CF932F(李超线段树+dp)

此题又是新玩法, 李超线段树合并优化\(dp\)

一个显然的\(\Theta(n^2)dp\): \(dp[x]\)表示从x出发到叶子节点的最小代价

\(dp[x] = \min(dp[y] + a[x] * b[y]) ~~(y \in subtree(x))\)

如果我们将\(b[y]\)看成斜率, \(dp[y]\)看成纵截距, \(a[x]\)看成横坐标, 那么问题转为了在平面上有一些直线, 选出与直线\(x = a[x]\)相交的最靠下的点吗, 李超线段树板题, 但这道题出到了树上所以要用上线段树合并因为有负数所以要整体右移一下, 相应的直线也需变换, 具体见代码

代码:

#pragma GCC optimize(3)
#include <iostream>
#include <cstdio>
#include <cstring>
#include <algorithm>
#define re register
#define ll long long
using namespace std;

template <typename T>
void read(T &x) {
    x = 0; bool f = 0;
    char c = getchar();
    for (;!isdigit(c);c=getchar()) if (c=='-') f=1;
    for (;isdigit(c);c=getchar()) x=x*10+(c^48);
    if (f) x=-x;
}

template <typename T>
void write(T x) {
    if (x < 0) putchar('-'), x = -x;
    if (x >= 10) write(x / 10);
    putchar('0' + x % 10);
}

const int N = 200500;
const int Delta = 100005;
const int Len = Delta << 1;


int h[N], ne[N<<1], to[N<<1];
int T[N], tot, cnt;
inline void add(int x, int y) {
    ne[++tot] = h[x], to[tot] = y, h[x] = tot;
}

int ls[N<<5], rs[N<<5], p[N<<5];
ll n, ans[N], a[N], b[N];
struct node {
    ll k, b;
}line[N];

inline ll calc(int num, ll x) {
    return line[num].k * x + line[num].b;
}

void insert(int &x, int l, int r, int num) {
    if (!x) return x = ++cnt, p[x] = num, void();
    int mid = (l + r) >> 1;
    if (calc(p[x], mid) > calc(num, mid)) swap(p[x], num);
    if (calc(p[x], l) <= calc(num, l) && 
        calc(p[x], r) <= calc(num, r)) return;
    if (calc(p[x], l) > calc(num, l)) insert(ls[x], l, mid, num);
    else insert(rs[x], mid + 1, r, num);
}

const ll INF = 0x7fffffffffff;
ll query(int now, int l, int r, ll x) {
    if (!now) return INF;
    int mid = (l + r) >> 1;
    return min(calc(p[now], x), x <= mid ?
        query(ls[now], l, mid, x) : query(rs[now], mid + 1, r, x));
}

int merge(int x, int y, int l, int r) {
    if (!x || !y) return x | y;
    insert(x, l, r, p[y]);
    int mid = (l + r) >> 1;
    ls[x] = merge(ls[x], ls[y], l, mid);
    rs[x] = merge(rs[x], rs[y], mid + 1, r);
    return x;
}

void dfs(int x, int fa) {
    for (re int i = h[x], y; i; i = ne[i]) {
        if ((y = to[i]) == fa) continue;
        dfs(y, x); 
        T[x] = merge(T[x], T[y], 1, Len);
    }
    ans[x] = query(T[x], 1, Len, a[x] + Delta);
    if (ans[x] == INF) ans[x] = 0;
    line[x] = (node){b[x], ans[x] - b[x] * Delta}; 
    insert(T[x], 1, Len, x);
}

int main() {
    read(n);
    for (re int i = 1;i <= n; i++) read(a[i]);
    for (re int j = 1;j <= n; j++) read(b[j]);
    for (re int i = 1;i < n; i++) {
        int x, y; read(x), read(y);
        add(x, y), add(y, x);
    } dfs(1, 0);
    for (re int i = 1;i <= n; i++) 
        write(ans[i]), putchar(' ');
    return 0;
}

来源：https://www.cnblogs.com/Hs-black/p/12271622.html

标签

线段树

李超

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