{"id":457,"date":"2023-02-02T00:03:09","date_gmt":"2023-02-01T16:03:09","guid":{"rendered":"https:\/\/blog.cirno.fun\/?p=457"},"modified":"2025-03-19T00:35:30","modified_gmt":"2025-03-18T16:35:30","slug":"%e9%9b%86%e5%90%88nim%e6%b8%b8%e6%88%8f%e3%80%81%e6%9c%89%e5%90%91%e5%9b%be%e6%b8%b8%e6%88%8f%e4%b8%8esg%e5%87%bd%e6%95%b0","status":"publish","type":"post","link":"https:\/\/blog.sssn.tech\/?p=457","title":{"rendered":"\u96c6\u5408NIM\u6e38\u620f\u2014\u2014\u6709\u5411\u56fe\u6e38\u620f\u4e0eSG\u51fd\u6570"},"content":{"rendered":"

\"file\"<\/p>\n

SG\u51fd\u6570\u662f\u89e3\u51b3\u535a\u5f08\u8bba\u95ee\u9898\u7684\u5229\u5668<\/p>\n

\u5728\u6b63\u5f0f\u89e3\u9898\u4e4b\u524d\u9996\u5148\u9700\u8981\u4e86\u89e3\u4ee5\u4e0b\u77e5\u8bc6\uff1a<\/p>\n

\u6709\u5411\u56fe\u6e38\u620f<\/h4>\n

\u7ed9\u5b9a\u4e00\u4e2a\u6709\u5411\u65e0\u73af\u56fe\uff0c\u56fe\u4e2d\u6709\u4e00\u4e2a\u552f\u4e00\u7684\u8d77\u70b9\uff0c\u5728\u8d77\u70b9\u4e0a\u653e\u6709\u4e00\u679a\u68cb\u5b50\u3002\u4e24\u540d\u73a9\u5bb6\u4ea4\u66ff\u5730\u628a\u8fd9\u679a\u68cb\u5b50\u6cbf\u6709\u5411\u8fb9\u8fdb\u884c\u79fb\u52a8\uff0c\u6bcf\u6b21\u53ef\u4ee5\u79fb\u52a8\u4e00\u6b65\uff0c\u65e0\u6cd5\u79fb\u52a8\u8005\u5224\u8d1f\u3002\u8be5\u6e38\u620f\u88ab\u79f0\u4e3a\u6709\u5411\u56fe\u6e38\u620f\u3002<\/p>\n

\u4efb\u4f55\u4e00\u4e2a\u516c\u5e73\u7ec4\u5408\u6e38\u620f\uff08ICG\uff09\u90fd\u53ef\u4ee5\u8f6c\u5316\u4e3a\u6709\u5411\u56fe\u6e38\u620f<\/strong>\u3002\u5177\u4f53\u65b9\u6cd5\u662f\uff0c\u628a\u6bcf\u4e2a\u5c40\u9762\u770b\u6210\u56fe\u4e2d\u7684\u4e00\u4e2a\u8282\u70b9\uff0c\u5e76\u4e14\u4ece\u6bcf\u4e2a\u5c40\u9762\u5411\u6cbf\u7740\u5408\u6cd5\u884c\u52a8\u80fd\u591f\u5230\u8fbe\u7684\u4e0b\u4e00\u4e2a\u5c40\u9762\u8fde\u6709\u5411\u8fb9\u3002<\/p>\n

\u4ecb\u7ecd\u4e0a\u8ff0\u7684\u6709\u5411\u56fe\u6e38\u620f\u662f\u4e3a\u4e86\u5f15\u5165\u4e0b\u9762\u4e24\u6761\u6982\u5ff5<\/p>\n

mex\u8fd0\u7b97<\/h4>\n

mex\u8fd0\u7b97\u662f\u5bf9\u4e00\u4e2a\u81ea\u7136\u6570\u96c6\u7684\u8fd0\u7b97\uff0c\u8fd4\u56de\u4e0d\u5c5e\u4e8e\u8be5\u96c6\u5408\u7684\u6700\u5c0f\u81ea\u7136\u6570<\/strong><\/p>\n

\u4f8b\u5982\u5bf9\u4e8e\u96c6\u5408G={0,1,2,4}<\/code>\uff0c$$mex$$(G)<\/code>\u7684\u7ed3\u679c\u662f3<\/p>\n

SG\u51fd\u6570<\/h4>\n

SG\u51fd\u6570\u662f\u4e00\u4e2a\u81ea\u53d8\u91cf\u4e3a\u6e38\u620f\u72b6\u6001\uff08\u5728\u6709\u5411\u65e0\u73af\u56fe\u4e2d\u662f\u4e00\u4e2a\u8282\u70b9\uff09\uff0c\u8fd4\u56de\u503c\u4e3a\u81ea\u7136\u6570\u7684\u7279\u6b8a\u51fd\u6570<\/p>\n

\u5bf9\u4e8e\u4e00\u4e2a\u516c\u5e73\u7ec4\u5408\u6e38\u620f\u5bf9\u5e94\u7684\u6709\u5411\u65e0\u73af\u56fe\uff0c\u6211\u4eec\u89c4\u5b9a\u6e38\u620f\u7ec8\u70b9\u7684SG\u503c\u4e3a0\uff0c\u5e76\u4e14\u5bf9\u4e8e\u5176\u4e2d\u4efb\u4f55\u4e00\u4e2a\u975e\u7ec8\u70b9\u7684\u8282\u70b9x\u6709<\/p>\n

sg(x) = mex\\{sg(y_1), sg(y_2),sg(y_3), \\cdot \\cdot \\cdot ,sg(y_n)\\}<\/code><\/pre>\n
\u5176\u4e2dy\u662fx\u7684\u6240\u6709\u540e\u7ee7\u8282\u70b9<\/p>\n
\u53ef\u4ee5\u4e3e\u4e00\u4e2a\u4f8b\u5b50\uff0c\u56fe\u4e2d\u9ed1\u8272\u6570\u5b57\u662f\u70b9\u7684\u7f16\u53f7\uff0c\u84dd\u8272\u6570\u5b57\u662fSG\u503c\uff1a<\/p>\n
\"file\"<\/p>\n
\u9996\u51484\uff0c8, 9\u53f7\u70b9\u4f5c\u4e3a\u7ec8\u70b9SG\u503c\u662f0\uff0c\u800c5\uff0c 7\u53f7\u70b9\u53ea\u6709\u901a\u5411\u7ec8\u70b9\u7684\u4e00\u4e2a\u540e\u7ee7\uff0cSG\u503c\u4e3a1 6\u53f7\u70b9\u53ea\u80fd\u901a\u5411SG\u503c\u4e3a1\u76847\u53f7\u70b9\uff0c\u6240\u4ee5SG(6) = 0 \u4e09\u70b9\u53ef\u4ee5\u901a\u5411SG\u503c\u5206\u522b\u4e3a0,1\u76846,7\u53f7\u70b9\uff0cSG(3)=2<\/p>\n
\u4ee5\u6b64\u7c7b\u63a8\u53ef\u4ee5\u5199\u51fa\u6240\u6709\u70b9\u7684SG\u503c<\/p>\n
\u5b9a\u7406\uff1a\u6709\u5411\u56fe\u6e38\u620f\u7684\u67d0\u4e2a\u5c40\u9762\u5fc5\u80dc\uff0c\u5f53\u4e14\u4ec5\u5f53\u8be5\u5c40\u9762\u5bf9\u5e94\u8282\u70b9\u7684SG\u51fd\u6570\u503c\u5927\u4e8e0\u3002<\/strong><\/p>\n
\u8bc1\u660e\u65b9\u6cd5\u548c\u7ecf\u5178NIM\u6e38\u620f\u4e00\u6837\uff0c\u53ef\u4ee5\u8bc1\u660e\uff1a<\/p>\n
    \n
  1. \u6e38\u620f\u7ec8\u70b9\uff08\u5fc5\u8d25\u5c40\u9762\uff09\u7684SG\u503c\u4e3a0<\/li>\n
  2. \u4efb\u4f55\u4e00\u4e2aSG\u503c\u4e0d\u4e3a0\u7684\u70b9\uff0c\u6211\u4eec\u90fd\u53ef\u4ee5\u901a\u8fc7\u4e00\u6b21\u64cd\u4f5c\u4f7f\u5176\u8d70\u5230\u4e00\u4e2aSG\u503c\u4e3a0\u7684\u70b9<\/li>\n
  3. \u5f53\u5904\u5728\u4e00\u4e2aSG\u4e3a0\u7684\u70b9\u65f6\uff0c\u65e0\u8bba\u5982\u4f55\u64cd\u4f5c\u90fd\u4e0d\u80fd\u8d70\u5230\u4e0b\u4e00\u4e2aSG\u503c\u4e3a0\u7684\u70b9<\/li>\n<\/ol>\n
    \u800c\u5bf9\u4e8e\u591a\u4e2a\u6709\u5411\u56fe\u7684\u6e38\u620f\uff08\u5728\u672c\u4f8b\u4e2d\u5177\u4f53\u4e3a\u591a\u5806\u77f3\u5b50\uff09\uff0c\u6709\u5b9a\u7406\uff1a<\/p>\n
    \u7531n\u4e2a\u6709\u5411\u56fe\u7ec4\u6210\u7684\u6e38\u620f\u7684\u67d0\u4e2a\u5c40\u9762\u5fc5\u80dc\uff0c\u5f53\u4e14\u4ec5\u5f53\u8be5\u6e38\u620f\u6240\u6709\u8d77\u70b9\u7684SG\u503c\u5f02\u6216\u548c\u4e0d\u4e3a0<\/strong><\/p>\n
    \u8fd9\u4e00\u5b9a\u7406\u88ab\u79f0\u4f5cSprague-Grundy<\/em>\u5b9a\u7406<\/p>\n
    \u8fd9\u91cc\u6709\u4e00\u7bc7CSDN\u7684\u8bc1\u660e\uff0c\u601d\u8def\u548c\u4e0a\u8ff0\u8bc1\u660e\u65b9\u6cd5\u76f8\u4f3c \"file\"<\/p>\n
    \u8bb2\u5b8c\u8fd9\u4e9b\u7ec8\u4e8e\u53ef\u4ee5\u770b\u672c\u9898\u4e86<\/p>\n
    \u672c\u9898\u53ef\u4ee5\u62bd\u8c61\u4e3a\u6709\u5411\u56fe\u6e38\u620f\uff0c\u5c06\u6bcf\u5806\u77f3\u5b50\u62bd\u8c61\u4e3a\u4e00\u4e2a\u56fe\uff0c\u72b6\u6001\u5c31\u662f\u77f3\u5b50\u5b58\u91cf<\/p>\n
    \u56e0\u4e3a\u53ef\u4ee5\u9009\u53d6\u7684\u65b9\u6cd5\u56fa\u5b9a\uff0c \u6240\u4ee5\u5bf9\u4e8e\u6bcf\u4e2a\u72b6\u6001\u7684SG\u503c\u90fd\u662f\u56fa\u5b9a\u7684\uff08\u8fd9\u4e00\u6b65\u6709\u4e9b\u96be\u60f3\uff1a\u4e3a\u4ec0\u4e48\u72b6\u6001\u4fe1\u606f\u53ea\u9700\u8981\u77f3\u5b50\u5b58\u91cf\uff1f\uff09<\/p>\n
    #include <bits\/stdc++.h>\n\nusing namespace std;\ntypedef long long ll;\n\nconst int N = 10100, K = 200;\n\nint s[K], f[N];\n\/\/ s\u662f\u6240\u6709\u53ef\u4f9b\u9009\u62e9\u7684\u53d6\u6cd5\u96c6\u5408\uff0c f\u662f\u5bf9\u4e00\u4e2a\u72b6\u6001\u7684sg\u503c\uff0c \u8fd9\u91cc\u7684\u4e00\u4e2a\u72b6\u6001\u5177\u4f53\u6765\u8bb2\u662f\u77f3\u5b50\u7684\u4f59\u91cf\n\/\/ \u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u77f3\u5b50\u4f59\u91cf\u548c\u64cd\u4f5c\u6b21\u6570\uff0c\u8f6e\u5230\u8c01\u64cd\u4f5c\u90fd\u6ca1\u6709\u5173\u7cfb\uff0c\u548c\u72b6\u6001\u7684\u51fa\u73b0\u4f4d\u7f6e\u7b49\u7b49\u4e5f\u6ca1\u6709\u5173\u7cfb\n\/\/ \u5c06\u672c\u4f8b\u62bd\u8c61\u6210\u6709\u5411\u65e0\u73af\u56fe\u4e4b\u540e\uff0c \u72b6\u6001\u4fe1\u606f\u53ea\u6709\u77f3\u5b50\u5b58\u91cf\uff0c \u6bcf\u5806\u77f3\u5b50\u90fd\u662f\u4e00\u4e2a\u72ec\u7acb\u7684\u56fe\n\/\/ sg(x)\u4e2d\u7684x\u5c31\u662f\u77f3\u5b50\u5b58\u91cf\nint n, k;\n\n\/\/ \u672c\u4f8b\u4e2d\u6c42\u53d6sg\u503c\u7684\u65b9\u6cd5\u662f\u8bb0\u5fc6\u5316\u641c\u7d22\nint sg(int x)\n{\n    if(f[x] != -1)\n        return f[x];\n\n    unordered_set<int> T; \/\/ T\u662fx\u70b9\u540e\u7ee7\u70b9\u7684sg\u503c\u96c6\u5408\n    for(int i = 1; i <= k; i++)\n    {\n        int d = s[i]; \/\/ \u53ef\u9009\u96c6\u5408\u7684\u7b2ci\u4e2a\u9009\u9879\uff1a\u62ffd\u4e2a\n        if(x >= d)\n        {\n            \/\/ \u5982\u679c\u53ef\u4ee5\u62ff\u5c31\u628a\u540e\u7ee7\u8282\u70b9\u7684sg\u503c\u52a0\u5165\u96c6\u5408\n            \/\/ \u8fd9\u6837\u770b\u6765\u5e94\u8be5\u662f\u540e\u5e8f\u904d\u5386\n            T.insert(sg(x - d));\n        }\n    }\n\n    \/\/ \u6c42sg\u503c\u7684\u90e8\u5206\n    \/\/ \u4e00\u4e2a\u70b9\u7684SG\u503c\u662f\u6700\u5c0f\u4e0d\u5c5e\u4e8e\u540e\u7ee7\u70b9SG\u503c\u7684\u81ea\u7136\u6570\n    \/\/ \u5982\u679c\u4ed6\u672c\u8eab\u662f\u7ec8\u70b9\uff0cSG\u503c\u5c31\u662f0\n    for(int i = 0; ; i++)\n        if(!T.count(i))\n            return f[x] = i; \/\/ \u8bb0\u5fc6\u5316\u4e0d\u80fd\u5fd8\u4e86\uff0c\u76f4\u63a5return i\u5c31\u98de\u4e86\n}\n\nint main()\n{\n    \/\/ ios::sync_with_stdio(false);\n    cin >> k;\n\n    for(int i = 1; i <= k; i++)\n        cin >> s[i];\n\n    int res = 0;\n    cin >> n;\n    memset(f, -1, sizeof f);\n    for(int i = 1; i <= n; i++)\n    {\n        int x;\n        cin >> x;\n        res ^= sg(x); \/\/ \u591a\u4e2a\u6709\u5411\u56fe\u6e38\u620f\u5fc5\u80dc\u5f53\u4e14\u4ec5\u5f53\u6240\u6709\u8d77\u70b9SG\u503c\u5f02\u6216\u548c\u4e0d\u4e3a0\n    }\n\n    if(res)\n        puts("Yes");\n    else\n        puts("No");\n\n    return 0;\n}<\/code><\/pre>\n","protected":false},"excerpt":{"rendered":"
    SG\u51fd\u6570\u662f\u89e3\u51b3\u535a\u5f08\u8bba\u95ee\u9898\u7684\u5229\u5668 \u5728\u6b63\u5f0f\u89e3\u9898\u4e4b\u524d\u9996\u5148\u9700\u8981\u4e86\u89e3\u4ee5\u4e0b\u77e5\u8bc6\uff1a \u6709\u5411\u56fe\u6e38\u620f […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6,7,10],"tags":[],"class_list":["post-457","post","type-post","status-publish","format-standard","hentry","category-nim","category-sg","category-10"],"_links":{"self":[{"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=\/wp\/v2\/posts\/457","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=457"}],"version-history":[{"count":2,"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=\/wp\/v2\/posts\/457\/revisions"}],"predecessor-version":[{"id":844,"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=\/wp\/v2\/posts\/457\/revisions\/844"}],"wp:attachment":[{"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=457"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=457"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=457"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}