\u79cd\u8be2\u95ee\u90fd\u626f\u4e0a\u5173\u7cfb\uff0c\u80cc\u5305\u5408\u5e76\u4e0d\u884c\u3002<\/p>\n
\u8003\u8651\u6574\u4f53\u4e8c\u5206\u548c\u5206\u6cbb\u3002<\/p>\n
\u5bf9\u4e8e\u6bcf\u4e00\u4e2a\u80cc\u5305\u505a\u524d\u7f00\u6700\u5927\u503c\u5904\u7406\uff0c\u540e\u7f00\u4e5f\u5904\u7406\u4e86\uff0c\u53ef\u4ee5 $O(m)$ \u67e5\u8be2\uff0c\u540e\u4f7f\u7528\u6574\u4f53\u4e8c\u5206\u3002<\/p>\n
T4<\/a><\/h1>\n
\"piYO4vF.png\"<\/div><\/a><\/p>\n
\u5206\u6790\u4e0e\u89e3\u7b54<\/h5>\n
\u8003\u70b9\uff1a\u6700\u5c0f\u751f\u6210\u6811\u7684\u7406\u89e3\u3002\uff08\u5bb9\u65a5\u539f\u7406\uff1f\uff09<\/p>\n
\u5f53 $m = n - 1$ \u65f6\uff0c\u6bcf\u4e00\u6761\u9053\u8def\u90fd\u662f\u9700\u8981\u53d6\uff0c\u5229\u7528\u8d2a\u5fc3\u6392\u5e8f\u540e\u679a\u4e3e\u6bcf\u4e00\u6761\u8fb9\u7684\u957f\u5ea6\u5373\u53ef\u901a\u8fc7\u4e00\u90e8\u5206\u5206\u3002\u51c6\u786e\u8bc1\u660e\u4e0d\u4f1a\u3002\u6811\u8fb9\u5747\u53ef\u8fd9\u6837\u64cd\u4f5c\u3002\u5c06\u8fd9\u4e2a\u65b9\u6cd5\u884d\u751f\u5230\u975e\u6811\u8fb9\uff0c\u4e5f\u662f\u56e0\u8be5\u8fb9\u6743\u4ece\u5c0f\u5230\u5927\u52a0\u3002<\/p>\n
\u8fd8\u662f\u5148\u8fdb\u884c\u6392\u5e8f\uff0c\u4f46\u662f\u5728\u9009\u62e9\u7684\u65f6\u5019\u9700\u8981\u6ce8\u610f\u4e4b\u524d\u5df2\u7ecf\u88ab\u9009\u8fc7\u7684\u60c5\u51b5\uff0c\uff0c\u7136\u540e\uff0c\u7136\u540e\u5c31\u505a\u51fa\u6765\u4e86\u3002<\/p>\n
\u603b\u7ed3<\/h1>\n
    \n
  • \u5bf9\u4e8e\u4e58\u6cd5\u9006\u5143\u7684\u8fd0\u7528\u8fd8\u662f\u4e0d\u719f\u6089<\/li>\n
  • \u7b2c\u56db\u9898\u867d\u7136\u6709\u601d\u8def\uff0c\u800c\u4e14\u662f\u6b63\u89e3\uff0c\u4f46\u662f\u6ca1\u6709\u5b9e\u73b0\u51fa\u6765\uff0c\u7801\u529b\u592a\u5f31\u3002<\/li>\n
  • \u6bcf\u9053\u9898\u8fd8\u662f\u8981\u5148\u6253\u4e00\u4e0b\u66b4\u529b\u518d\u53bb\u505a\u6b63\u89e3\u3002<\/li>\n
  • \u6ce8\u610f\u90e8\u5206\u5206\u63a8\u5e7f<\/li>\n<\/ul>\n
    \n
    \u9898\u89e3<\/h1>\n
    A. \u866b\u7fa4\u4e4b\u5fc3<\/h2>\n
    \u5a01\u5c14\u900a\u5b9a\u7406\u677f\u5b50\u9898\uff0c\u6ce8\u610f\u9898\u76ee\u6761\u4ef6\u3002<\/p>\n
    B. \u559d\u9189\u7684\u5154\u5b50<\/h2>\n
    \u7b97\u6cd5\u4e00<\/h3>\n
    \u5bf9\u4e8esubtask1\uff0c\u8fc7\u7a0b\u552f\u4e00\uff0c\u53ef\u4ee5\u76f4\u63a5\u66b4\u529b\u3002 <\/p>\n
    \u7b97\u6cd5\u4e8c<\/h3>\n
    \u4ee4 $f_i$ \u8868\u793a\u6a21 $n$ \u4f59 $i$ \u65f6\u7684\u6700\u5c11\u6b65\u6570\uff0c\u8003\u8651\u5efa\u56fe\u3002<\/p>\n
    \u5bf9\u4e8e $L = R$ \u7684\u70b9\u8fb9\u6570\u4e0d\u591a\uff0cbfs \u4e00\u904d\u5373\u53ef\u3002\u80fd\u8fc7 subtask1, 2, 5\u3002<\/p>\n
    \u7b97\u6cd5\u4e09<\/h3>\n
    \u8003\u8651 $L\\neq R$ \u65f6\u7684\u5efa\u56fe\uff0c\u53ef\u4ee5\u7528\u7ebf\u6bb5\u6811\u4e4b\u7c7b\u7684\u4f18\u5316\u5efa\u56fe\u3002<\/p>\n
    \u6700\u540e\u518d\u7528\u5355\u8c03\u961f\u5217\u4e4b\u7c7b\u7684\u65b9\u6cd5\u505a\u4e00\u4e0b\u5373\u53ef\u3002 <\/p>\n
    \u590d\u6742\u5ea6\u5dee\u4e0d\u591a\u662f $O(T (n \\lg n + q))$<\/p>\n
    \u80fd\u8fc7 subtask1, 2, 3, 4\uff0c\u6216\u8bb8\u80fd\u8fc7 subtask5, 6\u3002<\/p>\n
    \u7b97\u6cd5\u56db<\/h3>\n
    \u8003\u8651\u7ef4\u62a4 $gi = \\min<\/em>{0\\le j\\le R\u2212L}\\ f_{(i+j)} \\mod\\ n$\uff0c\u5efa\u56fe bfs\uff0c\u53ef\u4ee5\u4f18\u5316\u5e38\u6570\uff0c\u964d\u4f4e\u4ee3\u7801\u590d\u6742\u5ea6\u3002<\/p>\n
    \u7136\u540e\u53ef\u4ee5\u518d\u7528\u7ebf\u6bb5\u6811\u4e4b\u7c7b\u7684\u4f18\u5316\u5efa\u56fe\u3002<\/p>\n
    \u7b97\u6cd5\u4e94<\/h3>\n
    \u8003\u8651 bfs \u7684\u8fc7\u7a0b\uff0c\u6bcf\u4e2a\u70b9\u7b2c\u4e00\u6b21\u88ab\u904d\u5386\u5230\u7684\u65f6\u5019\u5b83\u7684\u6700\u7ec8\u7b54\u6848\u5c31\u786e\u5b9a\u4e86\u3002<\/p>\n
    \u6240\u4ee5\u6211\u4eec\u53ea\u9700\u8981\u6bcf\u6b21\u8003\u8651\u5728\u4e00\u6bb5\u533a\u95f4\u5185\u628a\u6240\u6709\u672a\u6807\u8bb0\u7684\u8282\u70b9\u627e\u51fa\u6765\u5373\u53ef\u3002<\/p>\n
    \u76f8\u5f53\u4e8e\u5220\u70b9\u67e5\u540e\u7ee7\uff0c\u4f7f\u7528 set \u5373\u53ef\u3002<\/p>\n
    \u590d\u6742\u5ea6\u548c\u7b97\u6cd5\u4e09\u7c7b\u4f3c\u3002<\/p>\n
    \u7b97\u6cd5\u516d<\/h3>\n
    \u5bf9\u4e8e\u5220\u70b9\u67e5\u540e\u7ee7\u95ee\u9898\uff0c\u53ef\u4ee5\u4f7f\u7528\u5e76\u67e5\u96c6\u6216\u8005\u7c7b\u4f3c\u56db\u6bdb\u5b50\u7684\u65b9\u6cd5\u7ef4\u62a4\u3002<\/p>\n
    \u590d\u6742\u5ea6 $O(Tn\\alpha(n))$ \u6216 $O(Tn)$\u3002<\/p>\n
    \u6ce8\u610f\u4e00\u4e0b\u5e38\u6570\u5c31\u80fdAC\u3002<\/p>\n
    \u7b97\u6cd5\u4e03<\/h3>\n
    \u6cbf\u7740\u7b97\u6cd5\u4e94\u7684\u601d\u8def\uff0c\u7ed9\u5e8f\u5217\u5206\u5757\uff0c$R-L$ \u4e2a\u4e00\u5757\uff0c\u5bb9\u6613\u53d1\u73b0\u6bcf\u4e2a\u5757\u5185\u4efb\u610f\u65f6\u523b\u90fd\u662f\u4e00\u4e2a\u524d\u7f00\u548c\u4e00\u4e2a\u540e\u7f00\u88ab\u5220\u9664\u3002\u6240\u4ee5\u53ea\u9700\u8981\u7ed9\u6bcf\u4e2a\u5757\u5b9a\u4e24\u4e2a\u6307\u9488\u3002\u4fee\u6539\u53ea\u4f1a\u6d89\u53ca\u5230 $O(1)$ \u4e2a\u5757\u3002\u6bcf\u4e2a\u5757\u5355\u72ec\u7ef4\u62a4\u4e00\u4e0b\u5373\u53ef\u3002<\/p>\n
    \u590d\u6742\u5ea6 $O(Tn)$\uff0c\u80fdAC\u3002<\/p>\n
    \u7b97\u6cd5\u516b<\/h3>\n
    \u56de\u5230\u4f18\u5316\u5efa\u56fe\u7684\u601d\u8def\u4e0a\u6765\uff0c\u518d\u7ed3\u5408\u4e00\u4e0b\u7b97\u6cd5\u4e03\u3002\u53ef\u4ee5\u5f97\u5230\u4e00\u4e2a $O(Tn)$ \u7684\u4f18\u5316\u5efa\u56fe\u7684\u505a\u6cd5\u3002<\/p>\n
    \u6ce8\u610f\u4e0b\u5e38\u6570\u80fdAC\u3002<\/p>\n
    \u5176\u5b83\u60f3\u6cd5<\/h3>\n
    \u5982\u679c\u8fd9\u4e2a\u9898\u6539\u6210\u6c42\u6700\u4f18\u89e3\u6982\u7387\uff0c\u4ecd\u7136\u53ef\u4ee5\u6269\u5c55\u7b97\u6cd5\u516b\u7684\u505a\u6cd5\u89e3\u51b3\u3002\u4f46\u4f7f\u7528\u7b97\u6cd5\u516d\u6216\u7b97\u6cd5\u4e03\u7684\u65b9\u6cd5\u4f3c\u4e4e\u4e0d\u592a\u53ef\u884c\u3002<\/p>\n
    C. \u76f2\u76d2\u6d41\u6c34\u7ebf<\/h2>\n
    \u7b97\u6cd5\u4e00<\/h3>\n
    \u76f4\u63a5\u66b4\u641c\u3002<\/p>\n
    \u53ef\u4ee5\u8fc7\u6389 subtask1\uff0c15\u5206\u3002<\/p>\n
    \u7b97\u6cd5\u4e8c<\/h3>\n
    \u6bcf\u6b21\u66b4\u529bdp\uff0c\u590d\u6742\u5ea6 $O(qnm)$\u3002<\/p>\n
    \u7b97\u6cd5\u4e09<\/h3>\n
    \u8003\u8651\u5230\u5408\u5e76\u4e24\u4e2a\u80cc\u5305\u590d\u6742\u5ea6\u4e3a $O(m^2)$\u3002<\/p>\n
    \u6240\u4ee5\u7528\u7ebf\u6bb5\u6811\u4e4b\u7c7b\u7684\u6570\u636e\u7ed3\u6784\u7ef4\u62a4\u53ef\u4ee5\u505a\u5230 $O(nm^2 + qm^2\\lg n)$\u3002<\/p>\n
    \u7b97\u6cd5\u56db<\/h3>\n
    \u6211\u4eec\u6709\u4e24\u79cd\u80cc\u5305\u7684\u64cd\u4f5c\u53ef\u4ee5\u505a\u5230\u590d\u6742\u5ea6 $O(m)$\uff1a\u4e00\u79cd\u662f\u5f80\u80cc\u5305\u91cc\u52a0\u5165\u4e00\u4e2a\u5143\u7d20\uff0c\u4e00\u79cd\u662f\u8be2\u95ee\u4e24\u4e2a\u80cc\u5305\u5408\u5e76\u540e\u4ef7\u683c\u603b\u548c\u4e0d\u8d85\u8fc7\u67d0\u4e2a\u5b9a\u503c\u7684\u7b54\u6848\u3002<\/p>\n
    \u6211\u4eec\u53ef\u4ee5\u8003\u8651\u5206\u6cbb\uff0c\u5bf9\u4e8e\u4e00\u6bb5\u80cc\u5305\u7684\u533a\u95f4 $[l, r]$\uff0c\u5982\u679c\u8db3\u591f\u5c0f\uff0c\u6211\u4eec\u53ef\u4ee5\u66b4\u529b\u6240\u6709\u7684\u8be2\u95ee\u3002<\/p>\n
    \u5426\u5219\u53ef\u4ee5\u53d6\u4e2d\u70b9 $mid = \\lfloor \\frac{l+r}2\\rfloor$\uff0c\u4ece $mid$ \u5f80\u4e24\u8fb9\u5206\u522b\u505a\u524d\u7f00\/\u540e\u7f00\u80cc\u5305\uff0c\u90a3\u4e48\u6b64\u65f6\u6bcf\u4e2a\u8fc7\u4e2d\u70b9\u7684\u8be2\u95ee\u53ef\u4ee5\u89c6\u4e3a\u4e24\u4e2a\u5df2\u5904\u7406\u51fa\u6765\u7684\u80cc\u5305\u5408\u5e76\u540e\u8be2\u95ee\u4ef7\u683c\u603b\u548c\u4e0d\u8d85\u8fc7 $m$ \u7684\u7b54\u6848\uff0c\u53ef\u4ee5\u76f4\u63a5 $O(m)$ \u89e3\u51b3\u3002<\/p>\n
    \u90a3\u4e48\u6b64\u65f6\u6ca1\u6709\u8fc7\u4e2d\u70b9\u7684\u8be2\u95ee\uff0c\u6240\u4ee5\u53ef\u4ee5\u5206\u6cbb\u89e3\u51b3 $[l, m]$ \u548c $[m + 1, r]$\u3002<\/p>\n
    \u590d\u6742\u5ea6 $O(nm \\lg n + qm)$\uff0c100\u5206\u3002<\/p>\n
    D. \u5931\u843d\u7684\u5e1d\u56fd<\/h2>\n
    \u5f53 $m=n-1$ \u65f6\uff0c\u95ee\u9898\u8f6c\u5316\u6210\u6709 $m$ \u4e2a\u53d8\u91cf\uff0c\u7b2c $i$ \u53d8\u91cf\u7684\u53d6\u503c\u4e3a $\\left[l_i, r_i\\right]$\uff0c\u4e14\u4efb\u610f\u4e24\u4e2a\u53d8\u91cf\u4e0d\u76f8\u7b49\uff0c\u6784\u9020\u4e00\u7ec4\u65b9\u6848\u3002\u6211\u4eec\u53ef\u4ee5\u5c06\u6240\u6709\u533a\u95f4\u6309\u7167\u5de6\u7aef\u70b9\u6392\u5e8f\uff0c\u968f\u540e\u626b $l_i$ \u65f6\u8d2a\u5fc3\u7684\u9009\u62e9 $r_i$ \u6700\u5c0f\u7684\u6765\u653e\u7f6e\u3002<\/p>\n
    \u5f53 $m>n-1$ \u65f6\uff0c\u6bcf\u6761\u975e\u6811\u8fb9\u4f1a\u6709\u989d\u5916\u7684\u9650\u5236\uff1a\u975e\u6811\u8fb9 $(u, v)$ \u7684\u957f\u5ea6\u5fc5\u987b\u4e25\u683c\u5927\u4e8e\u8def\u5f84 $u \\rightarrow v$ \u7684\u957f\u5ea6\u6700\u5c0f\u503c\u3002\u56e0\u6b64\uff0c\u5bf9\u4e8e\u4e00\u6761\u975e\u6811\u8fb9\uff0c\u6211\u4eec\u53ea\u6709\u5728\u8fd9\u6761\u8def\u5f84\u4e0a\u6240\u6709\u7684\u8fb9\u5747\u88ab\u8d4b\u503c\u540e\uff0c\u624d\u53ef\u4ee5\u8d4b\u975e\u6811\u8fb9\u7684\u503c\u3002\u56e0\u6b64\u6211\u4eec\u4ecd\u7136\u6309\u7167 $l_i$ \u6392\u5e8f\uff0c\u4f46\u5728\u5408\u5e76\u4e24\u4e2a\u8054\u901a\u5757\u65f6\uff0c\u66f4\u65b0\u8fd9\u4e00\u8f6e\u6240\u8054\u901a\u7684\u975e\u6811\u8fb9\uff0c\u5982\u679c\u4e00\u4e2a\u975e\u6811\u8fb9\u7684\u4e24\u7aef\u70b9\u8054\u901a\uff0c\u5219\u5c06\u5176\u52a0\u5165\u5806\u4e2d\u5373\u53ef\u3002<\/p>\n
    \u603b\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a $O\\left((n+m) \\log ^2 n\\right)$ \u6216 $O((n+m) \\log n)$ \u3002<\/p>\n","protected":false},"excerpt":{"rendered":"
    ${\\color{Red} \\mathrm{\u5199\u7684\u5f88\u5783\u573e\uff0c\u5f85\u8865\u5145} } $ Update on 2023.11. […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[12],"tags":[],"_links":{"self":[{"href":"http:\/\/ggapa.net:81\/wp-json\/wp\/v2\/posts\/911"}],"collection":[{"href":"http:\/\/ggapa.net:81\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/ggapa.net:81\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/ggapa.net:81\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/ggapa.net:81\/wp-json\/wp\/v2\/comments?post=911"}],"version-history":[{"count":11,"href":"http:\/\/ggapa.net:81\/wp-json\/wp\/v2\/posts\/911\/revisions"}],"predecessor-version":[{"id":929,"href":"http:\/\/ggapa.net:81\/wp-json\/wp\/v2\/posts\/911\/revisions\/929"}],"wp:attachment":[{"href":"http:\/\/ggapa.net:81\/wp-json\/wp\/v2\/media?parent=911"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/ggapa.net:81\/wp-json\/wp\/v2\/categories?post=911"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/ggapa.net:81\/wp-json\/wp\/v2\/tags?post=911"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}