Magic boy Bi Luo with his excited tree (树形dp）

Posted 2022-07-30 zhangbuang

Bi Luo is a magic boy, he also has a migic tree, the tree has NN nodes , in each node , there is a treasure, it‘s value is V[i]V[i], and for each edge, there is a cost C[i]C[i], which means every time you pass the edge ii , you need to pay C[i]C[i]. 

You may attention that every V[i]V[i] can be taken only once, but for some C[i]C[i] , you may cost severial times. 

Now, Bi Luo define ans[i]ans[i] as the most value can Bi Luo gets if Bi Luo starts at node ii. 

Bi Luo is also an excited boy, now he wants to know every ans[i]ans[i], can you help him?

InputFirst line is a positive integer T(T≤104)T(T≤104) , represents there are TT test cases. 

Four each test： 

The first line contain an integer NN(N≤105)(N≤105). 

The next line contains NN integers V[i]V[i], which means the treasure’s value of node i(1≤V[i]≤104)i(1≤V[i]≤104). 

For the next N−1N−1 lines, each contains three integers u,v,cu,v,c , which means node uuand node vv are connected by an edge, it‘s cost is c(1≤c≤104)c(1≤c≤104). 

You can assume that the sum of NN will not exceed 106106.OutputFor the i-th test case , first output Case #i: in a single line , then output NNlines , for the i-th line , output ans[i]ans[i] in a single line.Sample Input

1
5
4 1 7 7 7 
1 2 6
1 3 1
2 4 8
3 5 2

Sample Output

Case #1:
15
10
14
9
15

SOLUTION:
这算是一道比较难的dp了吧，
很容易想到dp i 0/1 代表只考虑i的子树后回到，不会到根节点的最大价值，
之后就要考虑此时的转移
求出dp之后，最难的步骤就是dp数组在树上换根时候的变换，感觉网上的题解写的不错

CODE:

//
//  Created by Running Photon
//  Copyright (c) 2015 Running Photon. All rights reserved.
//
#include <algorithm>
#include <cctype>
#include <cmath>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <iomanip>
#include <iostream>
#include <map>
#include <queue>
#include <string>
#include <sstream>
#include <set>
#include <vector>
#include <stack>
#define ALL(x) x.begin(), x.end()
#define INS(x) inserter(x, x,begin())
#define ll long long
#define CLR(x) memset(x, 0, sizeof x)
using namespace std;
const int inf = 0x3f3f3f3f;
const int MOD = 1e9 + 7;
const int maxn = 2e5 + 10;
const int maxv = 1e5 + 10;
const double eps = 1e-9;

int pnt[maxn], nxt[maxn], val[maxv], head[maxv], cost[maxn], cnt;
void add_edge(int u, int v, int value) 
    pnt[cnt] = v;
    cost[cnt] = value;
    nxt[cnt] = head[u];
    head[u] = cnt++;

int dp[2][maxv];
void dfs1(int u, int fa) 
    dp[0][u] = val[u];
    dp[1][u] = val[u];
    for(int i = head[u]; ~i; i = nxt[i]) 
        int v = pnt[i];
        if(v == fa) continue;
        dfs1(v, u);
        if(dp[0][v] - cost[i] * 2 > 0) 
            dp[0][u] += dp[0][v] - cost[i] * 2;
        
    
    for(int i = head[u]; ~i; i = nxt[i]) 
        int v = pnt[i];
        if(v == fa) continue;
        if(dp[0][u] - max(0, dp[0][v] - cost[i] * 2) + (dp[1][v] - cost[i]) >= dp[1][u]) 
            dp[1][u] = dp[0][u] - max(0, dp[0][v] - cost[i] * 2) + (dp[1][v] - cost[i]);
        
    

void dfs2(int u, int fa, int Cost) 
    int dir = u, tmp;
    if(fa != -1 && dp[0][fa] - Cost * 2 > 0) 
        dp[0][u] += dp[0][fa] - Cost * 2;
    

    tmp = dp[1][u] = dp[0][u];

    if(fa!=-1)
        for(int i=1;i<=1;i++) 
        int v = fa;

        if(dp[0][u] - max(0, dp[0][v] - Cost * 2) + (dp[1][v] - Cost) >= dp[1][u]) 
            dir = v;
            swap(tmp, dp[1][u]);
            dp[1][u] = dp[0][u] - max(0, dp[0][v] - Cost * 2) + (dp[1][v] - Cost);
        
        else
            if(dp[0][u] - max(0, dp[0][v] - Cost * 2) + (dp[1][v] - Cost) >= tmp) 
            tmp = dp[0][u] - max(0, dp[0][v] - Cost * 2) + (dp[1][v] - Cost);
        
    

    for(int i = head[u]; ~i; i = nxt[i]) 
        int v = pnt[i];

        if(dp[0][u] - max(0, dp[0][v] - cost[i] * 2) + (dp[1][v] - cost[i]) >= dp[1][u]) 
            dir = v;
            swap(tmp, dp[1][u]);
            dp[1][u] = dp[0][u] - max(0, dp[0][v] - cost[i] * 2) + (dp[1][v] - cost[i]);
        
        else if(dp[0][u] - max(0, dp[0][v] - cost[i] * 2) + (dp[1][v] - cost[i]) >= tmp) 
            tmp = dp[0][u] - max(0, dp[0][v] - cost[i] * 2) + (dp[1][v] - cost[i]);
        
    
    // printf("dp[0][%d] = %d dp[1][%d] = %d  tmp = %d\n", u, dp[0][u], u, dp[1][u], tmp);

    int L = dp[0][u], R = dp[1][u];
    for(int i = head[u]; ~i; i = nxt[i]) 
        int v = pnt[i];
        if(v == fa) continue;
        if(dp[0][v] - cost[i] * 2 > 0) 
            dp[0][u] = L - (dp[0][v] - cost[i] * 2);
        
        if(dir == v) 
            dp[1][u] = tmp;
            if(dp[0][v] - cost[i] * 2 > 0) dp[1][u] -= (dp[0][v] - cost[i] * 2);
        
        else if(dp[0][v] - cost[i] * 2 > 0) 
            dp[1][u] = R - (dp[0][v] - cost[i] * 2);
        
        // printf("dir = %d\n", dir);
        // printf("nxt%d = %d\n", v, dp[1][u]);
        dfs2(v, u, cost[i]);
        dp[0][u] = L, dp[1][u] = R;
    

int main() 

    int T;
    int cas = 0;
    scanf("%d", &T);
    while(T--) 
        int n;
        printf("Case #%d:\n", ++cas);
        scanf("%d", &n);
        cnt = 0;
        memset(head, -1, sizeof (int) * (n + 1));
        for(int i = 1; i <= n; i++) scanf("%d", val + i);
        for(int i = 1; i < n; i++) 
            int u, v, value;
            scanf("%d%d%d", &u, &v, &value);
            add_edge(u, v, value);
            add_edge(v, u, value);
        
        dfs1(1, -1);
        dfs2(1, -1, 0);
        for(int i = 1; i <= n; i++) 
            printf("%d\n", max(dp[0][i], dp[1][i]));
        
    

    return 0;