Problem Link

741B Arpa’s weak amphitheater and Mehrdad’s valuable Hoses

Analysis

It’s a simple knapsack problem. Let’s solve this version of knapsack problem first: we have n sets of items, each item has value and weight, find the maximum value we can earn if we can choose at most one item from each set and sum of the chosen items must be less than or equal to W. Let dpw be the max value we can earn if sum of weights of chosen items are less than or equal to w. Now iterate on sets one by one and update dp as follows: for each item X, and for each weight w, newDpw = max(newDpw, oldDpw - X.weight + X.value).

Run dfs and find groups at first. The problem is same with above problem, each group is some set in above problem, just add whole group as an item to the set that related to this group.

Time complexity: O(n·W).

AC code

#include<cstdio>
#include<cstring>
#include<algorithm>
#include<iostream>
#include<vector>using namespace std;
typedef long long LL;
const LL maxn = 1e3+10;int uf[maxn];
int w[maxn],b[maxn];
int ww[maxn],bb[maxn];
int dp[2][maxn];
int vis[maxn];
int find(int x){return uf[x] == x? x:uf[x] = find(uf[x]);}
void UNION(int x,int y){uf[find(x)] = find(y);
}
int n,m,W;int main()
{//freopen("H:\\c++\\file\\stdin.txt","r",stdin);//freopen("H:\\c++\\file\\stdout.txt","w",stdout);cin>>n>>m>>W;for(int i=1 ; i<=n ; ++i)cin>>w[i];for(int i = 1 ; i<=n ; ++i)cin>>b[i];for(int i=1 ; i<=n ; ++i)uf[i] = i;while(m--){int x,y;cin>>x>>y;UNION(x,y);}memset(ww,0,sizeof(ww));memset(bb,0,sizeof(bb));for(int i=1 ; i<=n ; ++i){ww[find(i)]+=w[i];bb[find(i)]+=b[i];}memset(dp,0,sizeof(dp));memset(vis,false,sizeof(vis));int t = 0;for(int i=1 ; i<=n ; ++i){int pp = find(i);if(vis[pp])continue;vis[pp] = true;t^=1;for(int j = 1; j<=W ; ++j){dp[t][j] = dp[t^1][j];if(j>=ww[pp])dp[t][j] = max(dp[t][j],dp[t^1][j-ww[pp]]+bb[pp]);}for(int j = 1 ; j<=n ; ++j){if(find(j)!=pp)continue;for(int k = 1; k<=W ; ++k)if(k>=w[j])dp[t][k] = max(dp[t][k],dp[t^1][k-w[j]]+b[j]);}}int ans =0;for(int i=1;  i<=W ; ++i)ans = max(ans,dp[t][i]);cout<<ans<<endl;return 0;
}

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codeforce 741 B. Arpa's weak amphitheater and Mehrdad's valuable Hoses(背包 dp)

Problem Link

Analysis

AC code

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