{"id":2629,"date":"2023-04-26T07:35:48","date_gmt":"2023-04-25T23:35:48","guid":{"rendered":"https:\/\/www.shuizilong.com\/house\/?p=2629"},"modified":"2023-04-27T16:43:24","modified_gmt":"2023-04-27T08:43:24","slug":"atcoder-regular-contest-159","status":"publish","type":"post","link":"https:\/\/www.shuizilong.com\/house\/archives\/atcoder-regular-contest-159\/","title":{"rendered":"AtCoder Regular Contest 159"},"content":{"rendered":"<ul>\n<li><a href=\"https:\/\/atcoder.jp\/contests\/arc159\/tasks\">https:\/\/atcoder.jp\/contests\/arc159\/tasks<\/a><\/li>\n<li><a href=\"https:\/\/www.bilibili.com\/video\/BV1M24y1L7gt\">https:\/\/www.bilibili.com\/video\/BV1M24y1L7gt<\/a><\/li>\n<li><a href=\"https:\/\/www.cnblogs.com\/cjjsb\/p\/17302607.html\">https:\/\/www.cnblogs.com\/cjjsb\/p\/17302607.html<\/a><\/li>\n<\/ul>\n<h1>Problem B. GCD Subtraction<\/h1>\n<p>\u548c\u524d\u4e00\u5929 <a href=\"https:\/\/www.shuizilong.com\/house\/archives\/atcoder-beginner-contest-297\/\">Beginner Contest 297<\/a> \u7684 D \u9898\u5f88\u50cf\uff01\u4f46\u662f\u9700\u8981\u66f4\u591a\u7684\u667a\u529b\u3002<\/p>\n<p>\u4e0d\u5931\u4e00\u822c\u6027\uff0c\u8003\u8651 a &lt;= b\uff0c\u9996\u5148\u6211\u4eec\u6784\u9020\u4e00\u4e2a\u6570\u636e\u8ba9\u66b4\u529b\u6302\u6389\uff0c\u6bd4\u5982 a=n, b=n+1\uff0c\u56e0\u4e3a\u6bcf\u6b21 d = gcd(a,b) \u90fd\u662f 1\u3002<br \/>\n\u5bf9\u4e8e d \u4e0d\u662f 1 \u7684\u60c5\u51b5\uff0c\u540e\u9762\u7684 d \u4e5f\u90fd\u662f d \u7684\u500d\u6570\uff0c\u56e0\u6b64\u53ef\u4ee5\u76f4\u63a5 a\/=d, b\/=d \u7136\u540e\u5f52\u6210 d=1 \u7684\u60c5\u51b5\u5904\u7406\u3002<\/p>\n<p>\u73b0\u5728\u5c31\u662f\u8003\u8651 d = 1 \u7684\u60c5\u51b5\u5177\u4f53\u4f1a\u6301\u7eed\u591a\u4e45\uff0c\u4e5f\u5c31\u662f\u6211\u4eec\u8981\u627e\u5230\u4e00\u4e2a\u6700\u5c0f\u7684 t\uff0c\u4f7f\u5f97\u65b0\u7684 d > 1\uff0c<br \/>\n\u5982\u679c\u6211\u4eec\u80fd O(1) \u7684\u7b97\u51fa\u8fd9\u4e2a\u503c\uff0c\u90a3\u4e48\u6bcf\u6b21\u81f3\u5c11 \/= 2\uff0c\u6211\u4eec\u5c31\u80fd\u5f97\u5230\u4e00\u4e2a O(logn) \u7684\u505a\u6cd5\u3002<\/p>\n<p>\u7531\u4e8e\u51cf\u6cd5\u64cd\u4f5c\u4e0d\u6539\u53d8 a\u3001b \u4e4b\u5dee\uff0c\u6240\u4ee5\u6211\u4eec\u6709\uff1a<br \/>\n1. d | b-a<br \/>\n2. a = b (mod d)<\/p>\n<p>\u53ea\u8981\u68c0\u67e5\u7684 d \u7684\u56e0\u5b50\u5373\u53ef\uff0c\u6bcf\u6b21\u7684 t \u53d6 a % d\uff0c\u4ece\u4e2d\u627e\u5230\u6700\u5c0f\u503c\u5373\u53ef\uff0c\u590d\u6742\u5ea6\u4e3a O(log(n)sqrt(n))\u3002<\/p>\n<pre class=\"brush: cpp; light: false; title: ; toolbar: true; notranslate\" title=\"\">\r\n#include &lt;lastweapon\/io&gt;\r\nusing namespace lastweapon;\r\n\r\nint main() {\r\n\r\n#ifndef ONLINE_JUDGE\r\n    \/\/freopen(&quot;in.txt&quot;, &quot;r&quot;, stdin);\r\n#endif\r\n\r\n    LL a, b; RD(a, b); if (a &gt; b) swap(a, b); LL z = 0;\r\n    while (a) {\r\n        LL d = __gcd(a, b); a \/= d; b \/= d;\r\n        LL m = b-a, t = a;\r\n        if (m &gt; 1) {\r\n            checkMin(t, a % m);\r\n            for (d=2;d*d&lt;=m;++d) if (m%d == 0) {\r\n                checkMin(t, a % d);\r\n                checkMin(t, a % (m\/d));\r\n            }\r\n        }\r\n        a -= t; b -= t; z += t;\r\n    }\r\n    cout &lt;&lt; z &lt;&lt; endl;\r\n}\r\n<\/pre>\n<p>\u8fdb\u4e00\u6b65 b-a \u4e2d\u95f4\u53ea\u4f1a\u9664\u4ee5 d\uff0c\u53c8\u56e0\u4e3a d \u4e00\u5b9a\u662f\u7d20\u56e0\u5b50\u7684\u60c5\u51b5\u4e0b t \u66f4\u4f18\uff0c\u6240\u4ee5\u53ef\u4ee5\u8fdb\u4e00\u6b65\u7528\u9884\u5904\u7406\u4f18\u5316\u590d\u6742\u5ea6\u5230 <a href=\"https:\/\/math.stackexchange.com\/questions\/145739\/prove-that-logn-o-sqrtn\">O(sqrt(n))<\/a>\u3002<\/p>\n<h1>Problem F. Good Division<\/h1>\n<p>\u867d\u7136\u662f F \u9898\uff0c\u505a\u6cd5\u5374\u610f\u5916\u7684\u7b80\u5355\u3002<\/p>\n<p>\u4e0d\u96be\u8bc1\u660e\uff0c\u4e00\u4e2a\u5e8f\u5217\u662f\u597d\u7684\uff0c\u5f53\u4e14\u4ec5\u5f53\u4e0d\u5b58\u5728 <a href=\"https:\/\/en.wikipedia.org\/wiki\/Majority\">majority<\/a>\u3002<br \/>\n\uff08\u8fd9\u4e2a\u6982\u5ff5\u662f\u4e0d\u662f\u6ca1\u4e2d\u6587\u5bf9\u5e94\uff1f= =\uff09<br \/>\n\u603b\u4e4b\u6211\u4eec\u7acb\u523b\u5f97\u5230\u4e00\u4e2a O(n2) \u7684\u52a8\u6001\u89c4\u5212\u3002<\/p>\n<pre class=\"brush: cpp; light: false; title: ; toolbar: true; notranslate\" title=\"\">\r\n#include &lt;lastweapon\/io&gt;\r\n#include &lt;lastweapon\/number&gt;\r\nusing namespace lastweapon;\r\nconst int N = int(1e6) + 9;\r\nInt f&#x5B;N]; int a&#x5B;N], c&#x5B;N];\r\nint n, maj;\r\n\r\nvoid add(int x) {\r\n    ++c&#x5B;x]; if (c&#x5B;x] &gt; c&#x5B;maj]) maj = x;\r\n}\r\n\r\nint main() {\r\n#ifndef ONLINE_JUDGE\r\n    freopen(&quot;in.txt&quot;, &quot;r&quot;, stdin);\r\n#endif\r\n\r\n    MOD = 998244353;\r\n    RD(n); REP(i, 2*n) RD(a&#x5B;i]);\r\n\r\n    f&#x5B;0] = 1;\r\n\r\n    REP_1(i, n) {\r\n        maj = 0;\r\n        DWN(j, i, 0) {\r\n            add(a&#x5B;j*2]); add(a&#x5B;j*2+1]);\r\n            if (!(c&#x5B;maj] &gt; i-j)) f&#x5B;i] += f&#x5B;j];\r\n        }\r\n        DWN(j, i, 0) c&#x5B;a&#x5B;j*2]] = c&#x5B;a&#x5B;j*2+1]] = 0;\r\n    }\r\n    cout &lt;&lt; f&#x5B;n] &lt;&lt; endl;\r\n}\r\n\r\n<\/pre>\n<p>\u7136\u540e\u600e\u4e48\u529e\uff1fcdq \u5206\u6cbb\u554a\uff01<br \/>\n\u53ef\u5177\u4f53\u600e\u4e48\u5206\u6cbb\u5462\uff0c\u597d\u50cf\u4e0d\u592a\u5bb9\u6613\u62ff\u6570\u636e\u7ed3\u6784\u7ef4\u62a4\u7684\u6837\u5b50\u3002\u3002\u3002<\/p>\n<p>\u5982\u679c\u8f93\u5165\u7684\u5143\u7d20\u4e92\u4e0d\u76f8\u540c\u7684\u8bdd\uff0c\u90a3\u4e48 dag \u7684\u8fb9\u5f88\u5bb9\u6613\u662f O(n2) \u7ea7\u522b\u7684\uff0c<br \/>\n\u4f46\u662f\u5982\u679c\u6211\u4eec\u8003\u8651\u8865\u56fe\uff0c\u5374\u4f1a\u53d1\u73b0\u8f6c\u79fb\u8981\u7a00\u758f\u5f88\u591a\uff0c\u4e8e\u662f\u8003\u8651\u8fdb\u884c\u5bb9\u65a5 &#8230; \u5148\u628a\u548c\u7b97\u4e86\u518d\u628a\u4e0d\u5408\u6cd5\u7684\u51cf\u6389\u3002\u3002<\/p>\n<p>\u8fdb\u4e00\u6b65\u601d\u8003 cdq \u5206\u6cbb\uff0c\u5206\u6cbb\u4e4b\u540e\uff0c\u6211\u4eec\u679a\u4e3e\u6bcf\u4e00\u4e2a\u53ef\u80fd\u7684 maj\uff0c\u4ece\u5de6\u4fa7\u5f71\u54cd\u53f3\u4fa7\u7684\u8f6c\u79fb\u73b0\u5728\u53ef\u4ee5\u7528\u7c7b\u4f3c<a href=\"https:\/\/en.wikipedia.org\/wiki\/Boyer%E2%80%93Moore_majority_vote_algorithm\">\u6469\u5c14\u6295\u7968\u6cd5<\/a>\u7684\u65b9\u6cd5\u7528\u524d\u7f00\u548c\u7ef4\u62a4\u6210\u7ebf\u6027\uff0c\u867d\u7136\u53ef\u80fd\u6709\u5f88\u591a maj\uff0c\u4f46\u662f\u6bcf\u6b21\u8981\u8003\u5bdf\u7684\u8303\u56f4\u548c maj \u51fa\u73b0\u7684\u6b21\u6570\u76f8\u5173\uff0c\u5408\u8d77\u6765\u8fd8\u662f\u7ebf\u6027\uff0c\u4e8e\u662f\u6211\u4eec\u5f97\u5230\u4e86\u4e00\u4e2a O(nlogn) \u7684\u505a\u6cd5\uff0c\u6700\u540e\u53ea\u8981\u5904\u7406\u597d\u7ec6\u8282\u5c31\u597d\u4e86\u3002<\/p>\n<p>\u5177\u4f53\u8bf4\u6765\uff0c\u5bf9\u4e8e\u6bcf\u4e2a\u6f5c\u5728\u7684 maj\uff0c\u6211\u4eec\u5148\u6839\u636e\u5176\u51fa\u73b0\u6b21\u6570\uff0c\u5f97\u5230\u5de6\u53f3\u53ef\u80fd\u5f71\u54cd\u5230\u7684\u6700\u8fdc\u8fb9\u754c\uff0c\u7136\u540e\u4ece\u4e2d\u95f4\u5f00\u59cb\u5de6\u53f3\u4e24\u8fb9\u5206\u522b\u9884\u5904\u7406\u6469\u5c14\u793a\u6570\uff0c<br \/>\n\u5c31\u662f\u6469\u5c14\u6295\u7968\u6cd5\u4e2d\u95f4\u7ef4\u62a4\u7684\u90a3\u4e2a maj \u7684\u51fa\u73b0\u6b21\u6570\uff0cmaj \u5b58\u5728\u7b49\u4ef7\u4e8e\u793a\u6570\u5927\u4e8e 0\uff0c\u6211\u4eec\u53ea\u8981\u627e\u5230\u8fd9\u4e9b\u533a\u95f4\u5373\u53ef\uff0c\u800c\u8fd9\u53ea\u9700\u8981\u679a\u4e3e\u53f3\u533a\u95f4\uff0c\u5bf9\u5de6\u533a\u95f4\u9884\u5904\u7406\u51fa\u6240\u6709\u793a\u6570 >= x \u7684\u4f4d\u7f6e\u7684 dp \u548c\u5373\u53ef\u3002<\/p>\n<ul>\n<li><a href=\"https:\/\/atcoder.jp\/contests\/arc159\/submissions\/40975837\">https:\/\/atcoder.jp\/contests\/arc159\/submissions\/40975837<\/a><\/li>\n<\/ul>\n<pre class=\"brush: cpp; light: false; title: ; toolbar: true; notranslate\" title=\"\">\r\n#include &lt;lastweapon\/io&gt;\r\n#include &lt;lastweapon\/number&gt;\r\nusing namespace lastweapon;\r\nconst int N = int(1e6) + 9;\r\nInt f&#x5B;N]; int a&#x5B;N], c&#x5B;N]; VI b;\r\nInt fs&#x5B;N*2]; int ps&#x5B;N];\r\nint n;\r\n\r\nvoid add(int i) {\r\n    REP(o, 2) {\r\n        int x = a&#x5B;i*2 + o];\r\n        if (!c&#x5B;x]++) b.PB(x);\r\n    }\r\n}\r\n\r\nint get_p(int i, int x) {\r\n    return (a&#x5B;i*2] == x ? 1 : -1) + (a&#x5B;i*2+1] == x ? 1 : -1);\r\n}\r\n\r\nvoid gao(int l, int m, int r, int x) {\r\n    ps&#x5B;m-1] = 0; FOR(i, m, r) ps&#x5B;i] = ps&#x5B;i-1] + get_p(i-1, x);\r\n    DWN(i, m-1, l) ps&#x5B;i] = ps&#x5B;i+1] + get_p(i, x);\r\n\r\n    int ll = INF, rr = -INF;\r\n    FOR(i, l, m) {\r\n        checkMin(ll, ps&#x5B;i]); checkMax(rr, ps&#x5B;i]+1);\r\n        fs&#x5B;N+ps&#x5B;i]] += f&#x5B;i];\r\n    }\r\n\r\n    ll += N; rr += N; DWN(i, rr, ll) fs&#x5B;i] += fs&#x5B;i+1];\r\n    FOR(i, m, r) f&#x5B;i] -= fs&#x5B;max(N+1-ps&#x5B;i], ll)];\r\n    FOR(i, ll, rr) fs&#x5B;i] = 0;\r\n}\r\n\r\nvoid cdq(int l, int r) {\r\n    if (l + 1 == r) return;\r\n    int m = (l+r+1)&gt;&gt;1;\r\n\tcdq(l, m);\r\n\tInt s = 0; FOR(i, l, m) s += f&#x5B;i]; FOR(i, m, r) f&#x5B;i] += s; FOR(i, l, r-1) add(i);\r\n\tfor (auto x: b) gao(max(l, m-c&#x5B;x]), m, min(r, m+c&#x5B;x]), x);\r\n\tfor (auto x: b) c&#x5B;x] = 0; b.clear();\r\n\tcdq(m, r);\r\n}\r\n\r\nint main() {\r\n#ifndef ONLINE_JUDGE\r\n    freopen(&quot;in.txt&quot;, &quot;r&quot;, stdin);\r\n#endif\r\n\r\n    MOD = 998244353;\r\n    RD(n); REP(i, 2*n) RD(a&#x5B;i]);\r\n\r\n    f&#x5B;0] = 1; cdq(0, n+1);\r\n    cout &lt;&lt; f&#x5B;n] &lt;&lt; endl;\r\n}\r\n\r\n<\/pre>\n<p>btw, <a href=\"https:\/\/atcoder.jp\/contests\/arc159\/editorial\/6171\">\u9898\u89e3<\/a> \u8bed\u6cd5\u597d\u602a\u3002\u3002\u5b8c\u5168\u770b\u4e0d\u61c2\u3002<br \/>\n\u4f46\u662f\u770b <a href=\"https:\/\/atcoder.jp\/contests\/arc159\/submissions\/40421276\">jly \u7684\u4ee3\u7801<\/a> \u54c7\uff0c\u597d\u50cf\u786e\u5b9e\u53ef\u4ee5\u76f4\u63a5\u7ebf\u6027\u505a\u3002\u3002\u3002<br \/>\n\u6838\u5fc3\u662f\u8003\u5bdf\u533a\u95f4\u66f4\u5c40\u90e8\u7684\u6027\u8d28\u3002\u3002<\/p>\n<p>\u8fd9\u8ba9\u6211\u60f3\u8d77\u6765 <a href=\"https:\/\/www.shuizilong.com\/house\/archives\/deltix-round-autumn-2021\/\">Deltix Round, Autumn 2021<\/a> \u7684 F \u9898\uff0c\u6bd4\u8d5b\u65f6\u4e5f\u662f\u4e00\u5927\u5806 cdq \u5206\u6cbb\u3002\u3002\u3002\u4f46\u662f\u5176\u5b9e\u6709\u66f4\u597d\u7684\u5355\u8c03\u6808\u7684\u505a\u6cd5\u3002\u3002\u3002\u53cd\u6b63\u6211\u89c9\u5f97\u80fd\u7528\u5355\u8c03\u6808\u7684\u90fd\u5e94\u8be5\u5c3d\u53ef\u80fd\u7528\u5355\u8c03\u6808\uff0c<a href=\"https:\/\/www.shuizilong.com\/house\/archives\/atcoder-beginner-contest-299\/\">\u6bd4\u5982\u524d\u4e00\u5929\u7684 G \u9898<\/a>\u3002\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>https:\/\/atcoder.jp\/contests\/arc159\/tasks https:\/\/www.bilibili.com\/video\/BV1M24y1L7gt https:\/\/www.cnblogs.com\/cjjsb\/p\/17302607.html Problem B. GCD Subtraction \u548c\u524d\u4e00\u5929 Beginner Contest 297 \u7684 D \u9898\u5f88\u50cf\uff01\u4f46\u662f\u9700\u8981\u66f4\u591a\u7684\u667a\u529b\u3002 \u4e0d\u5931\u4e00\u822c\u6027\uff0c\u8003\u8651 a &lt;= b\uff0c\u9996\u5148\u6211\u4eec\u6784\u9020\u4e00\u4e2a\u6570\u636e\u8ba9\u66b4\u529b\u6302\u6389\uff0c\u6bd4\u5982 a=n, b=n+1\uff0c\u56e0\u4e3a\u6bcf\u6b21 d = gcd(a,b) \u90fd\u662f 1\u3002 \u5bf9\u4e8e d \u4e0d\u662f 1 \u7684\u60c5\u51b5\uff0c\u540e\u9762\u7684 d \u4e5f\u90fd\u662f d \u7684\u500d\u6570\uff0c\u56e0\u6b64\u53ef\u4ee5\u76f4\u63a5 a\/=d, b\/=d \u7136\u540e\u5f52\u6210 d=1 \u7684\u60c5\u51b5\u5904\u7406\u3002 \u73b0\u5728\u5c31\u662f\u8003\u8651 d = 1 \u7684\u60c5\u51b5\u5177\u4f53\u4f1a\u6301\u7eed\u591a\u4e45\uff0c\u4e5f\u5c31\u662f\u6211\u4eec\u8981\u627e\u5230\u4e00\u4e2a\u6700\u5c0f\u7684 t\uff0c\u4f7f\u5f97\u65b0\u7684 d > 1\uff0c \u5982\u679c\u6211\u4eec\u80fd O(1) \u7684\u7b97\u51fa\u8fd9\u4e2a\u503c\uff0c\u90a3\u4e48\u6bcf\u6b21\u81f3\u5c11 \/= 2\uff0c\u6211\u4eec\u5c31\u80fd\u5f97\u5230\u4e00\u4e2a O(logn) \u7684\u505a\u6cd5\u3002 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","enabled":false}}},"categories":[1],"tags":[],"class_list":["post-2629","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p2tdP7-Gp","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/posts\/2629","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/comments?post=2629"}],"version-history":[{"count":34,"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/posts\/2629\/revisions"}],"predecessor-version":[{"id":2663,"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/posts\/2629\/revisions\/2663"}],"wp:attachment":[{"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/media?parent=2629"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/categories?post=2629"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/tags?post=2629"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}