This documentation is automatically generated by online-judge-tools/verification-helper
#include "graph/st_numbering.hpp"
#include "graph/base.hpp"
#include "ds/unionfind/unionfind.hpp"
#include "graph/block_cut.hpp"
#include "graph/shortest_path/bfs01.hpp"
#include "graph/shortest_path/restore_path.hpp"
// https://en.wikipedia.org/wiki/Bipolar_orientation
// 順列 p を求める. p[s]=0,p[t]=n-1.
// p[u]<p[v] となる向きに辺を向き付けると任意の v に対して svt パスが存在.
// 存在条件:BCT で全部の成分を通る st パスがある 不可能ならば empty をかえす.
template <typename GT>
vc<int> st_numbering(GT &G, int s, int t) {
static_assert(!GT::is_directed);
assert(G.is_prepared());
int N = G.N;
if (N == 1) return {0};
if (s == t) return {};
vc<int> par(N, -1), pre(N, -1), low(N, -1);
vc<int> V;
auto dfs = [&](auto &dfs, int v) -> void {
pre[v] = len(V), V.eb(v);
low[v] = v;
for (auto &e: G[v]) {
int w = e.to;
if (v == w) continue;
if (pre[w] == -1) {
dfs(dfs, w);
par[w] = v;
if (pre[low[w]] < pre[low[v]]) { low[v] = low[w]; }
}
elif (pre[w] < pre[low[v]]) { low[v] = w; }
}
};
pre[s] = 0, V.eb(s);
dfs(dfs, t);
if (len(V) != N) return {};
vc<int> nxt(N, -1), prev(N);
nxt[s] = t, prev[t] = s;
vc<int> sgn(N);
sgn[s] = -1;
FOR(i, 2, len(V)) {
int v = V[i];
int p = par[v];
if (sgn[low[v]] == -1) {
int q = prev[p];
if (q == -1) return {};
nxt[q] = v, nxt[v] = p;
prev[v] = q, prev[p] = v;
sgn[p] = 1;
} else {
int q = nxt[p];
if (q == -1) return {};
nxt[p] = v, nxt[v] = q;
prev[v] = p, prev[q] = v;
sgn[p] = -1;
}
}
vc<int> A = {s};
while (A.back() != t) { A.eb(nxt[A.back()]); }
// 作れているか判定
if (len(A) < N) return {};
assert(A[0] == s && A.back() == t);
vc<int> rk(N, -1);
FOR(i, N) rk[A[i]] = i;
assert(MIN(rk) != -1);
FOR(i, N) {
bool l = 0, r = 0;
int v = A[i];
for (auto &e: G[v]) {
if (rk[e.to] < rk[v]) l = 1;
if (rk[v] < rk[e.to]) r = 1;
}
if (i > 0 && !l) return {};
if (i < N - 1 && !r) return {};
}
vc<int> res(N);
FOR(i, N) res[A[i]] = i;
return res;
}
bool check_st_numbering(Graph<int, 0> G, int s, int t) {
int N = G.N;
assert(N >= 2);
if (s == t) return 0;
UnionFind uf(N);
for (auto &e: G.edges) uf.merge(e.frm, e.to);
if (uf.n_comp >= 2) return 0; // disconnected
// BCT において st パスがすべての block を通ることが必要
auto BCT = block_cut(G);
auto [dist, par] = bfs01<int>(BCT, s);
vc<int> path = restore_path(par, t);
vc<int> vis(BCT.N);
for (auto &x: path) vis[x] = 1;
FOR(i, N, BCT.N) {
if (!vis[i]) return 0;
}
return 1;
}
#line 1 "graph/st_numbering.hpp"
#line 2 "ds/hashmap.hpp"
// u64 -> Val
template <typename Val>
struct HashMap {
// n は入れたいものの個数で ok
HashMap(u32 n = 0) { build(n); }
void build(u32 n) {
u32 k = 8;
while (k < n * 2) k *= 2;
cap = k / 2, mask = k - 1;
key.resize(k), val.resize(k), used.assign(k, 0);
}
// size を保ったまま. size=0 にするときは build すること.
void clear() {
used.assign(len(used), 0);
cap = (mask + 1) / 2;
}
int size() { return len(used) / 2 - cap; }
int index(const u64& k) {
int i = 0;
for (i = hash(k); used[i] && key[i] != k; i = (i + 1) & mask) {}
return i;
}
Val& operator[](const u64& k) {
if (cap == 0) extend();
int i = index(k);
if (!used[i]) { used[i] = 1, key[i] = k, val[i] = Val{}, --cap; }
return val[i];
}
Val get(const u64& k, Val default_value) {
int i = index(k);
return (used[i] ? val[i] : default_value);
}
bool count(const u64& k) {
int i = index(k);
return used[i] && key[i] == k;
}
// f(key, val)
template <typename F>
void enumerate_all(F f) {
FOR(i, len(used)) if (used[i]) f(key[i], val[i]);
}
private:
u32 cap, mask;
vc<u64> key;
vc<Val> val;
vc<bool> used;
u64 hash(u64 x) {
static const u64 FIXED_RANDOM = std::chrono::steady_clock::now().time_since_epoch().count();
x += FIXED_RANDOM;
x = (x ^ (x >> 30)) * 0xbf58476d1ce4e5b9;
x = (x ^ (x >> 27)) * 0x94d049bb133111eb;
return (x ^ (x >> 31)) & mask;
}
void extend() {
vc<pair<u64, Val>> dat;
dat.reserve(len(used) / 2 - cap);
FOR(i, len(used)) {
if (used[i]) dat.eb(key[i], val[i]);
}
build(2 * len(dat));
for (auto& [a, b]: dat) (*this)[a] = b;
}
};
#line 3 "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() {
#ifdef LOCAL
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
}
#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}
// sum(deg(v)) の計算量になっていて、
// 新しいグラフの n+m より大きい可能性があるので注意
Graph<T, directed> rearrange(vc<int> V, bool keep_eid = 0) {
if (len(new_idx) != N) new_idx.assign(N, -1);
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 (len(used_e) <= e.id) used_e.resize(e.id + 1);
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;
}
Graph<T, true> to_directed_tree(int root = -1) {
if (root == -1) root = 0;
assert(!is_directed && prepared && M == N - 1);
Graph<T, true> G1(N);
vc<int> par(N, -1);
auto dfs = [&](auto& dfs, int v) -> void {
for (auto& e: (*this)[v]) {
if (e.to == par[v]) continue;
par[e.to] = v, dfs(dfs, e.to);
}
};
dfs(dfs, root);
for (auto& e: edges) {
int a = e.frm, b = e.to;
if (par[a] == b) swap(a, b);
assert(par[b] == a);
G1.add(a, b, e.cost);
}
G1.build();
return G1;
}
HashMap<int> MP_FOR_EID;
int get_eid(u64 a, u64 b) {
if (len(MP_FOR_EID) == 0) {
MP_FOR_EID.build(N - 1);
for (auto& e: edges) {
u64 a = e.frm, b = e.to;
u64 k = to_eid_key(a, b);
MP_FOR_EID[k] = e.id;
}
}
return MP_FOR_EID.get(to_eid_key(a, b), -1);
}
u64 to_eid_key(u64 a, u64 b) {
if (!directed && a > b) swap(a, b);
return N * a + b;
}
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 2 "ds/unionfind/unionfind.hpp"
struct UnionFind {
int n, n_comp;
vc<int> dat; // par or (-size)
UnionFind(int n = 0) { build(n); }
void build(int m) {
n = m, n_comp = m;
dat.assign(n, -1);
}
void reset() { build(n); }
int operator[](int x) {
while (dat[x] >= 0) {
int pp = dat[dat[x]];
if (pp < 0) { return dat[x]; }
x = dat[x] = pp;
}
return x;
}
ll size(int x) {
x = (*this)[x];
return -dat[x];
}
bool merge(int x, int y) {
x = (*this)[x], y = (*this)[y];
if (x == y) return false;
if (-dat[x] < -dat[y]) swap(x, y);
dat[x] += dat[y], dat[y] = x, n_comp--;
return true;
}
vc<int> get_all() {
vc<int> A(n);
FOR(i, n) A[i] = (*this)[i];
return A;
}
};
#line 2 "graph/block_cut.hpp"
/*
block-cut tree を、block に通常の頂点を隣接させて拡張しておく
https://twitter.com/noshi91/status/1529858538650374144?s=20&t=eznpFbuD9BDhfTb4PplFUg
[0, n):もとの頂点 [n, n + n_block):block
関節点:[0, n) のうちで、degree >= 2 を満たすもの
孤立点は、1 点だけからなる block
成分が欲しい場合:近傍を見ると点集合. 辺から成分を得るには tree.jump
と思ったが非連結なときに注意がいるな…
*/
template <typename GT>
Graph<int, 0> block_cut(GT& G) {
int n = G.N;
vc<int> low(n), ord(n), st;
vc<bool> used(n);
st.reserve(n);
int nxt = n;
int k = 0;
vc<pair<int, int>> edges;
auto dfs = [&](auto& dfs, int v, int p) -> void {
st.eb(v);
used[v] = 1;
low[v] = ord[v] = k++;
int child = 0;
for (auto&& e: G[v]) {
if (e.to == p) continue;
if (!used[e.to]) {
++child;
int s = len(st);
dfs(dfs, e.to, v);
chmin(low[v], low[e.to]);
if ((p == -1 && child > 1) || (p != -1 && low[e.to] >= ord[v])) {
edges.eb(nxt, v);
while (len(st) > s) {
edges.eb(nxt, st.back());
st.pop_back();
}
++nxt;
}
} else {
chmin(low[v], ord[e.to]);
}
}
};
FOR(v, n) if (!used[v]) {
dfs(dfs, v, -1);
for (auto&& x: st) { edges.eb(nxt, x); }
++nxt;
st.clear();
}
Graph<int, 0> BCT(nxt);
for (auto&& [a, b]: edges) BCT.add(a, b);
BCT.build();
return BCT;
}
#line 3 "graph/shortest_path/bfs01.hpp"
template <typename T, typename GT>
pair<vc<T>, vc<int>> bfs01(GT& G, int v) {
assert(G.is_prepared());
int N = G.N;
vc<T> dist(N, infty<T>);
vc<int> par(N, -1);
deque<int> que;
dist[v] = 0;
que.push_front(v);
while (!que.empty()) {
auto v = que.front();
que.pop_front();
for (auto&& e: G[v]) {
if (dist[e.to] == infty<T> || dist[e.to] > dist[e.frm] + e.cost) {
dist[e.to] = dist[e.frm] + e.cost;
par[e.to] = e.frm;
if (e.cost == 0)
que.push_front(e.to);
else
que.push_back(e.to);
}
}
}
return {dist, par};
}
// 多点スタート。[dist, par, root]
template <typename T, typename GT>
tuple<vc<T>, vc<int>, vc<int>> bfs01(GT& G, vc<int> vs) {
assert(G.is_prepared());
int N = G.N;
vc<T> dist(N, infty<T>);
vc<int> par(N, -1);
vc<int> root(N, -1);
deque<int> que;
for (auto&& v: vs) {
dist[v] = 0;
root[v] = v;
que.push_front(v);
}
while (!que.empty()) {
auto v = que.front();
que.pop_front();
for (auto&& e: G[v]) {
if (dist[e.to] == infty<T> || dist[e.to] > dist[e.frm] + e.cost) {
dist[e.to] = dist[e.frm] + e.cost;
root[e.to] = root[e.frm];
par[e.to] = e.frm;
if (e.cost == 0)
que.push_front(e.to);
else
que.push_back(e.to);
}
}
}
return {dist, par, root};
}
#line 1 "graph/shortest_path/restore_path.hpp"
vector<int> restore_path(vector<int> par, int t){
vector<int> pth = {t};
while (par[pth.back()] != -1) pth.eb(par[pth.back()]);
reverse(all(pth));
return pth;
}
#line 7 "graph/st_numbering.hpp"
// https://en.wikipedia.org/wiki/Bipolar_orientation
// 順列 p を求める. p[s]=0,p[t]=n-1.
// p[u]<p[v] となる向きに辺を向き付けると任意の v に対して svt パスが存在.
// 存在条件:BCT で全部の成分を通る st パスがある 不可能ならば empty をかえす.
template <typename GT>
vc<int> st_numbering(GT &G, int s, int t) {
static_assert(!GT::is_directed);
assert(G.is_prepared());
int N = G.N;
if (N == 1) return {0};
if (s == t) return {};
vc<int> par(N, -1), pre(N, -1), low(N, -1);
vc<int> V;
auto dfs = [&](auto &dfs, int v) -> void {
pre[v] = len(V), V.eb(v);
low[v] = v;
for (auto &e: G[v]) {
int w = e.to;
if (v == w) continue;
if (pre[w] == -1) {
dfs(dfs, w);
par[w] = v;
if (pre[low[w]] < pre[low[v]]) { low[v] = low[w]; }
}
elif (pre[w] < pre[low[v]]) { low[v] = w; }
}
};
pre[s] = 0, V.eb(s);
dfs(dfs, t);
if (len(V) != N) return {};
vc<int> nxt(N, -1), prev(N);
nxt[s] = t, prev[t] = s;
vc<int> sgn(N);
sgn[s] = -1;
FOR(i, 2, len(V)) {
int v = V[i];
int p = par[v];
if (sgn[low[v]] == -1) {
int q = prev[p];
if (q == -1) return {};
nxt[q] = v, nxt[v] = p;
prev[v] = q, prev[p] = v;
sgn[p] = 1;
} else {
int q = nxt[p];
if (q == -1) return {};
nxt[p] = v, nxt[v] = q;
prev[v] = p, prev[q] = v;
sgn[p] = -1;
}
}
vc<int> A = {s};
while (A.back() != t) { A.eb(nxt[A.back()]); }
// 作れているか判定
if (len(A) < N) return {};
assert(A[0] == s && A.back() == t);
vc<int> rk(N, -1);
FOR(i, N) rk[A[i]] = i;
assert(MIN(rk) != -1);
FOR(i, N) {
bool l = 0, r = 0;
int v = A[i];
for (auto &e: G[v]) {
if (rk[e.to] < rk[v]) l = 1;
if (rk[v] < rk[e.to]) r = 1;
}
if (i > 0 && !l) return {};
if (i < N - 1 && !r) return {};
}
vc<int> res(N);
FOR(i, N) res[A[i]] = i;
return res;
}
bool check_st_numbering(Graph<int, 0> G, int s, int t) {
int N = G.N;
assert(N >= 2);
if (s == t) return 0;
UnionFind uf(N);
for (auto &e: G.edges) uf.merge(e.frm, e.to);
if (uf.n_comp >= 2) return 0; // disconnected
// BCT において st パスがすべての block を通ることが必要
auto BCT = block_cut(G);
auto [dist, par] = bfs01<int>(BCT, s);
vc<int> path = restore_path(par, t);
vc<int> vis(BCT.N);
for (auto &x: path) vis[x] = 1;
FOR(i, N, BCT.N) {
if (!vis[i]) return 0;
}
return 1;
}