P4180 [BJWC2010]严格次小生成树

树上倍增记录最小生成树上最大次大

的边。

#include<bits/stdc++.h>
using namespace std;
#define int long long
#define ri register int
#define N 400005
#define M 900005
#define inf 0x2fffffffffffffff
struct node
{
	int u, v, w, Next;
};
node e[M], A[M << 1];
bool B[M << 1];
int fa[N];
int first[N], tot = 0;
int n, m, bz[N][20], maxi[N][20], mini[N][20], deep[N];
inline bool cmp(node a, node b)
{
	return a.w < b.w;
}
inline int getfa(int x)
{
	if(x == fa[x])
	{
		return x;
	}
	return fa[x] = getfa(fa[x]);
}
inline void add(int x, int y, int z)
{
	e[++tot].Next = first[x];
	first[x] = tot;
	e[tot].u = x;
	e[tot].v = y;
	e[tot].w = z;
	return;
}
inline void dfs(int u, int fa)
{
	bz[u][0] = fa;
	for(ri i = first[u]; i; i = e[i].Next)
	{
		int v = e[i].v;
		if(v == fa)
		{
			continue;
		}
		deep[v] = deep[u] + 1ll;
		maxi[v][0] = e[i].w;
		mini[v][0] = -inf;
		dfs(v, u);
	}
	return;
}
inline void init()
{
	for(ri i = 1; i <= 18; i++)
	{
		for(ri j = 1; j <= n; j++)
		{
			bz[j][i] = bz[bz[j][i - 1]][i - 1];
			maxi[j][i] = max(maxi[j][i - 1], maxi[bz[j][i - 1]][i - 1]);
			mini[j][i] = max(mini[j][i - 1], mini[bz[j][i - 1]][i - 1]);
			if(maxi[j][i - 1] > maxi[bz[j][i - 1]][i - 1])
			{
				mini[j][i] = max(mini[j][i], maxi[bz[j][i - 1]][i - 1]);
			}
			else if(maxi[j][i - 1] < maxi[bz[j][i - 1]][i - 1])
			{
				mini[j][i] = max(mini[j][i], maxi[j][i - 1]);
			}
		}
	}
	return;
}
inline int lca(int x, int y)
{
	if(deep[x] < deep[y])
	{
		swap(x, y);
	}
	for(ri i = 18; i >= 0; i--)
	{
		if(deep[bz[x][i]] >= deep[y])
		{
			x = bz[x][i];
		}
	}
	if(x == y)
	{
		return x;
	}
	for(ri i = 18; i >= 0; i--)
	{
		if(bz[x][i] ^ bz[y][i])
		{
			x = bz[x][i];
			y = bz[y][i];
		}
	}
	return bz[x][0];
}
inline int qmax(int u, int v, int maxx)
{
	int ans = -inf;
	for(ri i = 18; i >= 0; i--)
	{
		if(deep[bz[u][i]] >= deep[v])
		{
			if(maxx != maxi[u][i])
			{
				ans = max(ans, maxi[u][i]);
			}
			else
			{
				ans = max(ans, mini[u][i]);
			}
			u = bz[u][i];
		}
	}
	return ans;
}
signed main()
{
	scanf("%lld%lld", &n, &m);
	for(ri i = 1; i <= m; i++)
	{
		scanf("%lld%lld%lld", &A[i].u, &A[i].v, &A[i].w);
	}
	sort(A + 1, A + m + 1, cmp);
	for(ri i = 1; i <= n; i++)
	{
		fa[i] = i;
	}
	int cnt = 0;
	for(ri i = 1; i <= m; i++)
	{
		ri fx = getfa(A[i].u), fy = getfa(A[i].v);
		if(fx == fy)
		{
			continue;
		}
		fa[fy] = fx;
		cnt += A[i].w;
		add(A[i].u, A[i].v, A[i].w);
		add(A[i].v, A[i].u, A[i].w);
		B[i] = true;
	}
	mini[1][0] = -inf;
	deep[1] = 1;
	dfs(1, 0);
	init();
	ri ans = inf;
	for(ri i = 1; i <= m; i++)
	{
		if(!B[i])
		{
			ri lcf = lca(A[i].u, A[i].v), maxu = qmax(A[i].u, lcf, A[i].w), maxv = qmax(A[i].v, lcf, A[i].w);
			ans = min(ans, cnt - max(maxu, maxv) + A[i].w);
		}
	}
	printf("%lld", ans);
	return 0;
}