[洛谷P4949]最短距离

题目大意：给一棵基环树，两种操作：

1. $1\;x\;y：$把第$x$条边长度改成$y$
2. $2\;x\;y：$查询$x$到$y$的最短距离

题解：发现最短距离只有两种可能，第一个是树上的距离，第二种是经过多出来的一条边，都求出来比较一下就行了。修改时，若修改的是多出来的边，直接修改即可，若不是可以把它子树中所有点的距离修改，可以按$dfs$序变成区间修改

卡点：无

C++ Code：

#include <cstdio>
#include <algorithm>
#define maxn 100010
int head[maxn], cnt = 1;
struct Edge {
	int to, nxt, w;
} e[maxn << 1];
inline void add(int a, int b, int c) {
	e[++cnt] = (Edge) {b, head[a], c}; head[a] = cnt;
	e[++cnt] = (Edge) {a, head[b], c}; head[b] = cnt;
}

int n, m, circle;
int l[maxn], r[maxn], w[maxn];

#define M 17
int fa[maxn][M], dep[maxn], sz[maxn];
int dfn[maxn], idx;
bool vis[maxn];
void dfs(int u, int father = 0) {
	for (int i = 1; i < M; i++) fa[u][i] = fa[fa[u][i - 1]][i - 1];
	vis[u] = true;
	dfn[u] = ++idx;
	sz[u] = 1;
	for (int i = head[u]; i; i = e[i].nxt) if (i ^ father ^ 1) {
		int v = e[i].to;
		if (!vis[v]) {
			*fa[v] = u;
			dep[v] = dep[u] + 1;
			dfs(v, i);
			sz[u] += sz[v];
		} else circle = i >> 1;
	}
}
inline int LCA(int x, int y) {
	if (x == y) return x;
	if (dep[x] < dep[y]) std::swap(x, y);
	for (int i = dep[x] - dep[y]; i; i &= i - 1) x = fa[x][__builtin_ctz(i)];
	if (x == y) return x;
	for (int i = M - 1; ~i; i--) if (fa[x][i] != fa[y][i]) x = fa[x][i], y = fa[y][i];
	return *fa[x];
}
#undef M

int last[maxn];
namespace BIT {
	long long Tr[maxn], res;
	inline void add(int p, int num) {for (; p <= n; p += p & -p) Tr[p] += num;}
	inline int ask(int p) {for (res = 0; p; p &= p - 1) res += Tr[p]; return res;}
}

inline void __modify(int u, int v, int w) {BIT::add(u, w), BIT::add(v + 1, -w);}
inline void modify(int u, int v, int w) {
	if (dfn[u] > dfn[v]) std::swap(u, v);
	__modify(dfn[v], dfn[v] + sz[v] - 1, w - ::last[v]); ::last[v] = w;
}

inline long long dis(int x, int y) {return BIT::ask(dfn[x]) + BIT::ask(dfn[y]) - BIT::ask(dfn[LCA(x, y)]) * 2;}
inline long long solve(int x, int y) {
	long long ans = dis(x, y);
	ans = std::min(ans, dis(x, l[circle]) + dis(y, r[circle]) + w[circle]);
	return std::min(ans, dis(x, r[circle]) + dis(y, l[circle]) + w[circle]);
}

int main() {
	scanf("%d%d", &n, &m);
	for (int i = 1; i <= n; i++) {
		scanf("%d%d%d", l + i, r + i, w + i);
		add(l[i], r[i], w[i]);
	}
	dfs(1);
	for (int i = 1; i <= n; i++) if (i != circle) modify(l[i], r[i], w[i]);
	while (m --> 0) {
		int op, x, y;
		scanf("%d%d%d", &op, &x, &y);
		if (op == 1) {
			if (x == circle) w[circle] = y;
			else modify(l[x], r[x], y);
		} else printf("%lld\n", solve(x, y));
	}
	return 0;
}