BZOJ1791或洛谷4381 [IOI2008]Island

一道基环树的直径

BZOJ原题链接

洛谷原题链接

又是一道实现贼麻烦的题。。

显然公园其实是基环树森林，求的最长距离其实就是求每一棵基环树的直径的总和。

对于每棵基环树，其直径要么经过环，要么是某个环上点的子树的直径。所以我们可以先找出它的环，然后对环上的每个点进行\(dfs\)（不能经过环上的点），找出该点的子树的直径和数组\(D\)，表示该点到子树中的叶子节点的最大距离。

然后考虑经过环的距离，设当前枚举到两个点\(x,y\)，则长度为\(D[x]+D[y]+dis(x,y)\)，这个可以通过单调队列来优化，维护一个最大值即可。

#include<cstdio>
using namespace std;
const int N = 1e6 + 10;
typedef long long ll;
int fi[N], di[N << 1], da[N << 1], ne[N << 1], cr[N], cr_d[N], q[N << 1], po[N << 1], l, cb, fp, n;
ll D[N], dis[N << 1], fr;
bool v[N], egv[N << 1], vis[N];
inline int re()
{
	int x = 0;
	char c = getchar();
	bool p = 0;
	for (; c<'0' || c>'9'; c = getchar())
		p |= c == '-';
	for (; c >= '0'&&c <= '9'; c = getchar())
		x = x * 10 + (c - '0');
	return p ? -x : x;
}
inline void add(int x, int y, int z)
{
	di[++l] = y;
	da[l] = z;
	ne[l] = fi[x];
	fi[x] = l;
}
inline ll maxn(ll x, ll y)
{
	return x > y ? x : y;
}
void dfs(int x)
{
	int i, y;
	v[x] = 1;
	for (i = fi[x]; i; i = ne[i])
	{
		y = di[i];
		if (!egv[i])
		{
			cr[y] = x;
			cr_d[y] = da[i];
			egv[i] = egv[i & 1 ? i + 1 : i - 1] = 1;
			if (v[y])
				cb = y;
			dfs(y);
		}
	}
}
void dfs_2(int x, ll d, int fa)
{
	int i, y;
	if (fr < d)
	{
		fr = d;
		fp = x;
	}
	for (i = fi[x]; i; i = ne[i])
	{
		y = di[i];
		if (!vis[y] && y != fa)
		{
			dfs_2(y, d + da[i], x);
			D[x] = maxn(D[x], D[y] + da[i]);
		}
	}
}
void dfs_3(int x, ll d, int fa)
{
	int i, y;
	fr = maxn(fr, d);
	for (i = fi[x]; i; i = ne[i])
	{
		y = di[i];
		if (!vis[y] && y != fa)
			dfs_3(y, d + da[i], x);
	}
}
int main()
{
	int i, j, x, y, k, head, tail;
	ll s = 0, ma;
	n = re();
	for (i = 1; i <= n; i++)
	{
		x = re();
		y = re();
		add(i, x, y);
		add(x, i, y);
	}
	for (i = 1; i <= n; i++)
		if (!v[i])
		{
			ma = 0;
			dfs(i);
			j = cb;
			k = 0;
			do
			{
				vis[j] = 1;
				k++;
				dis[k + 1] = dis[k] + cr_d[j];
				po[k] = j;
				j = cr[j];
			} while (j^cb);
			do
			{
				vis[j] = 0;
				fr = 0;
				dfs_2(j, 0, 0);
				fr = 0;
				dfs_3(fp, 0, 0);
				vis[j] = 1;
				ma = maxn(ma, fr);
				k++;
				dis[k + 1] = dis[k] + cr_d[j];
				po[k] = j;
				j = cr[j];
			} while (j^cb);
			head = 1;
			tail = 0;
			for (j = 1; j <= k; j++)
			{
				while (j - q[head] >= (k >> 1))
					head++;
				if (head <= tail)
					ma = maxn(ma, D[po[j]] + D[po[q[head]]] + dis[j] - dis[q[head]]);
				else
					ma = maxn(ma, D[po[j]]);
				while (head < tail&&D[po[j]] - dis[j] >= D[po[q[tail]]] - dis[q[tail]])
					tail--;
				q[++tail] = j;
			}
			s += ma;
		}
	printf("%lld", s);
	return 0;
}