{"id":195,"date":"2025-02-10T17:48:50","date_gmt":"2025-02-10T09:48:50","guid":{"rendered":"https:\/\/blog.seekrealthing.net\/blog\/?p=195"},"modified":"2025-02-10T18:29:45","modified_gmt":"2025-02-10T10:29:45","slug":"691-e-xor-sequences-%e8%a7%a3%e6%9e%90%e6%80%9d%e7%b6%ad%e3%80%81%e7%9f%a9%e9%99%a3%e5%bf%ab%e9%80%9f%e5%86%aa","status":"publish","type":"post","link":"https:\/\/blog.seekrealthing.net\/blog\/?p=195","title":{"rendered":"E. Xor-sequences \u89e3\u6790(\u601d\u7dad\u3001\u77e9\u9663\u5feb\u901f\u51aa)"},"content":{"rendered":"\n<div class=\"wp-block-jetpack-markdown\"><p>\u4eca\u5929\u6211\u5011\u4f86\u770b\u770bCF691E\n<a href=\"https:\/\/codeforces.com\/contest\/691\/problem\/E\">\u984c\u76ee\u9023\u7d50<\/a><\/p>\n<h3>\u984c\u76ee\u63cf\u8ff0<\/h3>\n<p>\u7d66\u5b9a $n$ \u500b\u6574\u6578 $a_1, a_2, \\dots, a_n$\uff0c\u6c42\u9577\u5ea6\u70ba $k$ \u7684\u5e8f\u5217 $(x_1, x_2, \\dots, x_k)$ \u7684\u6578\u91cf\uff0c\u5176\u4e2d\u6bcf\u500b $x_i$ \u90fd\u5f9e\u7d66\u5b9a\u7684\u6578\u5b57\u4e2d\u9078\u53d6\uff08\u76f8\u540c\u6578\u5b57\u82e5\u51fa\u73fe\u591a\u6b21\u8996\u70ba\u4e0d\u540c\uff09\u3002\u5e8f\u5217\u5fc5\u9808\u6eff\u8db3\uff1a<br>\n$$\n\\forall, 1 \\le i &lt; k,\\quad \\mathrm{popcount}(x_i \\oplus x_{i+1}) \\equiv 0 \\pmod{3},\n$$\n\u5176\u4e2d $\\oplus$ \u8868\u793a\u4f4d\u5143\u7570\u6216\uff08XOR\uff09\u904b\u7b97\uff0c$\\mathrm{popcount}(y)$ \u8868\u793a $y$ \u7684\u4e8c\u9032\u5236\u8868\u793a\u4e2d 1 \u7684\u500b\u6578\u3002<\/p>\n<hr>\n<h3><strong>\u8f38\u5165\u683c\u5f0f<\/strong><\/h3>\n<ul>\n<li>\u7b2c\u4e00\u884c\uff1a\u5169\u500b\u6574\u6578 $n$ \u548c $k$ ($1 \\le n \\le 100$, $1 \\le k \\le 10^{18}$)\u3002<\/li>\n<li>\u7b2c\u4e8c\u884c\uff1a$n$ \u500b\u6574\u6578 $a_i$ ($0 \\le a_i \\le 10^{18}$)\u3002<\/li>\n<\/ul>\n<hr>\n<h3><strong>\u8f38\u51fa\u683c\u5f0f<\/strong><\/h3>\n<ul>\n<li>\u8f38\u51fa\u4e00\u500b\u6574\u6578\uff0c\u4ee3\u8868\u6eff\u8db3\u689d\u4ef6\u7684\u5e8f\u5217\u6578\u91cf\u6a21 $10^9+7$ \u7684\u7d50\u679c\u3002<\/li>\n<\/ul>\n<h2>\u524d\u8a00<\/h2>\n<p>\u6211\u771f\u7684\u4ee5\u70ba\u9019\u984c\u662f\u8981\u5148\u5efa\u4e00\u5f35\u5716\uff0c$a_i\\oplus a_j\\equiv0\\pmod3$\u7684\u8a71\u5c31\u9023\u8d77\u4f86\uff0c\u7136\u5f8c\u554f\u4f60\u8aaa\u6709\u5e7e\u7a2e\u65b9\u5f0f\u53ef\u4ee5\u5728\u9019\u5f35\u5716\u8d70$k$\u6b65\uff0c\u4f46\u662f\u5b8c\u5168\u60f3\u4e0d\u51fa\u4f86\u600e\u9ebc\u505a\uff0c\u770b\u89e3\u7b54\u624d\u767c\u73fe\u662f\u77e9\u9663\u5feb\u901f\u51aa\uff0c\u592a\u795e\u5947\u4e86<\/p>\n<h3>\u60f3\u6cd5<\/h3>\n<p>\u6211\u4e5f\u4e0d\u77e5\u9053\u9019\u600e\u9ebc\u60f3\u5230\u7684\uff0c\u4f46\u662f\u5982\u679c\u4f60\u60f3\u5230\u8aaa\uff0c\u4ee4\u4e00\u500b$z[i][j]$\u4ee3\u8868\u76ee\u524d\u8003\u616e\u5230\u9577\u5ea6\u70ba$j$\u7684XOR-sequence\uff0c\u4e26\u4e14\u7d50\u5c3e\u662f$a[i]$\u7684\u53ef\u80fd\u6027\u7684\u6578\u91cf\u3002<br>\n\u90a3\u9ebc\u5982\u679c$a[k]$\u662f\u53ef\u4ee5\u63a5\u5728$a[i]$\u5f8c\u9762\u7684\u6578\u5b57\uff0c$z[k][j+1]$\u5c31\u53ef\u4ee5\u5f9e$z[i][j]$\u63a8\u51fa\u4f86\u3002<br>\n\u5177\u9ad4\u4f86\u8aaa\uff1a<br>\n$$\nz[k][j+1] = \\sum_{\\text{avaliable }i}z[i][j]\n$$\n\u9032\u4e00\u6b65\u8aaa\uff0c\u5982\u679c\u4e0d\u8981\u9650\u5236$i$\u4e00\u5b9a\u53ef\u4ee5\u5728$k$\u524d\u9762\uff0c\u516c\u5f0f\u6703\u8b8a\u6210\u9019\u6a23\uff1a<br>\n$$\nz[k][j+1] = \\sum_{i}G_{i,k}z[i][j]\n$$\nwhere $G_{ik}=1$ for $a[i]$\u53ef\u4ee5\u63a5\u5728$a[k]$\u524d\u9762<br>\n$G_{ik}=0$ otherwise<br>\n\u73fe\u5728\u4f60\u5c31\u6709\u53ef\u80fd\u770b\u51fa\u4f86\uff0c\u9019\u500b\u64cd\u4f5c\u662f\u628a$z[][j]$\u66f4\u65b0\u6210$z[][j+1]$\uff0c\u6240\u4ee5\u6211\u5011\u53ef\u4ee5\u4e0d\u7528\u5b58\u6bcf\u4e00\u500b\u9577\u5ea6$j$\u7684$z[][j]$\uff0c\u800c\u53ef\u4ee5\u53ea\u4fdd\u7559\u4e00\u500b$z[]$\uff0c\u4ee3\u8868\u76ee\u524d\u8003\u616e\u5230\u7684XOR-sequence\u9577\u5ea6\u7684\uff0c\u6bcf\u4e00\u500b$a[i]$\u7d50\u5c3e\u7684\u53ef\u80fd\u6578\u91cf\u3002<br>\n\u4e5f\u5c31\u662f\u8aaa\uff1a<br>\n$$\nz\u2019[k] = \\sum_{i}G_{i,k}z[i]\n$$\n\u7136\u5f8c\u56e0\u70ba$G$\u662fsymmetric\u7684($G_{ij} = G_{ji}$)\u6240\u4ee5<br>\n$$\nz\u2019<em>k = \\sum<\/em>{i}G_{k,i}z_i\n$$\n\u4e0d\u904e\u56e0\u70ba\u984c\u76ee\u7684$k$\u8d85\u5927\uff0c\u6240\u4ee5\u5982\u679c\u4f60\u60f3\u8aaa\uff0c\u9019\u7a2e\u65b9\u6cd5\u53ea\u80fdsequential\u5730\u66f4\u65b0\u800c\u653e\u68c4\u9019\u7a2e\u505a\u6cd5\uff0c\u5c31\u5931\u6557\u4e86\u3002<br>\n\u9019\u5c31\u662f\u77e9\u9663$G$\u4e58\u4e0a\u4e00\u500bvector $z$\u7684\u5f62\u5f0f\u3002\u6240\u4ee5\u5c0d\u65bc\u66f4\u65b0$k$\u6b21$z$\uff0c\u53ef\u4ee5\u7528\u77e9\u9663\u5feb\u901f\u51aa\uff0c\u53ea\u8981$\\lg(k)$\u6b21\u77e9\u9663\u4e58\u6cd5\u5c31\u53ef\u4ee5\u4e86(\u77e9\u9663\u4e58\u6cd5$M_{n\\times n}\\cdot M_{n\\times n}$\u7684\u8907\u96dc\u5ea6\u662f$n^3$)<br>\n\u5be6\u4f5c\u4e0a\u8981\u6ce8\u610f\uff0c\u8a18\u5f97\u8a08\u7b97\u7684\u6642\u5019\u8981$\\pmod{10^9+7}$\uff0c\u9084\u6709<code>__builtin_popcount<\/code>\u662f\u5c0d<code>int<\/code>\uff0c\u8981\u5c0d<code>long long<\/code>\u7528\uff0c\u8981\u6539\u7528<code>__builtin_popcountll<\/code><\/p>\n<h3>\u7a0b\u5f0f\u78bc:<\/h3>\n<pre><code class=\"language-cpp\">const int _n = 100 + 10;\nll n, k, a[_n];\nstruct Mat {\n public:\n  int rlen, clen;\n  vector&lt;vector&lt;int&gt;&gt; _a;\n  Mat(int rlen, int clen) : rlen(rlen), clen(clen) {\n    _a = vector&lt;vector&lt;int&gt;&gt;(rlen, vector&lt;int&gt;(clen, 0));\n  }\n  Mat operator*(const Mat&amp; rhs) const {\n    Mat res(this-&gt;rlen, rhs.clen);\n    rep(i, 0, res.rlen) rep(j, 0, res.clen) {\n      res._a[i][j] = 0;\n      rep(k, 0, this-&gt;clen) res._a[i][j] = (1ll * res._a[i][j] + 1ll * (_a[i][k] % mod) * (rhs._a[k][j] % mod)) % mod;\n    }\n    return res;\n  }\n  Mat operator^(ll b) {\n    Mat res(rlen, clen), tmp = *this;\n    assert(rlen == clen);\n    rep(i, 0, rlen) {\n      res._a[i][i] = 1;\n    }\n    while (b) {\n      if (b &amp; 1ll) res = res * tmp;\n      b &gt;&gt;= 1ll;\n      tmp = tmp * tmp;\n    }\n    return res;\n  }\n};\nmain(void) {\n  ios_base::sync_with_stdio(0);\n  cin.tie(0);\n  cout.tie(0);\n  cin &gt;&gt; n &gt;&gt; k;\n  Mat G(n, n);\n  Mat z(n, 1);\n  rep(i, 0, n) {\n    cin &gt;&gt; a[i];\n    z._a[i][0] = 1;\n  }\n\n  rep(i, 0, n) {\n    rep(j, 0, n) {\n      if (__builtin_popcountll(a[i] ^ a[j]) % 3 == 0) {\n        G._a[i][j] = 1;\n      }\n    }\n  }\n  z = (G ^ (k - 1)) * z;\n  ll ans = 0;\n  rep(i, 0, n) {\n    ans = (1ll * ans + 1ll * z._a[i][0]) % mod;\n  }\n  cout &lt;&lt; ans &lt;&lt; '\\n';\n  return 0;\n}\n<\/code><\/pre>\n<p>\u6a19\u982d\u3001\u6a21\u677f\u8acb\u9edeSubmission\u770b\n<a href=\"https:\/\/codeforces.com\/contest\/691\/submission\/304796709\">Submission<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_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},"version":2}},"categories":[3],"tags":[10,9,8,49],"class_list":["post-195","post","type-post","status-publish","format-standard","hentry","category-ojcode","tag-codeforce","tag-ojcode","tag-8","tag-49"],"jetpack_publicize_connections":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v24.4 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>E. 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