{"id":81,"date":"2026-01-14T21:39:35","date_gmt":"2026-01-14T13:39:35","guid":{"rendered":"https:\/\/tblog.zeabur.app\/?p=81"},"modified":"2026-01-14T21:40:22","modified_gmt":"2026-01-14T13:40:22","slug":"%e9%a2%98%e8%a7%a3-acgo-a67278-%e6%b1%89%e6%98%8e%e8%b7%9d%e7%a6%bb%e2%85%b1","status":"publish","type":"post","link":"https:\/\/tblog.zeabur.app\/?p=81","title":{"rendered":"\u9898\u89e3 ACGO A67278 \u6c49\u660e\u8ddd\u79bb\u2161"},"content":{"rendered":"\n
\n

\u5728\u770b\u9898\u89e3\u524d\uff0c\u5efa\u8bae\u81ea\u5df1\u5148\u5c1d\u8bd5\u5206\u6790\u672c\u9898\uff0c\u4e2a\u4eba\u8ba4\u4e3a\u8fd8\u662f\u633a\u6709\u610f\u4e49\u7684\u3002<\/p>\n<\/blockquote>\n\n\n\n

\u9898\u76ee\u5206\u6790<\/h2>\n\n\n\n

\u6839\u636e\u6c49\u660e\u8ddd\u79bb\u7684\u5b9a\u4e49:
$\\displaystyle d_H(x,y)=\\sum_{i=0}^{\\infty}I((x\\&2^i)\\neq(y\\&2^i))$
\u4e0d\u96be\u5f97\u51fa\uff0c\u9898\u76ee\u5927\u610f\u5c31\u662f\u5728\u6c42\u533a\u95f4 $[0,n]$ \u5185\u6709\u591a\u5c11\u4e2a\u6570\u5bf9 $(x,y)$ \u6ee1\u8db3\uff1a$0\\le x<y\\le n,\\ d_H(x,y)\\le k$\u3002
\u5982\u679c\u76f4\u63a5\u66b4\u529b\uff0c\u5728 $n\\le 10^6$ \u7684\u6761\u4ef6\u4e0b\u4e00\u5b9a\u8d85\u65f6\uff0c\u4f46\u5728 $10^6$ \u7684\u60c5\u51b5\u4e0b\uff0c\u4e0d\u96be\u53d1\u73b0\u5982\u679c\u6309\u4f4d\u7edf\u8ba1\u4e8c\u8fdb\u5236\u4f4d\u4e0a\u7684 $1$ \u7684\u4e2a\u6570\uff0c\u4e0d\u4ec5\u53ef\u4ee5\u8ba1\u7b97\u6570\u5bf9\u95f4\u7684\u6c49\u660e\u8ddd\u79bb\uff0c\u4e5f\u53ea\u9700\u8981 $\\mathcal{O}(|n|)$ \u7684\u590d\u6742\u5ea6\uff0c\u518d\u52a0\u4e0a\u4e8c\u8fdb\u5236\u4f4d\u8fd0\u7b97\u4e0e\u6768\u8f89\u4e09\u89d2\uff08\u7ec4\u5408\u6570\uff09\u7684\u4f18\u5316\uff0c\u5b8c\u5168\u53ef\u4ee5\u5728 $\\mathcal{O}(t)$ \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e0b\u8dd1\u5b8c\u3002
\u56e0\u4e3a\u8981\u6c42\u7edf\u8ba1\u6c49\u660e\u8ddd\u79bb $d_H\\le k$ \u7684\u65b9\u6848\u6570\uff0c\u53ef\u4ee5\u60f3\u5230\u8fd0\u7528\u7ec4\u5408\u6570\u6765\u8fdb\u884c\u5feb\u901f\u8ba1\u7b97\uff0c\u53c8\u7531\u4e8e\u6bcf\u4e00\u6b21\u90fd\u662f\u7edf\u8ba1\u4ece $0$ \u81f3 $n$ \u7684\u60c5\u51b5\uff0c\u53ef\u4ee5\u901a\u8fc7\u9884\u5904\u7406\u7ec4\u5408\u6570\u7684\u524d\u7f00\u548c\u6765\u5b9e\u73b0 $\\mathcal{O}(1)$ \u67e5\u8be2\u3002<\/p>\n\n\n\n

\u524d\u7f6e\u829d\u58eb<\/h2>\n\n\n\n

\u7ec4\u5408\u6570\u5b66<\/h3>\n\n\n\n

\u7ec4\u5408\uff1a\u4ecen\u4e2a\u4e0d\u540c\u5143\u7d20\u4e2d\u53d6\u51fa $m\uff08m\\le n\uff09$ \u4e2a\u5143\u7d20\u5e76\u6210\u4e00\u7ec4\uff0c\u4e0d\u8003\u8651\u987a\u5e8f\u7684\u65b9\u6848\u6570\u7528\u7b26\u53f7 $C_n^m$ \u6216 $\\binom{n}{m}$ \u8868\u793a\u3002<\/p>\n\n\n\n

\u7ec4\u5408\u6570\u6768\u8f89\u4e09\u89d2\u4f18\u5316<\/h3>\n\n\n\n

\u9012\u63a8\u516c\u5f0f\uff1a$\\binom{n}{m}=\\binom{n-1}{m-1}+\\binom{n-1}{m}$\u3002
\u5176\u5b9e\u4e5f\u4e0d\u80fd\u7b97\u662f\u4f18\u5316\u3002<\/p>\n\n\n\n

\u524d\u7f00\u548c<\/h3>\n\n\n\n

\u8fd9\u5e94\u8be5\u4e0d\u7528\u8bf4\u4e86\u5427\u3002<\/p>\n\n\n\n

\u6a21\u9006\u5143<\/h3>\n\n\n\n

\u4e3a\u4e86\u5b9e\u73b0\u6b63\u786e\u8ba1\u7b97\u5728\u6a21\u8fd0\u7b97\u4e0b\u7684\u9664\u6cd5\u8ba1\u7b97\uff0c\u9700\u8981\u4f7f\u7528\u5bf9\u5e94\u6a21\u9006\u5143\u7684\u4e58\u6cd5\u6765\u8fbe\u6210\uff08\u53c2\u8003\u6a21\u53cd\u5143\u7d20<\/a>\uff09\u3002<\/p>\n\n\n\n

\u4ee3\u7801\u5b9e\u73b0<\/h2>\n\n\n\n
#include<bits\/stdc++.h>\n#define int long long\nusing namespace std;\nconst int MOD=998244353;\/\/\u6a21\u6570\nconst int inv=(MOD+1)\/2;\/\/\u9664 2 \u7684\u6a21\u9006\u5143\nint C[25][25],pref[25][25];\/\/C \u4e3a\u7ec4\u5408\u6570\uff0cpref \u8bb0\u5f55\u4e86\u524d\u7f00\u548c\ninline int cntb(int  n,int b)\/\/\u7528\u4e8e\u7edf\u8ba1\u4e8c\u8fdb\u5236\u4f4d\u4e0a\u6570\u7684\u4e2a\u6570\uff08\u6216\u8005\u8bf4\u8d21\u732e\uff09\n{\n    int l=1LL<<(b+1),h=1LL<<b;\n    int r=(n+1)%l;\n    int f=n-r+1;\n    if(r>h)f+=(r-h)*2;\n    return f;\n}\ninline int addmod(int a,int b)\/\/\u52a0\u6cd5\u53d6\u6a21\uff08\u4e3b\u8981\u662f\u6015\u7206\u4e86\uff0c\u867d\u7136\u51e0\u4e4e\u4e0d\u53ef\u80fd\uff09\n{\n    a+=b;\n    if(a>=MOD)a-=MOD;\n    return a;\n}\ninline int mulmod(int a,int b)\/\/\u4e58\u6cd5\u53d6\u6a21\n{\n    return (a%MOD)*(b%MOD)%MOD;\n}\n\/*\ninline int mulmod(int a,int b)\/\/\u672c\u6765\u60f3\u7528\u8fd9\u4e00\u7248\u66f4\u5b89\u5168\u7684\u53d6\u6a21\u65b9\u5f0f\u7684\uff0c\u7ed3\u679c\u590d\u6742\u5ea6\u592a\u9ad8\uff0cTLE \u4e86\uff0cQwQ\n{\n    a=a%MOD;\n    b=b%MOD;\n    int res=0;\n    while(b)\n    {\n        if(b&1)res=(res+a)%MOD;\n        a=(a+a)%MOD;\n        b>>=1;\n    }\n    return res;\n}\n*\/\ninline void init()\/\/\u9884\u5904\u7406\u7ec4\u5408\u6570\u4e0e\u524d\u7f00\u548c\n{\n    for(int i=0;i<=21;i++)\n    {\n        C[i][0]=C[i][i]=1;\n        for(int j=1;j<i;j++)\n        {\n            C[i][j]=(C[i-1][j-1]+C[i-1][j])%MOD;\n        }\n    }\n    for(int i=0;i<=20;i++)\n    {\n        pref[i][0]=0;\n        for(int j=1;j<=21;j++)\n        {\n            int minn=min(i,j-1);\n            if(minn>=0)\n            {\n                int t=0;\n                for(int k=0;k<=minn;k++)t=(t+C[i][k])%MOD;\n                pref[i][j]=t;\n            }\n            else pref[i][j]=0;\n        }\n    }\n}\nsigned main() \n{\n    init();\n    int t;\n    cin>>t;\n    while(t--)\n    {\n        int n,k;\n        cin>>n>>k;\n        k=min(k,21LL);\n        int ans=(n+1)%MOD;\n        for(int b=0;b<=20;b++)\n        {\n            ans=addmod(ans,mulmod(cntb(n,b)%MOD,pref[b][k]%MOD));\/\/\u7528\u4e00\u4e0b\u53d6\u6a21\u5373\u53ef\n        }\n        ans-=((n+1)%MOD);\n        \/\/while(ans<0)ans+=MOD;\/\/\u9632\u6b62\u7206\u6210\u8d1f\u6570\u4e86\uff0c\u5176\u5b9e\u4e0d\u52a0\u4e5f\u53ef\u4ee5\uff0c\u4f46\u6bd4\u8d5b\u7684\u65f6\u5019\u6015\u524d\u9762\u591a\u51cf\u4e86\uff0c\u8fd9\u91cc\u5224\u4e00\u4e0b\n        ans=mulmod(ans,inv);\n        cout<<ans%MOD<<endl;\/\/\u4e4b\u540e\u5c31\u662f\u5feb\u4e50\u7684\u8f93\u51fa\u4e86\n    }\n    return 0;\n}<\/code><\/pre>\n\n\n\n
\u770b\u7740\u601d\u8def\u5f88\u7b80\u5355\uff0c\u53ef\u662f\u6211\u597d\u51e0\u4e2a\u5c0f\u65f6\u7684\u6210\u679c\uff0cQwQ\uff0c\u82e5\u54ea\u91cc\u601d\u8def\u9519\u4e86\u6216\u6ca1\u8bb2\u6e05\u695a\uff0c\u52ff\u55b7\u3002<\/p>\n\n\n\n
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