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01背包

it2026-08-30  3

所有背包思想来自闫总的ACWing

问题链接：https://www.acwing.com/problem/content/2/ 思路：

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.edge-pattern-dashed{stroke-dasharray:3}#mermaid-svg-TaFKafRpsgTakoLE .edge-pattern-dotted{stroke-dasharray:2}#mermaid-svg-TaFKafRpsgTakoLE .marker{fill:#333}#mermaid-svg-TaFKafRpsgTakoLE .marker.cross{stroke:#333} :root { --mermaid-font-family: "trebuchet ms", verdana, arial;} #mermaid-svg-TaFKafRpsgTakoLE { color: rgba(0, 0, 0, 0.75); font: ; } Dp 状态表示 计算 从集合前i个进行选择，耗费不超过j的方案 属性：Max 第i个不选 变化部分:dp[i-1][j] 确定部分：dp[i][j] 第i个选 变化部分:dp[i-1][j-vi] 确定部分：dp[i-1][j-vi]+wi

dp公式：dp[i][j]=max{dp[i-1][j],dp[i-1][j-vi]+wi} 朴素代码：

#include<iostream> using namespace std; const int maxn=1e3+5; int v[maxn], w[maxn]; int N,V,dp[maxn][maxn]; int main(){ cin>>N>>V; for(int i=1;i<=N;i++) cin>>v[i]>>w[i]; for(int i=1;i<=N;i++){ for(int j=0;j<=V;j++){ dp[i][j]=dp[i-1][j];//左半边子集 if(j>=v[i]) dp[i][j]=max(dp[i][j],dp[i-1][j-v[i]]+w[i]); } } cout<<dp[N][V]; }

优化为一维数组的代码：

#include<iostream> using namespace std; int N,V,dp[1010]; int v[1010], w[1010]; int main(){ cin>>N>>V; for(int i=1;i<=N;i++) cin>>v[i]>>w[i]; for(int i=1;i<=N;i++){ for(int j=V;j>=0;j--){//从大到小循环为了确保dp[j-v[i]]为第i-1次的结果 if(j>=v[i]){ dp[j]=max(dp[j],dp[j-v[i]]+w[i]); } } } cout<<dp[V]; }
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