Shopping（树形背包+点分治）

Posted 2020-12-06 zcr-blog

Solution

我们发现要解决一个树上的连通块问题，解决这种问题的时候我们不妨先随便选一个根，如果要选某两个点则他们到n的路径上的点都会被选就变成了一个树形背包问题。

注意这里是多重背包，所以我们可以用单调队列优化，时间复杂度$O(N^2M)$。

考虑暴力选根的时候会把很多重复的情况算进去，所以我们可以用点分治，只计算根的孩子之间的贡献，递归子树时其余兄弟节点就不用管了。

因为每次选的是重心，所以子树大小必然减一半，时间复杂度$O(NMlog{N})$。

点分治+树形背包，这是一种常见的处理树上连通块的方法。

Code

因为加了单调队列优化，所以要注意树形背包时倒着做（即从叶节点开始）。

#include <iostream>
#include <cstdio>
#include <cstring>
#include <algorithm>
using namespace std;
const int N = 5010, M = 40010, inf = 0x3f3f3f3f;
struct node{
    int pre, to;
}edge[N << 1];
int head[N], tot;
int T;
int n, m;
int sz[N], mx[N], rt;
int w[N], v[N], d[N];
int dfn[N], bl[N], dep;
int dp[N][M];
int ans;
bool vis[N];
void get_root(int x, int tot_size, int fa) {
    sz[x] = 1;
    mx[x] = 0;
    for (int i = head[x]; i; i = edge[i].pre) {
        int y = edge[i].to;
        if (y == fa || vis[y]) continue;
        get_root(y, tot_size, x);
        sz[x] += sz[y];
        if (mx[x] < sz[y]) {
            mx[x] = sz[y];
        }
    }
    mx[x] = max(mx[x], tot_size - sz[x]);
    if (mx[x] < mx[rt]) {
        rt = x;
    }
}
void dfs(int x, int fa) {
    sz[x] = 1;
    dfn[++dep] = x;
    bl[x] = dep;
    for (int i = head[x]; i; i = edge[i].pre) {
        int y = edge[i].to;
        if (y == fa || vis[y]) continue;
        dfs(y, x);
        sz[x] += sz[y];
    }
}
void cmax(int &x, int y) {
    x = max(x, y);
}
int q[N];
void solve(int x) {
    vis[x] = 1;
    dep = 0;
    dfs(x, 0);
    for (int i = 0; i <= dep + 1; i++) {
        for (int j = 0; j <= m; j++) {
            dp[i][j] = 0;
        }
    }
    for (int i = dep; i >= 1; i--) {
        for (int j = 0; j <= m; j++) {
            cmax(dp[i][j], dp[i + sz[dfn[i]]][j]);
        }
        int a = v[dfn[i]];
        int b = w[dfn[i]];
        int c = d[dfn[i]];
        for (int j = 0; j < a; j++) {
            int heead = 1, tail = 0;
            for (int k = 0; j + k * a <= m; k++) {
                while (heead <= tail && q[heead] < k - c) heead++;
                if (heead <= tail) cmax(dp[i][j + k * a], dp[i + 1][j + q[heead] * a] - q[heead] * b + k * b);
                while (heead <= tail && dp[i + 1][j + a * q[tail]] - q[tail] * b <= dp[i + 1][j + a * k] - k * b) tail--;
                q[++tail] = k;
            }
        }
    }
    for (int i = 1; i <= m; i++) {
        ans = max(ans, dp[1][i]);
    }
    for (int i = head[x]; i; i = edge[i].pre) {
        int y = edge[i].to;
        if (vis[y]) continue;
        rt = 0;
        get_root(y, sz[y], x);
        solve(rt);
    }
}
int read() {
    int ret = 0, f = 1;
    char ch = getchar();
    while (!isdigit(ch)) {
        if (ch == ‘-‘) f = -1;
        ch = getchar();
    }
    while (isdigit(ch)) {
        ret = (ret << 1) + (ret << 3) + ch - ‘0‘;
        ch = getchar();
    }
    return ret * f;
}
void write(int x) {
    if (x > 9) write(x / 10);
    putchar(x % 10 + ‘0‘);
}
void print(int x) {
    if (x < 0) {
        x = -x;
        putchar(‘-‘);
    }
    write(x);
    putchar(‘
‘);
}
void init() {
    ans = 0;
    memset(vis, 0, sizeof(vis));
    memset(head, 0, sizeof(head));
    tot = 0;
}
void add(int u, int vv) {
    edge[++tot] = node{head[u], vv};
    head[u] = tot;
} 
int main() {
    mx[0] = inf;
    T = read();
    while (T--) {
        init();
        n = read(); m = read();
        for (int i = 1; i <= n; i++) {
            w[i] = read();
        }
        for (int i = 1; i <= n; i++) {
            v[i] = read();
        }
        for (int i = 1; i <= n; i++) {
            d[i] = read();
        }
        for (int i = 1, u, vv; i < n; i++) {
            u = read();
            vv = read();
            add(u, vv);
            add(vv, u);
        }
        rt = 0;
        get_root(1, n, 0);
        solve(rt);
        print(ans);
    }
    return 0;
}