{"id":66,"date":"2025-01-11T23:08:03","date_gmt":"2025-01-11T15:08:03","guid":{"rendered":"http:\/\/150.230.63.147\/blog\/?p=66"},"modified":"2025-01-11T23:08:03","modified_gmt":"2025-01-11T15:08:03","slug":"b-once-again-%e8%a7%a3%e6%9e%90%e6%80%9d%e7%b6%ad%e3%80%81dp%e3%80%81lis%e3%80%81%e7%9f%a9%e9%99%a3%e5%86%aa","status":"publish","type":"post","link":"https:\/\/blog.seekrealthing.net\/blog\/?p=66","title":{"rendered":"B. Once Again\u2026 \u89e3\u6790(\u601d\u7dad\u3001DP\u3001LIS\u3001\u77e9\u9663\u51aa)"},"content":{"rendered":"\n

Codeforce 582 B. Once Again\u2026 \u89e3\u6790(\u601d\u7dad\u3001DP\u3001LIS\u3001\u77e9\u9663\u51aa)<\/h1>\n

\u4eca\u5929\u6211\u5011\u4f86\u770b\u770bCF582B
\n\u984c\u76ee\u9023\u7d50<\/a><\/p>\n

\n

\u984c\u76ee<\/strong>
\n\u7d66\u4f60\u4e00\u500b\u9577\u5ea6\u70ba$n$\u7684\u6578\u5217$a$\uff0c\u6c42$a$\u5faa\u74b0$T$\u6b21\u4ee5\u5f8c\u7684\u6700\u5927\u905e\u589e\u5b50\u5e8f\u5217(LIS)\u3002$n\\le100,T\\le10^7$<\/p>\n<\/blockquote>\n

\u524d\u8a00<\/h3>\n

\u9019\u984c\u5be6\u5728\u662f\u641e\u4e86\u975e\u5e38\u975e\u5e38\u4e45\uff0c\u7d93\u9a57\u904e\u65bc\u4e0d\u8db3\uff0c\u7121\u8ad6\u662f\u77e9\u9663\u5feb\u901f\u51aa\u7684\u5947\u602a$dp$\u5f0f\u6216\u8005\u662f\u504f\u601d\u7dad\u4f5c\u6cd5\uff0c\u90fd\u641e\u4e0d\u592a\u51fa\u4f86<\/p>\n

<div class=\u201cVVVcopyrightAAA\u201d;>@copyright petjelinux \u7248\u6b0a\u6240\u6709<br>
\n<a href=\u201chttps:\/\/www.cnblogs.com\/petjelinux\/\u201d>\u89c0\u770b\u66f4\u591a\u6b63\u7248\u539f\u59cb\u6587\u7ae0\u8acb\u81f3petjelinux\u7684blog<\/a><\/div><\/p>\n

\u60f3\u6cd5<\/h3>\n

\u9996\u5148\uff0c\u8981\u4f86\u8b1b\u4e00\u4e0b$LIS$\u7684\u4f5c\u6cd5:<\/p>\n

    \n
  1. $dp:$ \u6211\u5011\u7dad\u8b77\u4ee5\u7b2c$i$\u500b\u5143\u7d20\u70ba\u7d50\u5c3e\u7684$LIS$\u9577\u5ea6\u70ba$dp[i]$\uff0c\u8f49\u79fb\u5f0f$:dp[i]=\\max\\limits_{j=0,a[j]\\le a[i]}^n{dp[i],dp[j]+1}$ \u4e14\u8981\u8003\u616e$LIS$\u53ea\u6709\u55ae\u4e00\u500b\u5143\u7d20\u6642\u9577\u5ea6\u70ba1\uff0c\u6240\u4ee5$dp[i]$\u6709\u500b\u6700\u5c0f\u503c$1$\u3002
    \n\u8907\u96dc\u5ea6:$O(n^2)$<\/li>\n
  2. \u4e8c\u5206\u641c$:$ \u6211\u5011\u7dad\u8b77\u4e00\u500b$stack$\uff0c\u4e14\u6bcf\u6b21\u52a0\u5165\u4e00\u500b\u5143\u7d20\u8003\u616e$LIS$\u3002$stack$\u9802\u7aef\u653e\u76ee\u524d\u7dad\u8b77\u7684$LIS$\u7684\u6700\u5f8c\u4e00\u500b\u5143\u7d20\uff0c\u5982\u679c\u65b0\u5143\u7d20$\\ge$\u9802\u7aef\u7684\u5143\u7d20\uff0c\u90a3\u5c31\u52a0\u5165\u3002\u5982\u679c\u65b0\u5143\u7d20$<$\u9802\u7aef\u7684\u5143\u7d20\uff0c\u90a3\u9ebc\u6211\u5011\u53ef\u4ee5\u4e8c\u5206\u641c\u76ee\u524d\u7dad\u8b77\u7684$LIS$\u4e2d\u7b2c\u4e00\u500b\u5927\u65bc\u65b0\u5143\u7d20\u7684\u5143\u7d20\uff0c\u4e26\u4e14\u66ff\u63db\u70ba\u65b0\u5143\u7d20\uff0c\u5982\u6b64\u4e00\u4f86\uff0c\u56e0\u70ba\u6211\u5011\u63db\u4e86\u4e00\u500b\u6bd4\u8f03\u5c0f\u7684\u5143\u7d20\u4e0a\u53bb\uff0c\u4e14\u4e0d\u5f71\u97ff\u76ee\u524d\u7684<\/strong>$LIS$\u9577\u5ea6\uff0c\u56e0\u6b64\u6574\u500b$stack$\u5728\u4e4b\u5f8c\u80fd\u7372\u5f97\u66f4\u597d\u7684\u905e\u589e\u5b50\u5e8f\u5217\u7684\u300c\u6f5b\u529b<\/strong>\u300d\u5c31\u8b8a\u9ad8\u4e86\u3002\u975e\u5e38\u5efa\u8b70\u76f4\u63a5\u770bcode\u601d\u8003\u70ba\u4ec0\u9ebc\u9019\u6a23\u505a<\/strong>
    \n\u8907\u96dc\u5ea6:$O(n\\lg(n))$<\/li>\n<\/ol>\n
    int LIS(){  \n  int sta[_n*_n],top=0;  \n  sta[top++]=a[0];  \n  rep(i,1,nn){  \n    if(a[i]>=sta[top-1])sta[top++]=a[i];  \n    else *upper_bound(sta,sta+top,a[i])=a[i];  \n  }  \n  return top;  \n}  \n<\/code><\/pre>\n
    \u56de\u6b78\u6b63\u984c\uff0c\u9019\u984c\u6709\u5169\u7a2e\u505a\u6cd5\uff0c\u504f\u601d\u7dad\u4f5c\u6cd5\u6216\u8005\u662f\u5947\u602a$dp$\u5f0f\u505a\u6cd5\u3002<\/p>\n
      \n
    1. \n
      \u504f\u601d\u7dad\u4f5c\u6cd5
      \n\u7562\u7adf\u5faa\u74b0($T$)\u592a\u9577\u4e86\uff0c\u800c$n$\u53c8\u5f88\u5c0f\uff0c\u6240\u4ee5\u81ea\u7136\u5730\u611f\u89ba\u61c9\u8a72\u6703\u7528\u5230\u7d93\u5178\u7684$LIS$(\u6700\u5927\u905e\u589e\u5b50\u5e8f\u5217)\u554f\u984c\u518d\u52a0\u4e0a\u4e00\u4e9b\u89c0\u5bdf\u3002
      \n\u6ce8\u610f\u5230\uff0c\u6700\u6162\u6700\u6162\u5728\u6578\u5217$a$\u5faa\u74b0$n$\u6b21\u4ee5\u5f8c\uff0c$LIS$\u6703\u91cd\u8907\u67d0\u500b\u5b57\u5143\u3002\u4e5f\u5c31\u662f\u5982\u679c\u6bcf\u6b21\u5faa\u74b0\u6211\u5011\u90fd\u53d6\u4e00\u500b\u76f8\u7570\u7684\u5143\u7d20\uff0c\u90a3\u9ebc$n$\u6b21\u5faa\u74b0\u4ee5\u5f8c\u5c31\u4e0d\u53ef\u80fd\u518d\u53d6\u5230\u76f8\u7570\u7684\u5143\u7d20\u4e86\u3002\u800c\u56e0\u6b64\uff0c\u6211\u5011\u6703\u60f3\u8981\u53bb\u7dad\u8b77\u9577\u5ea6\u70ba$n^2$\u7684$LIS$(\u5982\u679c$T>n$)\u3002
      \n\u73fe\u5728\uff0c\u5982\u679c$T\\le n$\uff0c\u90a3\u9ebc\u6211\u5011\u53ea\u8981\u66b4\u529b\u7b97\u51fa$LIS$\u5c31\u884c\uff0c\u7562\u7adf$n$\u5f88\u5c0f\u3002
      \n\u5982\u679c$T>n$\uff0c\u89c0\u5bdf\u5230\uff0c\u56e0\u70ba$T>n$\uff0c\u6240\u4ee5\u6211\u5011\u53ef\u4ee5\u60f3\u50cf\u5f97\u51fa\uff0c\u6574\u4e32$LIS$\u4e00\u5b9a\u6709\u975e\u5e38\u591a\u91cd\u8907\u7684\u6578\u5b57\uff0c\u6574\u4e32$LIS$\u6703\u5148\u905e\u589e\uff0c\u7136\u5f8c\u5230\u67d0\u500b\u6578\u5b57\u958b\u59cb\u91cd\u8907\uff0c\u63a5\u8457\u5728\u5c3e\u7aef\u7e7c\u7e8c\u905e\u589e\u3002
      \n\u90a3\u9ebc\u6211\u5011\u53ea\u8981\u5148\u7b97\u51fa\u6578\u5217$a$\u4e2d\u6700\u591a\u91cd\u8907\u51fa\u73fe\u5e7e\u6b21\u540c\u6a23\u7684\u6578\u5b57\uff0c\u4e26\u4e14\u7b97\u51fa\u9577\u5ea6$n^2$\u7684$LIS$\u4ee5\u5f8c\uff0c\u7531\u65bc\u6211\u5011\u77e5\u9053\u7576$T>n$\u6642\uff0c\u81f3\u5c11\u6703\u6709$T-n$\u6b21$a$\u6578\u5217\u7684\u5faa\u74b0\u6703\u8ca2\u737b\u7d66$LIS$\u540c\u6a23\u7684\u6578\u5b57\uff0c\u56e0\u6b64\uff0c\u89e3\u7b54\u5c31\u662f:$LIS(\u9577\u5ea6n^2\u7684)+(T-n)\\times(\u51fa\u73fe\u6b21\u6578\u6700\u591a\u7684\u6578\u5b57)$\u3002(\u60f3\u6cd5\u4f86\u6e90<\/a>)
      \n\u9084\u6709\u53e6\u4e00\u500b\u53ef\u80fd\u6703\u60f3\u5230\u7684\u4f5c\u6cd5\uff0c\u4f46\u662f\u8907\u96dc\u5ea6\u6703\u9ad8\u4e00\u4e9b\uff0c\u76f4\u63a5\u653e      \u9023\u7d50<\/a><\/p>\n<\/li>\n
    2. \n
      \u5947\u602a$dp$\u5f0f\u505a\u6cd5:
      \n\u5176\u5be6\u9019\u500b\u505a\u6cd5\u662f\u77e9\u9663\u5feb\u901f\u51aa\uff0c\u4f46\u6211\u4e0d\u77e5\u9053\u70ba\u4ec0\u9ebc\u60f3\u5f97\u5230\u9019\u7a2e\u72c0\u614b
      \n\u8003\u616e$dp$\u72c0\u614b: $dp[i][j]$\u8868\u793a\u300c\u76ee\u524d\u70ba\u6b62\u8003\u616e\u904e\u7684\u5faa\u74b0\u7bc0\u300d(\u8003\u616e\u904e\u5e7e\u500b\u5faa\u74b0\u7bc0\u4e26\u6c92\u6709\u5728\u72c0\u614b\u4e2d)\u7684\uff0c\u958b\u982d\u5143\u7d20$\\ge a[i]$\u4e14\u7d50\u5c3e\u70ba\u6700\u5f8c\u4e00\u500b\u5faa\u74b0\u7684\u7b2c$j$\u500b\u5143\u7d20\u7684$LIS$\u9577\u5ea6\u3002
      \n\u800c\u6703\u6709\u4e00\u500b\u8f49\u79fb\u5f0f$:dp_{p+q}[i][j]=\\max\\limits_{k=0}^n{dp_p[i][k]+dp_q[k][j]}$(\u5c31\u662f\u628a\u5169\u500b\u5c0d\u65bc(\u4e0d\/\u76f8)\u540c\u9577\u5ea6\u5faa\u74b0\u7bc0\u7684$LIS$\u63a5\u8d77\u4f86)
      \n\u5176\u4e2d$dp_p$\u8868\u793a\u76ee\u524d\u9019\u500b$dp$\u72c0\u614b\u8003\u616e\u7684\u6578\u5217\u9577\u5ea6\u662f$p\\times n$(\u4e5f\u5c31\u662f$p$\u500b\u5faa\u74b0\u7bc0)
      \n\u6211\u5011\u60f3\u8981\u6c42\u7684\u662f$dp_{nT}$\uff0c\u800c\u6211\u5011\u53ef\u4ee5\u7528\u6a38\u7d20\u7684$dp$\u7b97\u6cd5($O(n^2)$\u7684\u65b9\u6cd5)\u4f86\u7b97\u51fa$dp_1$\u3002
      \n\u628a$dp[][]$\u770b\u6210\u662f\u77e9\u9663\uff0c\u800c\u4e00\u6b21\u8f49\u79fb\u770b\u6210\u662f\u4e00\u6b21\u77e9\u9663\u4e58\u6cd5\uff0c\u90a3\u9ebc\u5c31\u53ef\u4ee5\u7528\u77e9\u9663\u5feb\u901f\u51aa\u7b97\u51fa\u4f86\u4e86\u3002
      \n\u8981\u6ce8\u610f\u5982\u679c\u5728\u7b97$dp[i][j]$\u6642\u5982\u679c$a[i]>a[j]$\u4ee3\u8868\u9019\u7a2e\u53ef\u80fd\u4e0d\u53ef\u80fd\u767c\u751f\uff0c\u8981\u7d66\u4e00\u500b\u6975\u5c0f\u503c\u3002<\/p>\n<\/li>\n<\/ol>\n
      \u8907\u96dc\u5ea6:\u504f\u601d\u7dad\u4f5c\u6cd5\u4e2d\u8f03\u6162\u7684$:O(n^4)$\u3001\u504f\u601d\u7dad\u4f5c\u6cd5\u4e2d\u8f03\u5feb\u7684\u60f3\u6cd5\u4f46\u662f\u6c92\u7528\u4e8c\u5206\u641c$LIS:O(n^4)$\u3001\u77e9\u9663\u5feb\u901f\u51aa$:O(n^3\\lg(T))$\u3001\u504f\u601d\u7dad\u4f5c\u6cd5\u4e2d\u8f03\u5feb\u7684\u60f3\u6cd5\u4e14\u6709\u7528\u4e8c\u5206\u641c$:O(n^2\\lg(n^2))$<\/p>\n
      \u7a0b\u5f0f\u78bc(\u504f\u601d\u7dad\u4f5c\u6cd5\u4e2d\u8f03\u6162\u7684):<\/h3>\n
      const int _n=110;  \nint tt,a[_n*_n],st[_n*_n],ed[_n*_n],num[310];  \nll n,nn,t;  \nmain(void) {ios_base::sync_with_stdio(0);cin.tie(0);cout.tie(0);  \n  cin>>n>>t;rep(i,0,n){cin>>a[i];num[a[i]]++;} nn=min(n*n,n*t);  \n  rep(i,n,nn)a[i]=a[i%n];  \n  rep(i,0,nn){ed[i]=1;{rep(j,0,i)if(a[j]<=a[i])ed[i]=max(ed[i],ed[j]+1);}}  \n  per(i,0,nn){st[i]=1;{per(j,i+1,nn)if(a[j]>=a[i])st[i]=max(st[i],st[j]+1);}}  \n  int maxx=-1;  \n  if(t<n)rep(i,0,nn)maxx=max(maxx,ed[i]);  \n  else rep(i,0,nn)maxx=max(maxx,ed[i]+st[i]-1+num[a[i]]*(t-n));  \n  cout<<maxx<<'\\n';  \n  return 0;  \n}  \n<\/code><\/pre>\n
      \u6a19\u982d\u3001\u6a21\u677f\u8acb\u9edeSubmission\u770b
      \n      Submission<\/a><\/p>\n
      \u7a0b\u5f0f\u78bc(\u504f\u601d\u7dad\u4f5c\u6cd5\u4e2d\u8f03\u5feb\u7684\u60f3\u6cd5\u4f46\u662f\u6c92\u7528\u4e8c\u5206\u641c):<\/h3>\n
      const int _n=110;  \nint tt,a[_n*_n],dp[_n*_n],num[310];  \nll n,nn,t,k;  \nmain(void) {ios_base::sync_with_stdio(0);cin.tie(0);cout.tie(0);  \n  cin>>n>>t;rep(i,0,n){cin>>a[i];num[a[i]]++;k=max(k,num[a[i]]);}  \n  nn=min(n*n,n*t);  \n  rep(i,n,nn)a[i]=a[i%n]; dp[0]=1;  \n  rep(i,1,nn){  \n    dp[i]=1;  \n    rep(j,0,i)if(a[j]<=a[i])dp[i]=max(dp[i],dp[j]+1);  \n  }  \n  if(t>n){  \n    int ans=k*(t-n);  \n    int maxx=0;rep(i,0,nn)maxx=max(maxx,dp[i]);  \n    ans+=maxx;  \n    cout<<ans<<'\\n';  \n  }else{  \n    int maxx=0;rep(i,0,nn)maxx=max(maxx,dp[i]);  \n    cout<<maxx<<'\\n';  \n  }  \n  return 0;  \n}  \n<\/code><\/pre>\n
      \u6a19\u982d\u3001\u6a21\u677f\u8acb\u9edeSubmission\u770b
      \n      Submission<\/a><\/p>\n
      \u7a0b\u5f0f\u78bc(\u77e9\u9663\u5feb\u901f\u51aa):<\/h3>\n
      const int _n=110;  \nint t,n,a[_n];  \nstruct mat{  \n  int a[_n][_n];  \n  mat(){memset(a,0,sizeof a);}  \n  mat operator*(const mat& rhs)const{  \n    mat res;  \n    rep(i,0,n)rep(j,0,n){  \n      res.a[i][j]=-1e5;  \n      rep(k,0,n)res.a[i][j]=max(res.a[i][j],a[i][k]+rhs.a[k][j]);  \n    }  \n    return res;  \n  }  \n  mat operator^(int b){  \n    mat res,tmp=*this;  \n    while(b){  \n      if(b&1)res=res*tmp;  \n      b>>=1;tmp=tmp*tmp;  \n    }  \n    return res;  \n  }  \n};  \nmain(void) {ios_base::sync_with_stdio(0);cin.tie(0);cout.tie(0);  \n  cin>>n>>t;rep(i,0,n)cin>>a[i]; mat ans;  \n  rep(i,0,n)rep(j,0,n){  \n    if(a[i]>a[j]){ans.a[i][j]=-1e5;continue;}  \n    ans.a[i][j]=1;  \n    rep(k,0,j)if(a[k]<=a[j])ans.a[i][j]=max(ans.a[i][j],ans.a[i][k]+1);  \n  }ans=ans^t;  \n  int maxx=0;rep(i,0,n)rep(j,0,n)maxx=max(maxx,ans.a[i][j]);  \n  cout<<maxx<<'\\n';  \n  return 0;  \n}  \n<\/code><\/pre>\n
      \u6a19\u982d\u3001\u6a21\u677f\u8acb\u9edeSubmission\u770b
      \n      Submission<\/a><\/p>\n
      \u7a0b\u5f0f\u78bc(\u504f\u601d\u7dad\u4f5c\u6cd5\u4e2d\u8f03\u5feb\u7684\u60f3\u6cd5\u4e14\u6709\u7528\u4e8c\u5206\u641c):<\/h3>\n
      const int _n=110;  \nint tt,a[_n*_n],num[310];  \nll n,nn,t,k;  \nint LIS(){  \n  int sta[_n*_n],top=0;  \n  sta[top++]=a[0];  \n  rep(i,1,nn){  \n    if(a[i]>=sta[top-1])sta[top++]=a[i];  \n    else *upper_bound(sta,sta+top,a[i])=a[i];  \n  }  \n  return top;  \n}  \nmain(void) {ios_base::sync_with_stdio(0);cin.tie(0);cout.tie(0);  \n  cin>>n>>t;rep(i,0,n){cin>>a[i];num[a[i]]++;k=max(k,num[a[i]]);}  \n  nn=min(n*n,n*t);  \n  rep(i,n,nn)a[i]=a[i%n];  \n  if(t>n){  \n    int ans=k*(t-n);  \n    ans+=LIS();  \n    cout<<ans<<'\\n';  \n  }else cout<<LIS()<<'\\n';  \n  return 0;  \n}  \n<\/code><\/pre>\n
      \u6a19\u982d\u3001\u6a21\u677f\u8acb\u9edeSubmission\u770b
      \n      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":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","enabled":false},"version":2}},"categories":[3],"tags":[4,26,8,27],"class_list":["post-66","post","type-post","status-publish","format-standard","hentry","category-ojcode","tag-dp","tag-lis","tag-8","tag-27"],"jetpack_publicize_connections":[],"yoast_head":"\nB. 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