线段树

给群里红石镐的题解备份，免得翻记录翻两百条。 题意 一棵树，支持两种操作： 路径加：把 $u$ 到 $v$ 路径上每个点权值加 $w$ 区间和：求 $u$ 到 $v$ 路径上的点权和 思路 板子题，树链剖分 + 线段树。两遍 DFS 剖出重链，把树上问题变成区间问题，再用线段树维护 dfn 序。 代码 #include <bits/stdc++.h> using namespace std; typedef long long ll; const int N = 100005; int n, m; int head[N], to[N << 1], nxt[N << 1], etot; int fa[N], dep[N], siz[N], son[N], top[N], dfn[N], tot; ll sum[N << 2], lazy[N << 2]; void add(int u, int v) { to[++etot] = v; nxt[etot] = head[u]; head[u] = etot; } // 第一遍 DFS：求 fa, dep, siz, son void dfs1(int u, int f) { fa[u] = f; dep[u] = dep[f] + 1; siz[u] = 1; for (int e = head[u]; e; e = nxt[e]) { int v = to[e]; if (v == f) continue; dfs1(v, u); siz[u] += siz[v]; if (siz[v] > siz[son[u]]) son[u] = v; } } // 第二遍 DFS：剖重链，求 top, dfn void dfs2(int u, int t) { top[u] = t; dfn[u] = ++tot; if (!son[u]) return; dfs2(son[u], t); for (int e = head[u]; e; e = nxt[e]) { int v = to[e]; if (v == fa[u] || v == son[u]) continue; dfs2(v, v); } } void pushup(int p) { sum[p] = sum[p << 1] + sum[p << 1 | 1]; } void pushdown(int p, int l, int r) { if (lazy[p]) { int mid = (l + r) >> 1; lazy[p << 1] += lazy[p]; lazy[p << 1 | 1] += lazy[p]; sum[p << 1] += lazy[p] * (mid - l + 1); sum[p << 1 | 1] += lazy[p] * (r - mid); lazy[p] = 0; } } void build(int p, int l, int r) { if (l == r) { sum[p] = 0; return; } int mid = (l + r) >> 1; build(p << 1, l, mid); build(p << 1 | 1, mid + 1, r); pushup(p); } void update(int p, int l, int r, int ql, int qr, ll w) { if (ql <= l && r <= qr) { sum[p] += w * (r - l + 1); lazy[p] += w; return; } pushdown(p, l, r); int mid = (l + r) >> 1; if (ql <= mid) update(p << 1, l, mid, ql, qr, w); if (qr > mid) update(p << 1 | 1, mid + 1, r, ql, qr, w); pushup(p); } ll query(int p, int l, int r, int ql, int qr) { if (ql <= l && r <= qr) return sum[p]; pushdown(p, l, r); int mid = (l + r) >> 1; ll res = 0; if (ql <= mid) res += query(p << 1, l, mid, ql, qr); if (qr > mid) res += query(p << 1 | 1, mid + 1, r, ql, qr); return res; } void upd_path(int u, int v, ll w) { while (top[u] != top[v]) { if (dep[top[u]] < dep[top[v]]) swap(u, v); update(1, 1, n, dfn[top[u]], dfn[u], w); u = fa[top[u]]; } if (dep[u] > dep[v]) swap(u, v); update(1, 1, n, dfn[u], dfn[v], w); } ll qry_path(int u, int v) { ll res = 0; while (top[u] != top[v]) { if (dep[top[u]] < dep[top[v]]) swap(u, v); res += query(1, 1, n, dfn[top[u]], dfn[u]); u = fa[top[u]]; } if (dep[u] > dep[v]) swap(u, v); res += query(1, 1, n, dfn[u], dfn[v]); return res; } int main() { ios::sync_with_stdio(false); cin.tie(nullptr); cin >> n >> m; for (int i = 1; i < n; i++) { int x, y; cin >> x >> y; add(x, y); add(y, x); } dfs1(1, 0); dfs2(1, 1); build(1, 1, n); while (m--) { string op; cin >> op; if (op == "Add") { int u, v; ll w; cin >> u >> v >> w; upd_path(u, v, w); } else if (op == "Query") { int u, v; cin >> u >> v; cout << qry_path(u, v) << '\n'; } } return 0; }