poj 3728 The merchant(离线LCA 的应用，5级)

The merchant

Time Limit: 3000MS		Memory Limit: 65536K
Total Submissions: 2399		Accepted: 790

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

4
1 
5 
3 
2
1 3
3 2
3 4
9
1 2
1 3
1 4
2 3
2 1
2 4
3 1
3 2
3 4

Sample Output

4
2
2
0
0
0
0
2
0

Source

POJ Monthly Contest – 2009.04.05, GaoYihan

思路：可以使用离线，离线可以计算出查询的一个点到祖先的最优值，然后记录结果，等整个该祖先的子树都遍历完，就可以知道                祖先到达查询的另一点的最优值，两者合并就是查询结果。

     知道以上，就比较容易了，设ix[x],ax[x]分别为x到祖先的最小最大值，up[x],dw[x]分别为x到祖先的正向最优，反向最优。

    ans=MAX（ax[x]-ix[y],up[y],dw[x] ）

最早题意理解有误，认为是，可以多次买，卖，的最优值，结果居然是只能一次。哎。

多次买卖的话，直接一个在线LCA就可以轻松搞定了。

#include<iostream>
#include<cstring>
#include<cstdio>
#define ll(x) (1<<x)
#define FOR(i,a,b) for(int i=a;i<=b;++i)
#define clr(f,z) memset(f,z,sizeof(f))
using namespace std;
const int mm=7e5+9;
int head[mm],edge,val[mm],qh[mm],N;
int rt[mm],ax[mm],ix[mm],ans[mm],up[mm],dw[mm],lca[mm];
bool vis[mm];
class Edge
{
  public:int u,v,next,w;
}e[mm];
void data()
{ clr(qh,-1);clr(lca,-1);
  clr(head,-1);edge=0;
}
void add(int u,int v,int w,int*head)
{ e[edge].w=w;e[edge].u=u;
  e[edge].v=v;e[edge].next=head[u];head[u]=edge++;
}
int find(int x)
{ int y;
  if(x!=rt[x])
  {
    y=rt[x];
    rt[x]=find(rt[x]);///从根开始到x的正向最优，反向最优
    up[x]=max(ax[y]-ix[x],max(up[x],up[y]));
    dw[x]=max(ax[x]-ix[y],max(dw[x],dw[y]));
    ax[x]=max(ax[x],ax[y]);
    ix[x]=min(ix[x],ix[y]);
  }
  return rt[x];
}
void dfs(int u)
{ int v;
  rt[u]=u;
  vis[u]=1;
  for(int i=head[u];~i;i=e[i].next)
  { v=e[i].v;
    if(!vis[v])
    {
      dfs(v);
      rt[v]=u;
    }
  }

  for(int i=qh[u];~i;i=e[i].next)
  {
    v=e[i].v;
    if(vis[v])
    {
      int t=find(v);///找到最近公共祖先，更新祖先到达v的正向最优和反向最优
      add(t,v,i,lca);///不过祖先到大u的正反向最优没更新，也无法更新故现在没法计算值
    }
  }
  for(int i=lca[u];~i;i=e[i].next)///只更新以u为最近祖先的查询，因为所有u的子树都遍历完了
  {
    int ki=e[i].w;
    int uu,vv,dd;
    uu=e[ki].u;vv=e[ki].v;
    dd=e[ki].w;
    find(uu);///vv已经更新过了，但uu没更新
    if(dd<0)//反向了
    {
      uu^=vv;vv^=uu;uu^=vv;
      dd=-dd;
    }
    ans[dd]=max(max(up[uu],dw[vv]),ax[vv]-ix[uu]);
  }
}
void find_bcc()
{ clr(vis,0);
  dfs(1);
}
int main()
{ int a,b,Q;
  while(~scanf("%d",&N))
  { data();
    FOR(i,1,N)scanf("%d",&ix[i]),ax[i]=ix[i],up[i]=dw[i]=0;
    FOR(i,2,N)
    scanf("%d%d",&a,&b),add(a,b,1,head),add(b,a,1,head);
    scanf("%d",&Q);
    FOR(i,1,Q)
    {
      scanf("%d%d",&a,&b);
      add(a,b,i,qh);
      add(b,a,-i,qh);
    }
    find_bcc();
    FOR(i,1,Q)
    printf("%d\n",ans[i]);
  }
}