JZOJ18992010集训队出题剪枝

Posted 2022-07-12 zjlcnblogs

题目大意

给出一个有根树，\(1\)为根，若某个节点的儿子全是叶子，你可以将该节点的儿子全部剪掉，这样的操作可以进行多次。定义这棵树的价值为：将树上所有叶子按照\(dfs\)序排序后，所有叶子点权之和-相邻两叶子路径上点权最大值。现在你要通过剪枝使得这棵树价值最大。

\(n\leq 100000\)

分析

设\(f_i\)表示\(i\)作为最后一个叶子时的最大价值。暴力枚举原树（没有剪枝）相邻的两个叶子，显然左链上每个点的\(f\)都可以转移到右链上，我们暴力处理出这条路径，分类讨论点权最大值在左链还是右链，进行状态转移。可以发现，一条边被枚举的次数最多是\(2\)，一次是从上个子树进入这个子树，一次是从这个子树进入下个子树，所以暴力枚举复杂度其实是\(O(n)\)的，就可以随便做了。

重点！！！！！！
暴力枚举一棵树相邻两叶子的路径复杂度是\(O(n)\)。

Code

#include <vector>
#include <cstdio>
#include <cstring>
#include <algorithm>
using namespace std;

const int N = 100007;
inline int read()

    int x = 0, f = 0;
    char c = getchar();
    for (; c < '0' || c > '9'; c = getchar()) if (c == '-') f = 1;
    for (; c >= '0' && c <= '9'; c = getchar()) x = (x << 1) + (x << 3) + (c ^ '0');
    return f ? -x : x;


int n, dfn, tot, ans, w[N], p[N], fa[N], dep[N], ord[N], leaf[N], f[N];
int len, arr[N], mx1[N], mx2[N];
vector<int> son[N];

void dfs(int u)

    ord[++dfn] = u;
    int sz = son[u].size();
    for (int i = 0; i < sz; i++) fa[son[u][i]] = u, dep[son[u][i]] = dep[u] + 1, dfs(son[u][i]);

int getlca(int u, int v)

    if (dep[u] < dep[v]) swap(u, v);
    while (dep[fa[u]] >= dep[v]) u = fa[u];
    if (u == v) return u;
    while (fa[u] != fa[v]) u = fa[u], v = fa[v];
    return fa[u];


int main()

    //freopen("cut.in", "r", stdin);
    n = read();
    for (int i = 1; i <= n; i++)
    
        w[i] = read(), p[i] = read();
        for (int j = 1, a; j <= p[i]; j++) a = read(), son[i].push_back(a);
    
    dep[1] = 1, dfs(1);
    for (int i = 1; i <= n; i++) if (!son[ord[i]].size()) leaf[++tot] = ord[i];
    for (int i = 1; i <= n; i++) f[i] = -0x3f3f3f3f;
    int x = leaf[1];
    while (x != 1) f[x] = w[x], x = fa[x];
    for (int i = 2; i <= tot; i++)
    
        int a = leaf[i - 1], b = leaf[i], c = getlca(a, b), dist = dep[a] + dep[b] - 2 * dep[c] + 1;
        arr[len = 1] = a; while (fa[a] != c) a = fa[a], arr[++len] = a;
        arr[++len] = c;
        arr[dist--] = b; while (fa[b] != c) b = fa[b], arr[dist--] = b;
        mx1[len] = w[c], dist = dep[leaf[i - 1]] + dep[leaf[i]] - 2 * dep[c] + 1;
        for (int j = len - 1; j >= 1; j--) mx1[j] = max(mx1[j + 1], w[arr[j]]);
        for (int j = len + 1; j <= dist; j++) mx1[j] = max(mx1[j - 1], w[arr[j]]);
        mx2[0] = -0x3f3f3f3f; for (int j = 1; j <= len - 1; j++) mx2[j] = max(mx2[j - 1], f[arr[j]] - mx1[j + 1]);
        for (int j = len + 1, k = len, maxf = -0x3f3f3f3f; j <= dist; j++)
        
            while (k > 1 && mx1[k] <= mx1[j - 1]) k--, maxf = max(maxf, f[arr[k]]);
            f[arr[j]] = max(f[arr[j]], maxf - mx1[j - 1] + w[arr[j]]);
            if (k > 1) f[arr[j]] = max(f[arr[j]], mx2[k - 1] + w[arr[j]]);
        
    
    x = leaf[tot];
    while (x != 1) ans = max(ans, f[x]), x = fa[x];
    printf("%d\n", ans);
    return 0;