poj3728之离线LCA+dp思想/RMQ+LCA(非常好的题目)

The merchant

Time Limit: 3000MS Memory Limit: 65536KTotal Submissions: 2740 Accepted: 913

Description

There are N cities in a country, and there is one and only one simple path between each pair of cities. A merchant has chosen some paths and wants to earn as much money as possible in each path. When he move along a path, he can choose one city to buy some goods and sell them in a city after it. The goods in all cities are the same but the prices are different. Now your task is to calculate the maximum possible profit on each path.

Input

The first line contains N, the number of cities.
Each of the next N lines contains w_i the goods' price in each city.
Each of the next N-1 lines contains labels of two cities, describing a road between the two cities.
The next line contains Q, the number of paths.
Each of the next Q lines contains labels of two cities, describing a path. The cities are numbered from 1 to N.

1 ≤ N, w_i, Q ≤ 50000 

Output

The output contains Q lines, each contains the maximum profit of the corresponding path. If no positive profit can be earned, output 0 instead.

Sample Input

41 5 3 21 33 23 491 21 31 42 32 12 43 13 23 4

Sample Output

422000020

本体非常好，建议多做几遍，分别用离线LCA+dp和在线LCA+RMQ做个几遍

以下是分析+代码：

LCA+DP

/*分析:先求出点u,v的最近公共祖先f,然后求u->f->v的利润最大值maxval对于这个maxval可能有三种情况:1:maxval是u->f的maxval2:maxval是f->v的maxval3:maxval是u->f的最小w[i]减去f->v的最大w[i]分析到这很明显需要设置4个变量来求maxval:up[u]表示u->f的最大maxvaldown[u]表示f->u的最大maxvalmaxw[u]表示u-f的最大w[i]minw[u]表示u-f的最小w[i]所以maxval=max(max(up[u],down[v]),maxw[v]-minw[u]);现在问题就是如何快速的求出这四个变量,在这里我们可以对u,v的LCA(u,v)进行分类解决对于LCA(u,v)是f的询问全部求出,然后再求LCA(u,v)是f的父亲的询问这样当我们求LCA(u,v)是f的父亲的询问的时候就可以借用已经求出的LCA(u,v)是f的询问 的结果,这样就不用反复去求u->f的那四个变量值,u->father[f]也能快速求出 这个变化主要在寻找father[v]这个过程中进行,具体看代码 */#include <iostream>#include <cstdio>#include <cstdlib>#include <cstring>#include <string>#include <queue>#include <algorithm>#include <map>#include <cmath>#include <iomanip>#define INF 99999999typedef long long LL;using namespace std;const int MAX=50000+10;int n,m,size;int uu[MAX],vv[MAX],ww[MAX],sum[MAX];int up[MAX],down[MAX],maxw[MAX],minw[MAX],father[MAX];int head[MAX],head2[MAX],head3[MAX];bool mark[MAX];struct Edge{int v,id,next;Edge(){}Edge(int V,int ID,int NEXT):v(V),id(ID),next(NEXT){}}edge[MAX*2],edge2[MAX*2],edge3[MAX*2];void Init(int num){for(int i=0;i<=num;++i)head[i]=head2[i]=head3[i]=-1,mark[i]=false;size=0; }void InsertEdge(int u,int v,int id){edge[size]=Edge(v,id,head[u]);head[u]=size++;}void InsertEdge2(int u,int v,int id){edge2[size]=Edge(v,id,head2[u]);head2[u]=size++;}void InsertEdge3(int u,int v,int id){edge3[size]=Edge(v,id,head3[u]);head3[u]=size++;}int findset(int v){if(v == father[v])return father[v];int fa=father[v];father[v]=findset(father[v]);up[v]=max(max(up[v],up[fa]),maxw[fa]-minw[v]);down[v]=max(max(down[v],down[fa]),maxw[v]-minw[fa]);maxw[v]=max(maxw[v],maxw[fa]);minw[v]=min(minw[v],minw[fa]);return father[v];}void LCA(int u){mark[u]=true;father[u]=u;for(int i=head2[u];i != -1;i=edge2[i].next){//对LCA(u,v)进行分类 int v=edge2[i].v,id=edge2[i].id;if(!mark[v])continue;int f=findset(v);InsertEdge3(f,v,id);}for(int i=head[u];i != -1;i=edge[i].next){int v=edge[i].v;if(mark[v])continue;LCA(v);father[v]=u;}for(int i=head3[u];i != -1;i=edge3[i].next){int id=edge3[i].id;findset(uu[id]);findset(vv[id]);sum[id]=max(max(up[uu[id]],down[vv[id]]),maxw[vv[id]]-minw[uu[id]]);}}int main(){int u,v;while(~scanf("%d",&n)){Init(n);for(int i=1;i<=n;++i){scanf("%d",ww+i);up[i]=down[i]=0;maxw[i]=minw[i]=ww[i];}for(int i=1;i<n;++i){scanf("%d%d",&u,&v);InsertEdge(u,v,i);InsertEdge(v,u,i); }size=0;scanf("%d",&m);for(int i=0;i<m;++i){scanf("%d%d",&uu[i],&vv[i]);InsertEdge2(uu[i],vv[i],i);InsertEdge2(vv[i],uu[i],i); }size=0;LCA(1);for(int i=0;i<m;++i)printf("%d\n",sum[i]);}return 0;}

RMQ+LCA:

#include <iostream>#include <cstdio>#include <cstdlib>#include <cstring>#include <string>#include <queue>#include <algorithm>#include <map>#include <cmath>#include <iomanip>#define INF 99999999typedef long long LL;using namespace std;const int MAX=50000+10;int n,m,size,top;int uu[MAX],vv[MAX],ww[MAX],anc[MAX];int up[MAX][20],down[MAX][20],maxw[MAX][20],minw[MAX][20],deep[MAX];int head[MAX],head2[MAX],bin[MAX],stack[MAX],mp[MAX][20],father[MAX];bool mark[MAX];struct Edge{int v,id,next;Edge(){}Edge(int V,int ID,int NEXT):v(V),id(ID),next(NEXT){}}edge[MAX*2],edge2[MAX*2];void Init(int num){for(int i=0;i<=num;++i)head[i]=head2[i]=-1,mark[i]=false;size=top=0;}void InsertEdge(int u,int v,int id){edge[size]=Edge(v,id,head[u]);head[u]=size++;} void InsertEdge2(int u,int v,int id){edge2[size]=Edge(v,id,head2[u]);head2[u]=size++;}void dfs(int u,int father,int k){deep[u]=k;for(int i=head[u];i != -1;i=edge[i].next){int v=edge[i].v;if(v == father)continue;dfs(v,u,k+1);}}void RMQ(int u,int father){stack[++top]=u;int fa=stack[top-1];up[u][0]=down[u][0]=0;maxw[u][0]=minw[u][0]=ww[u];for(int i=1;bin[i]<=top;++i){//2^i<=topfa=stack[top-bin[i-1]];up[u][i]=max(max(up[u][i-1],up[fa][i-1]),maxw[fa][i-1]-minw[u][i-1]);down[u][i]=max(max(down[u][i-1],down[fa][i-1]),maxw[u][i-1]-minw[fa][i-1]);maxw[u][i]=max(maxw[u][i-1],maxw[fa][i-1]);minw[u][i]=min(minw[u][i-1],minw[fa][i-1]);mp[u][i]=stack[top-bin[i]];}for(int i=head[u];i != -1;i=edge[i].next){int v=edge[i].v;if(v == father)continue;RMQ(v,u);}--top;}int findset(int v){if(v != father[v])father[v]=findset(father[v]);return father[v];}void LCA(int u){mark[u]=true;father[u]=u;for(int i=head2[u];i != -1;i=edge2[i].next){int v=edge2[i].v,id=edge2[i].id;if(!mark[v])continue;anc[id]=findset(v);}for(int i=head[u];i != -1;i=edge[i].next){int v=edge[i].v;if(mark[v])continue;LCA(v);father[v]=u;}}int search(int x){int i=0;while(bin[i+1]<=x)++i;return i;}int Minw(int u,int anc){int i=search(deep[u]-deep[anc]+1);if(bin[i] == deep[u]-deep[anc]+1)return minw[u][i];return min(minw[u][i],Minw(mp[u][i],anc));}int Maxw(int u,int anc){int i=search(deep[u]-deep[anc]+1);if(bin[i] == deep[u]-deep[anc]+1)return maxw[u][i];return max(maxw[u][i],Maxw(mp[u][i],anc));}int Down(int u,int anc){int i=search(deep[u]-deep[anc]+1);if(bin[i] == deep[u]-deep[anc]+1)return down[u][i];int downfa=Down(mp[u][i],anc);downfa=max(downfa,down[u][i]);int minwfa=Minw(mp[u][i],anc);return max(downfa,maxw[u][i]-minwfa); }int UP(int u,int anc){int i=search(deep[u]-deep[anc]+1);if(bin[i] == deep[u]-deep[anc]+1)return up[u][i];int upfa=UP(mp[u][i],anc);upfa=max(upfa,up[u][i]);int maxwfa=Maxw(mp[u][i],anc);return max(upfa,maxwfa-minw[u][i]);}int main(){bin[0]=1;for(int i=1;bin[i-1]<MAX;++i)bin[i]=bin[i-1]*2;int u,v;while(~scanf("%d",&n)){Init(n);for(int i=1;i<=n;++i)scanf("%d",ww+i);for(int i=1;i<n;++i){scanf("%d%d",&u,&v);InsertEdge(u,v,i);InsertEdge(v,u,i);}size=0;scanf("%d",&m);for(int i=0;i<m;++i){scanf("%d%d",uu+i,vv+i);InsertEdge2(uu[i],vv[i],i);InsertEdge2(vv[i],uu[i],i);}dfs(1,-1,1);RMQ(1,-1);LCA(1);for(int i=0;i<m;++i){int upmax=UP(uu[i],anc[i]),downmax=Down(vv[i],anc[i]);int Minww=Minw(uu[i],anc[i]),Maxww=Maxw(vv[i],anc[i]);printf("%d\n",max(max(upmax,downmax),Maxww-Minww));}}return 0;}/*730011112110312221 22 33 44 52 61 715 6*/