UVa 11354 Bond 最小生成树+LCA倍增

题目来源：UVa 11354 Bond

题意：n个点m条边的图 q次询问 找到一条从s到t的一条边 使所有边的最大危险系数最小

思路：使最大的危险系数尽量小 答案是最小生成树上的边 然后用LCA倍增法记录s和t到他们最近公共祖先的最大值

#include <cstdio>
#include <cstring>
#include <algorithm>
using namespace std;
const int maxn = 50010;
const int INF = 0x3f3f3f3f;
int anc[maxn][30], maxcost[maxn][30];
int fa[maxn], L[maxn], cost[maxn], vis[maxn], f[maxn];
int n, m;
int first[maxn], cnt;
struct edge
{
	int u, v, w, next;
}e[maxn*2], e2[maxn*8];

bool cmp(edge a, edge b)
{
	return a.w < b.w;
}

void AddEdge(int u, int v, int w)
{
	e2[cnt].v = v;
	e2[cnt].w = w;
	e2[cnt].next = first[u];
	first[u] = cnt++;
	e2[cnt].v = u;
	e2[cnt].w = w;
	e2[cnt].next = first[v];
	first[v] = cnt++;

}
void pre()
{
	for(int i = 1; i <= n; i++)
	{
		anc[i][0] = fa[i]; maxcost[i][0] = cost[i];
		for(int j = 1; (1<<j) < n; j++)
			anc[i][j] = -1;
	}
	for(int j = 1; (1<<j) < n; j++)
		for(int i = 1; i <= n; i++)
			if(anc[i][j-1] != -1)
			{
				int a = anc[i][j-1];
				anc[i][j] = anc[a][j-1];
				maxcost[i][j] = max(maxcost[i][j-1], maxcost[a][j-1]);
			}

}

int query(int p, int q)
{
	int tmp, log, i;
	if(L[p] < L[q])
		swap(p, q);
	for(log = 1; (1<<log) <= L[p]; log++);
	log--;

	int ans = -INF;
	for(int i = log; i >= 0; i--)
		if(L[p] - (1<<i) >= L[q])
		{
			ans = max(ans, maxcost[p][i]);
			p = anc[p][i];
		}
	if(p == q)
		return ans;
	for(int i = log; i >= 0; i--)
	{
		if(anc[p][i] != -1 && anc[p][i] != anc[q][i])
		{
			ans = max(ans, maxcost[p][i]);
			ans = max(ans, maxcost[q][i]);
			p = anc[p][i];
			q = anc[q][i];
		}
	}
	ans = max(ans, cost[p]);
	ans = max(ans, cost[q]);
	return ans;
}

void dfs(int u)
{
	for(int i = first[u]; i != -1; i = e2[i].next)
	{
		int v = e2[i].v;
		if(vis[v])
			continue;
		vis[v] = 1;
		fa[v] = u;
		cost[v] = e2[i].w;
		L[v] = L[u]+1;
		dfs(v);
	}
}

int find(int x)
{
	if(f[x] != x)
		return f[x] = find(f[x]);
	return x;
}
int main()
{
	int cas = 0;
	while(scanf("%d %d", &n, &m) != EOF)
	{
		memset(first, -1, sizeof(first));
		memset(vis, 0, sizeof(vis));
		cnt = 0;
		for(int i = 0; i < m; i++)
		{
			scanf("%d %d %d", &e[i].u, &e[i].v, &e[i].w);
		}
		sort(e, e+m, cmp);
		for(int i = 0; i <= n; i++)
			f[i] = i;
		int sum = 0;
		for(int i = 0; i < m; i++)
		{
			int u = e[i].u;
			int v = e[i].v;
			int x = find(u);
			int y = find(v);
			if(x != y)
			{
				sum++;
				f[x] = y;
				AddEdge(e[i].u, e[i].v, e[i].w);
			}
		}
		fa[1] = -1;
		cost[1] = 0;
		L[1] = 0;
		vis[1] = 1;
		dfs(1);
		pre();
		int q;
		scanf("%d", &q);
		if(cas++)
			printf("\n");
		while(q--)
		{
			int u, v;
			scanf("%d %d", &u, &v);
			printf("%d\n", query(u, v));
		}
	}
	return 0;
}

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