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#include "graph/ds/tree_abelgroup.hpp"
#include "ds/fenwicktree/fenwicktree.hpp"
#include "graph/tree.hpp"
template <typename TREE, typename AbelGroup, bool edge, bool path_query,
bool subtree_query>
struct Tree_AbelGroup {
using MX = AbelGroup;
using X = typename MX::value_type;
TREE &tree;
int N;
FenwickTree<MX> bit, bit_subtree;
Tree_AbelGroup(TREE &tree) : tree(tree), N(tree.N) {
build([](int i) -> X { return MX::unit(); });
}
Tree_AbelGroup(TREE &tree, vc<X> &dat) : tree(tree), N(tree.N) {
build([&](int i) -> X { return dat[i]; });
}
template <typename F>
Tree_AbelGroup(TREE &tree, F f) : tree(tree), N(tree.N) {
build(f);
}
template <typename F>
void build(F f) {
vc<X> bit_raw_1(2 * N);
vc<X> bit_raw_2(N);
FOR(v, N) {
X x = MX::unit();
if (!edge) x = f(v);
if (edge) x = (v == 0 ? MX::unit() : f(tree.v_to_e(v)));
bit_raw_1[tree.ELID(v)] = x;
bit_raw_1[tree.ERID(v)] = MX::inverse(x);
bit_raw_2[tree.LID[v]] = x;
}
if constexpr (path_query) bit.build(bit_raw_1);
if constexpr (subtree_query) bit_subtree.build(bit_raw_2);
}
void add(int i, X x) {
int v = (edge ? tree.e_to_v(i) : i);
if constexpr (path_query) {
bit.add(tree.ELID(v), x);
bit.add(tree.ERID(v), MX::inverse(x));
}
if constexpr (subtree_query) bit_subtree.add(tree.LID[v], x);
}
X prod_path(int frm, int to) {
static_assert(path_query);
int lca = tree.LCA(frm, to);
// [frm, lca)
X x1 = bit.prod(tree.ELID(lca) + 1, tree.ELID(frm) + 1);
// edge なら (lca, to]、vertex なら [lca, to]
X x2 = bit.prod(tree.ELID(lca) + edge, tree.ELID(to) + 1);
return MX::op(x1, x2);
}
X prod_subtree(int u) {
static_assert(subtree_query);
int l = tree.LID[u], r = tree.RID[u];
return bit_subtree.prod(l + edge, r);
}
};
#line 2 "alg/monoid/add.hpp"
template <typename X>
struct Monoid_Add {
using value_type = X;
static constexpr X op(const X &x, const X &y) noexcept { return x + y; }
static constexpr X inverse(const X &x) noexcept { return -x; }
static constexpr X power(const X &x, ll n) noexcept { return X(n) * x; }
static constexpr X unit() { return X(0); }
static constexpr bool commute = true;
};
#line 3 "ds/fenwicktree/fenwicktree.hpp"
template <typename Monoid>
struct FenwickTree {
using G = Monoid;
using E = typename G::value_type;
int n;
vector<E> dat;
E total;
FenwickTree() {}
FenwickTree(int n) { build(n); }
template <typename F>
FenwickTree(int n, F f) {
build(n, f);
}
FenwickTree(const vc<E>& v) { build(v); }
void build(int m) {
n = m;
dat.assign(m, G::unit());
total = G::unit();
}
void build(const vc<E>& v) {
build(len(v), [&](int i) -> E { return v[i]; });
}
template <typename F>
void build(int m, F f) {
n = m;
dat.clear();
dat.reserve(n);
total = G::unit();
FOR(i, n) { dat.eb(f(i)); }
for (int i = 1; i <= n; ++i) {
int j = i + (i & -i);
if (j <= n) dat[j - 1] = G::op(dat[i - 1], dat[j - 1]);
}
total = prefix_sum(m);
}
E prod_all() { return total; }
E sum_all() { return total; }
E sum(int k) { return prefix_sum(k); }
E prod(int k) { return prefix_prod(k); }
E prefix_sum(int k) { return prefix_prod(k); }
E prefix_prod(int k) {
chmin(k, n);
E ret = G::unit();
for (; k > 0; k -= k & -k) ret = G::op(ret, dat[k - 1]);
return ret;
}
E sum(int L, int R) { return prod(L, R); }
E prod(int L, int R) {
chmax(L, 0), chmin(R, n);
if (L == 0) return prefix_prod(R);
assert(0 <= L && L <= R && R <= n);
E pos = G::unit(), neg = G::unit();
while (L < R) { pos = G::op(pos, dat[R - 1]), R -= R & -R; }
while (R < L) { neg = G::op(neg, dat[L - 1]), L -= L & -L; }
return G::op(pos, G::inverse(neg));
}
void add(int k, E x) { multiply(k, x); }
void multiply(int k, E x) {
static_assert(G::commute);
total = G::op(total, x);
for (++k; k <= n; k += k & -k) dat[k - 1] = G::op(dat[k - 1], x);
}
template <class F>
int max_right(const F check) {
assert(check(G::unit()));
int i = 0;
E s = G::unit();
int k = 1;
while (2 * k <= n) k *= 2;
while (k) {
if (i + k - 1 < len(dat)) {
E t = G::op(s, dat[i + k - 1]);
if (check(t)) { i += k, s = t; }
}
k >>= 1;
}
return i;
}
// check(i, x)
template <class F>
int max_right_with_index(const F check) {
assert(check(0, G::unit()));
int i = 0;
E s = G::unit();
int k = 1;
while (2 * k <= n) k *= 2;
while (k) {
if (i + k - 1 < len(dat)) {
E t = G::op(s, dat[i + k - 1]);
if (check(i + k, t)) { i += k, s = t; }
}
k >>= 1;
}
return i;
}
int kth(E k) {
return max_right([&k](E x) -> bool { return x <= k; });
}
};
#line 2 "graph/tree.hpp"
#line 2 "graph/base.hpp"
template <typename T>
struct Edge {
int frm, to;
T cost;
int id;
};
template <typename T = int, bool directed = false>
struct Graph {
static constexpr bool is_directed = directed;
int N, M;
using cost_type = T;
using edge_type = Edge<T>;
vector<edge_type> edges;
vector<int> indptr;
vector<edge_type> csr_edges;
vc<int> vc_deg, vc_indeg, vc_outdeg;
bool prepared;
class OutgoingEdges {
public:
OutgoingEdges(const Graph* G, int l, int r) : G(G), l(l), r(r) {}
const edge_type* begin() const {
if (l == r) { return 0; }
return &G->csr_edges[l];
}
const edge_type* end() const {
if (l == r) { return 0; }
return &G->csr_edges[r];
}
private:
const Graph* G;
int l, r;
};
bool is_prepared() { return prepared; }
Graph() : N(0), M(0), prepared(0) {}
Graph(int N) : N(N), M(0), prepared(0) {}
void build(int n) {
N = n, M = 0;
prepared = 0;
edges.clear();
indptr.clear();
csr_edges.clear();
vc_deg.clear();
vc_indeg.clear();
vc_outdeg.clear();
}
void add(int frm, int to, T cost = 1, int i = -1) {
assert(!prepared);
assert(0 <= frm && 0 <= to && to < N);
if (i == -1) i = M;
auto e = edge_type({frm, to, cost, i});
edges.eb(e);
++M;
}
#ifdef FASTIO
// wt, off
void read_tree(bool wt = false, int off = 1) { read_graph(N - 1, wt, off); }
void read_graph(int M, bool wt = false, int off = 1) {
for (int m = 0; m < M; ++m) {
INT(a, b);
a -= off, b -= off;
if (!wt) {
add(a, b);
} else {
T c;
read(c);
add(a, b, c);
}
}
build();
}
#endif
void build() {
assert(!prepared);
prepared = true;
indptr.assign(N + 1, 0);
for (auto&& e: edges) {
indptr[e.frm + 1]++;
if (!directed) indptr[e.to + 1]++;
}
for (int v = 0; v < N; ++v) { indptr[v + 1] += indptr[v]; }
auto counter = indptr;
csr_edges.resize(indptr.back() + 1);
for (auto&& e: edges) {
csr_edges[counter[e.frm]++] = e;
if (!directed)
csr_edges[counter[e.to]++] = edge_type({e.to, e.frm, e.cost, e.id});
}
}
OutgoingEdges operator[](int v) const {
assert(prepared);
return {this, indptr[v], indptr[v + 1]};
}
vc<int> deg_array() {
if (vc_deg.empty()) calc_deg();
return vc_deg;
}
pair<vc<int>, vc<int>> deg_array_inout() {
if (vc_indeg.empty()) calc_deg_inout();
return {vc_indeg, vc_outdeg};
}
int deg(int v) {
if (vc_deg.empty()) calc_deg();
return vc_deg[v];
}
int in_deg(int v) {
if (vc_indeg.empty()) calc_deg_inout();
return vc_indeg[v];
}
int out_deg(int v) {
if (vc_outdeg.empty()) calc_deg_inout();
return vc_outdeg[v];
}
#ifdef FASTIO
void debug() {
print("Graph");
if (!prepared) {
print("frm to cost id");
for (auto&& e: edges) print(e.frm, e.to, e.cost, e.id);
} else {
print("indptr", indptr);
print("frm to cost id");
FOR(v, N) for (auto&& e: (*this)[v]) print(e.frm, e.to, e.cost, e.id);
}
}
#endif
vc<int> new_idx;
vc<bool> used_e;
// G における頂点 V[i] が、新しいグラフで i になるようにする
// {G, es}
Graph<T, directed> rearrange(vc<int> V, bool keep_eid = 0) {
if (len(new_idx) != N) new_idx.assign(N, -1);
if (len(used_e) != M) used_e.assign(M, 0);
int n = len(V);
FOR(i, n) new_idx[V[i]] = i;
Graph<T, directed> G(n);
vc<int> history;
FOR(i, n) {
for (auto&& e: (*this)[V[i]]) {
if (used_e[e.id]) continue;
int a = e.frm, b = e.to;
if (new_idx[a] != -1 && new_idx[b] != -1) {
history.eb(e.id);
used_e[e.id] = 1;
int eid = (keep_eid ? e.id : -1);
G.add(new_idx[a], new_idx[b], e.cost, eid);
}
}
}
FOR(i, n) new_idx[V[i]] = -1;
for (auto&& eid: history) used_e[eid] = 0;
G.build();
return G;
}
private:
void calc_deg() {
assert(vc_deg.empty());
vc_deg.resize(N);
for (auto&& e: edges) vc_deg[e.frm]++, vc_deg[e.to]++;
}
void calc_deg_inout() {
assert(vc_indeg.empty());
vc_indeg.resize(N);
vc_outdeg.resize(N);
for (auto&& e: edges) { vc_indeg[e.to]++, vc_outdeg[e.frm]++; }
}
};
#line 4 "graph/tree.hpp"
// HLD euler tour をとっていろいろ。
template <typename GT>
struct Tree {
using Graph_type = GT;
GT &G;
using WT = typename GT::cost_type;
int N;
vector<int> LID, RID, head, V, parent, VtoE;
vc<int> depth;
vc<WT> depth_weighted;
Tree(GT &G, int r = 0, bool hld = 1) : G(G) { build(r, hld); }
void build(int r = 0, bool hld = 1) {
if (r == -1) return; // build を遅延したいとき
N = G.N;
LID.assign(N, -1), RID.assign(N, -1), head.assign(N, r);
V.assign(N, -1), parent.assign(N, -1), VtoE.assign(N, -1);
depth.assign(N, -1), depth_weighted.assign(N, 0);
assert(G.is_prepared());
int t1 = 0;
dfs_sz(r, -1, hld);
dfs_hld(r, t1);
}
void dfs_sz(int v, int p, bool hld) {
auto &sz = RID;
parent[v] = p;
depth[v] = (p == -1 ? 0 : depth[p] + 1);
sz[v] = 1;
int l = G.indptr[v], r = G.indptr[v + 1];
auto &csr = G.csr_edges;
// 使う辺があれば先頭にする
for (int i = r - 2; i >= l; --i) {
if (hld && depth[csr[i + 1].to] == -1) swap(csr[i], csr[i + 1]);
}
int hld_sz = 0;
for (int i = l; i < r; ++i) {
auto e = csr[i];
if (depth[e.to] != -1) continue;
depth_weighted[e.to] = depth_weighted[v] + e.cost;
VtoE[e.to] = e.id;
dfs_sz(e.to, v, hld);
sz[v] += sz[e.to];
if (hld && chmax(hld_sz, sz[e.to]) && l < i) { swap(csr[l], csr[i]); }
}
}
void dfs_hld(int v, int ×) {
LID[v] = times++;
RID[v] += LID[v];
V[LID[v]] = v;
bool heavy = true;
for (auto &&e: G[v]) {
if (depth[e.to] <= depth[v]) continue;
head[e.to] = (heavy ? head[v] : e.to);
heavy = false;
dfs_hld(e.to, times);
}
}
vc<int> heavy_path_at(int v) {
vc<int> P = {v};
while (1) {
int a = P.back();
for (auto &&e: G[a]) {
if (e.to != parent[a] && head[e.to] == v) {
P.eb(e.to);
break;
}
}
if (P.back() == a) break;
}
return P;
}
int heavy_child(int v) {
int k = LID[v] + 1;
if (k == N) return -1;
int w = V[k];
return (parent[w] == v ? w : -1);
}
int e_to_v(int eid) {
auto e = G.edges[eid];
return (parent[e.frm] == e.to ? e.frm : e.to);
}
int v_to_e(int v) { return VtoE[v]; }
int ELID(int v) { return 2 * LID[v] - depth[v]; }
int ERID(int v) { return 2 * RID[v] - depth[v] - 1; }
// 目標地点へ進む個数が k
int LA(int v, int k) {
assert(k <= depth[v]);
while (1) {
int u = head[v];
if (LID[v] - k >= LID[u]) return V[LID[v] - k];
k -= LID[v] - LID[u] + 1;
v = parent[u];
}
}
int la(int u, int v) { return LA(u, v); }
int LCA(int u, int v) {
for (;; v = parent[head[v]]) {
if (LID[u] > LID[v]) swap(u, v);
if (head[u] == head[v]) return u;
}
}
// root を根とした場合の lca
int LCA_root(int u, int v, int root) {
return LCA(u, v) ^ LCA(u, root) ^ LCA(v, root);
}
int lca(int u, int v) { return LCA(u, v); }
int lca_root(int u, int v, int root) { return LCA_root(u, v, root); }
int subtree_size(int v, int root = -1) {
if (root == -1) return RID[v] - LID[v];
if (v == root) return N;
int x = jump(v, root, 1);
if (in_subtree(v, x)) return RID[v] - LID[v];
return N - RID[x] + LID[x];
}
int dist(int a, int b) {
int c = LCA(a, b);
return depth[a] + depth[b] - 2 * depth[c];
}
WT dist_weighted(int a, int b) {
int c = LCA(a, b);
return depth_weighted[a] + depth_weighted[b] - WT(2) * depth_weighted[c];
}
// a is in b
bool in_subtree(int a, int b) { return LID[b] <= LID[a] && LID[a] < RID[b]; }
int jump(int a, int b, ll k) {
if (k == 1) {
if (a == b) return -1;
return (in_subtree(b, a) ? LA(b, depth[b] - depth[a] - 1) : parent[a]);
}
int c = LCA(a, b);
int d_ac = depth[a] - depth[c];
int d_bc = depth[b] - depth[c];
if (k > d_ac + d_bc) return -1;
if (k <= d_ac) return LA(a, k);
return LA(b, d_ac + d_bc - k);
}
vc<int> collect_child(int v) {
vc<int> res;
for (auto &&e: G[v])
if (e.to != parent[v]) res.eb(e.to);
return res;
}
vc<int> collect_light(int v) {
vc<int> res;
bool skip = true;
for (auto &&e: G[v])
if (e.to != parent[v]) {
if (!skip) res.eb(e.to);
skip = false;
}
return res;
}
vc<pair<int, int>> get_path_decomposition(int u, int v, bool edge) {
// [始点, 終点] の"閉"区間列。
vc<pair<int, int>> up, down;
while (1) {
if (head[u] == head[v]) break;
if (LID[u] < LID[v]) {
down.eb(LID[head[v]], LID[v]);
v = parent[head[v]];
} else {
up.eb(LID[u], LID[head[u]]);
u = parent[head[u]];
}
}
if (LID[u] < LID[v]) down.eb(LID[u] + edge, LID[v]);
elif (LID[v] + edge <= LID[u]) up.eb(LID[u], LID[v] + edge);
reverse(all(down));
up.insert(up.end(), all(down));
return up;
}
vc<int> restore_path(int u, int v) {
vc<int> P;
for (auto &&[a, b]: get_path_decomposition(u, v, 0)) {
if (a <= b) {
FOR(i, a, b + 1) P.eb(V[i]);
} else {
FOR_R(i, b, a + 1) P.eb(V[i]);
}
}
return P;
}
};
#line 3 "graph/ds/tree_abelgroup.hpp"
template <typename TREE, typename AbelGroup, bool edge, bool path_query,
bool subtree_query>
struct Tree_AbelGroup {
using MX = AbelGroup;
using X = typename MX::value_type;
TREE &tree;
int N;
FenwickTree<MX> bit, bit_subtree;
Tree_AbelGroup(TREE &tree) : tree(tree), N(tree.N) {
build([](int i) -> X { return MX::unit(); });
}
Tree_AbelGroup(TREE &tree, vc<X> &dat) : tree(tree), N(tree.N) {
build([&](int i) -> X { return dat[i]; });
}
template <typename F>
Tree_AbelGroup(TREE &tree, F f) : tree(tree), N(tree.N) {
build(f);
}
template <typename F>
void build(F f) {
vc<X> bit_raw_1(2 * N);
vc<X> bit_raw_2(N);
FOR(v, N) {
X x = MX::unit();
if (!edge) x = f(v);
if (edge) x = (v == 0 ? MX::unit() : f(tree.v_to_e(v)));
bit_raw_1[tree.ELID(v)] = x;
bit_raw_1[tree.ERID(v)] = MX::inverse(x);
bit_raw_2[tree.LID[v]] = x;
}
if constexpr (path_query) bit.build(bit_raw_1);
if constexpr (subtree_query) bit_subtree.build(bit_raw_2);
}
void add(int i, X x) {
int v = (edge ? tree.e_to_v(i) : i);
if constexpr (path_query) {
bit.add(tree.ELID(v), x);
bit.add(tree.ERID(v), MX::inverse(x));
}
if constexpr (subtree_query) bit_subtree.add(tree.LID[v], x);
}
X prod_path(int frm, int to) {
static_assert(path_query);
int lca = tree.LCA(frm, to);
// [frm, lca)
X x1 = bit.prod(tree.ELID(lca) + 1, tree.ELID(frm) + 1);
// edge なら (lca, to]、vertex なら [lca, to]
X x2 = bit.prod(tree.ELID(lca) + edge, tree.ELID(to) + 1);
return MX::op(x1, x2);
}
X prod_subtree(int u) {
static_assert(subtree_query);
int l = tree.LID[u], r = tree.RID[u];
return bit_subtree.prod(l + edge, r);
}
};