Nowcoder城市网络 树上倍增

Posted 2021-09-05 kaka0010

原题链接：https://ac.nowcoder.com/acm/problem/13331

目录

• 题意
• 分析
• Code

题意

有一棵树，树上每个节点都有一个权值，1为根节点。每次选择两个节点u,v保证v在u通往根节点的路径上，每次从u出发，身上携带价格为w的珠宝，每次遇到权值比你大的节点就可以交换一次，问一共会交换多少次。

分析

看到这类问题一般就会往倍增上面思考，难点就是如何找出第一个大于你权值的祖先节点。在一个序列上我们知道用单调栈来实现，在树上其实可以用更简单的倍增解决

1. 如果父节点就比你大，直接将f[x][0]=fa
2. 如果父节点不满足，将当前节点设为fa，然后一直倍增找到第一个大于你的点

思路是比较好想的，实现起来会有一些细节需要处理。

Code

#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
typedef unsigned int ul;
typedef pair<int, int> PII;
const int inf = 0x3f3f3f3f;
const int N = 5e5 + 10;
const int M = 1e6 + 10;
const ll mod = 1e9 + 7;
const double eps = 1e-8;

#define lowbit(i) (i & -i)
#define Debug(x) cout << (x) << endl
#define fi first
#define se second
#define mem memset
#define endl '\\n'

int a[N], f[N][21], dep[N];
vector<int> g[N];
void dfs(int x, int fa) {
    dep[x] = dep[fa] + 1;
    if (a[fa] > a[x]) f[x][0] = fa;
    else {
        int now = fa;
        for (int i = 20; ~i; i--) {
            if (f[now][i] && a[f[now][i]] <= a[x]) {
                now = f[now][i];
            }
        }
        f[x][0] = f[now][0];
    }
    for (int i = 1; i <= 20; i++) f[x][i] = f[f[x][i-1]][i-1];
    for (auto v : g[x]) {
        if (v == fa) continue;
        dfs(v, x);
    }
}
inline void solve() {
    int n, m; cin >> n >> m;
    for (int i = 1; i <= n; i++) cin >> a[i];
    for (int i = 1; i <= n-1; i++) {
        int u, v; cin >> u >> v;
        g[u].push_back(v);
        g[v].push_back(u);
    }
    dfs(1, 0);
    while (m--) {
        int u, v, w;
        cin >> u >> v >> w;
        int now;
        if (a[u] > w) {
            now = u;
        } else {
            now = u;
            for (int i = 20; ~i; i--) {
                if (f[now][i] && a[f[now][i]] <= w) {
                    now = f[now][i];
                }
            }
            if (dep[f[now][0]] < dep[v]) {
                cout << 0 << endl;
                continue;
            }
            now = f[now][0];
        }
        int ans = 1;
        for (int i = 20; ~i; i--) {
            if (f[now][i] && dep[f[now][i]] >= dep[v]) {
                ans += (1 << i);
                now = f[now][i];
            }
        }
        cout << ans << endl;
    }
}

signed main() {
    ios_base::sync_with_stdio(false);
    cin.tie(0);
    cout.tie(0);
#ifdef ACM_LOCAL
    freopen("input", "r", stdin);
    freopen("output", "w", stdout);
    signed test_index_for_debug = 1;
    char acm_local_for_debug = 0;
    do {
        if (acm_local_for_debug == '$') exit(0);
        if (test_index_for_debug > 20)
            throw runtime_error("Check the stdin!!!");
        auto start_clock_for_debug = clock();
        solve();
        auto end_clock_for_debug = clock();
        cout << "Test " << test_index_for_debug << " successful" << endl;
        cerr << "Test " << test_index_for_debug++ << " Run Time: "
             << double(end_clock_for_debug - start_clock_for_debug) / CLOCKS_PER_SEC << "s" << endl;
        cout << "--------------------------------------------------" << endl;
    } while (cin >> acm_local_for_debug && cin.putback(acm_local_for_debug));
#else
    solve();
#endif
    return 0;
}