LightOJ - 1162 Min Max Roads

LightOJ - 1162 Min Max Roads

题解：在线倍增LCA和模拟ST表

让我们求从\(u->v\)最短路径上的边权最大值和边权最小值，那么我们可以利用倍增思想，类似其\(fa[u][i]\)数组代表从\(u\)往上跳\(2^i\)步的点这一思想，我们可以建立两个二维数组\(dmax[u][i]，dmin[u][i]\)，代表从\(u\)往上跳\(2^i\)步到达的点和\(u\)之间路径的最大权值和最小权值，同时我们可以知道dmax和dmin的初始状态：\(dmax[u][0]=dmin[u][0]=w_i\),最后类似\(fa[u][i] = fa[fa[u[i-1]]][i-1]\)我们列出状态方程：\(dmax[u][i]=max(dmax[u][i-1],dmax[fa[u][i-1]][i-1])\),\(dmin[u][i]=min(dmin[u][i-1],dmin[fa[u][i-1]][i-1])\),

这就类似ST表的思想，一个区间的最大值是可重复的贡献，所以我们各取区间的一半分别取\(max/min\)，所以这个方程就代表：从\(u\)往上跳\(2^i\)步到达的点和\(u\)之间路径的最大权值和最小权值，\([u,2^i] = max/min([u,2^{i-1}],[u+2^{i-1},u+2^i])\)

#include <bits/stdc++.h>
#define Zeoy std::ios::sync_with_stdio(false), std::cin.tie(0), std::cout.tie(0)
#define all(x) (x).begin(), (x).end()
#define endl '\n'
using namespace std;
typedef long long ll;
typedef pair<int, int> pii;
typedef pair<ll, ll> pll;
const int inf = 0x3f3f3f3f;
const ll INF = 0x3f3f3f3f3f3f3f3f;
const int mod = 1e9 + 7;
const double eps = 1e-9;
const int N = 1e5 + 10;
int n, m;
int fa[N][22], dmax[N][22], dmin[N][22];
int du[N], dep[N];
vector<pii> g[N];
void init()
{
    memset(fa, 0, sizeof fa);
    memset(dmax, 0, sizeof dmax);
    memset(dmin, 0, sizeof dmin);
    for (int i = 1; i <= n; ++i)
        du[i] = 0, dep[i] = 0, g[i].clear();
}
void dfs(int u, int par, int w)
{
    dep[u] = dep[par] + 1;
    fa[u][0] = par;
    dmax[u][0] = w;
    dmin[u][0] = w;
    for (int i = 1; i <= 20; ++i)
    {
        fa[u][i] = fa[fa[u][i - 1]][i - 1];
        dmax[u][i] = max(dmax[u][i - 1], dmax[fa[u][i - 1]][i - 1]);
        dmin[u][i] = min(dmin[u][i - 1], dmin[fa[u][i - 1]][i - 1]);
    }
    for (auto &[v, W] : g[u])
    {
        if (v == par)
            continue;
        dfs(v, u, W);
    }
}
void lca(int u, int v)
{
    int maxx = -inf, minn = inf;
    if (dep[u] < dep[v])
        swap(u, v);
    for (int i = 20; i >= 0; i--)
    {
        if (dep[fa[u][i]] >= dep[v])
        {
            maxx = max(maxx, dmax[u][i]);       //注意一定要先去取区间max和min，不然u就会改变
            minn = min(minn, dmin[u][i]);
            u = fa[u][i];
        }
    }
    if (u == v)
    {
        cout << minn << " " << maxx << endl;
        return;
    }
    for (int i = 20; i >= 0; --i)
    {
        if (fa[u][i] != fa[v][i])
        {
            maxx = max({maxx, dmax[u][i], dmax[v][i]});
            minn = min({minn, dmin[u][i], dmin[v][i]});
            u = fa[u][i], v = fa[v][i];
        }
    }
    maxx = max({maxx, dmax[u][0], dmax[v][0]});
    minn = min({minn, dmin[u][0], dmin[v][0]});
    cout << minn << " " << maxx << endl;
}
int main(void)
{
    Zeoy;
    int t = 1;
    cin >> t;
    int tot = 1;
    while (t--)
    {
        cin >> n;
        cout << "Case " << tot++ << ":\n";
        init();
        for (int i = 1, u, v, w; i < n; ++i)
        {
            cin >> u >> v >> w;
            du[v]++;
            g[u].push_back({v, w});
            g[v].push_back({u, w});
        }
        int st;
        for (int i = 1; i <= n; ++i)
        {
            if (du[i] == 0)
            {
                st = i;
                break;
            }
        }
        dfs(st, 0, 0);
        cin >> m;
        for (int i = 1, u, v; i <= m; ++i)
        {
            cin >> u >> v;
            lca(u, v);
        }
    }
    return 0;
}