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#include "Graph/dmdecomp.hpp"
#pragma once #include "Graph/bimatching.hpp" #include "Graph/scc.hpp" vector<vector<int>> DMdecomposition(int n, int m, vector<vector<int>> &g, vector<int> match) { if (match.empty()) match = BiMatching(n, m, g); vector G(n + m, vector<int>()), REV(n + m, vector<int>()); rep(i, 0, n) for (auto &j : g[i]) { G[i].push_back(j + n); REV[j + n].push_back(i); } vector<int> R(m, -1); rep(i, 0, n) if (match[i] != -1) { G[match[i] + n].push_back(i); REV[i].push_back(match[i] + n); R[match[i]] = i; } vector<int> V0, Vinf; queue<int> que; vector<int> used(n + m); rep(i, 0, n) if (match[i] == -1) { used[i] = 1; que.push(i); } while (!que.empty()) { int v = que.front(); que.pop(); Vinf.push_back(v); for (auto &to : G[v]) if (!used[to]) { used[to] = 1; que.push(to); } } rep(i, 0, m) if (R[i] == -1) { used[i + n] = 1; que.push(i + n); } while (!que.empty()) { int v = que.front(); que.pop(); V0.push_back(v); for (auto &to : REV[v]) if (!used[to]) { used[to] = 1; que.push(to); } } SCC scc(n + m); rep(i, 0, n + m) for (auto &to : G[i]) { if (!used[i] and !used[to]) scc.add_edge(i, to); } scc.run(); vector group(scc.m, vector<int>()); rep(i, 0, n + m) if (!used[i]) group[scc.id[i]].push_back(i); vector<vector<int>> ret; ret.push_back(V0); for (auto &v : group) ret.push_back(v); ret.push_back(Vinf); return ret; } /** * @brief DM decomposition */
#line 2 "Utility/random.hpp" namespace Random { mt19937_64 randgen(chrono::steady_clock::now().time_since_epoch().count()); using u64 = unsigned long long; u64 get() { return randgen(); } template <typename T> T get(T L) { // [0,L] return get() % (L + 1); } template <typename T> T get(T L, T R) { // [L,R] return get(R - L) + L; } double uniform() { return double(get(1000000000)) / 1000000000; } string str(int n) { string ret; rep(i, 0, n) ret += get('a', 'z'); return ret; } template <typename Iter> void shuffle(Iter first, Iter last) { if (first == last) return; int len = 1; for (auto it = first + 1; it != last; it++) { len++; int j = get(0, len - 1); if (j != len - 1) iter_swap(it, first + j); } } template <typename T> vector<T> select(int n, T L, T R) { // [L,R] if (n * 2 >= R - L + 1) { vector<T> ret(R - L + 1); iota(ALL(ret), L); shuffle(ALL(ret)); ret.resize(n); return ret; } else { unordered_set<T> used; vector<T> ret; while (SZ(used) < n) { T x = get(L, R); if (!used.count(x)) { used.insert(x); ret.push_back(x); } } return ret; } } void relabel(int n, vector<pair<int, int>> &es) { shuffle(ALL(es)); vector<int> ord(n); iota(ALL(ord), 0); shuffle(ALL(ord)); for (auto &[u, v] : es) u = ord[u], v = ord[v]; } template <bool directed, bool simple> vector<pair<int, int>> genGraph(int n) { vector<pair<int, int>> cand, es; rep(u, 0, n) rep(v, 0, n) { if (simple and u == v) continue; if (!directed and u > v) continue; cand.push_back({u, v}); } int m = get(SZ(cand)); vector<int> ord; if (simple) ord = select(m, 0, SZ(cand) - 1); else { rep(_, 0, m) ord.push_back(get(SZ(cand) - 1)); } for (auto &i : ord) es.push_back(cand[i]); relabel(n, es); return es; } vector<pair<int, int>> genTree(int n) { vector<pair<int, int>> es; rep(i, 1, n) es.push_back({get(i - 1), i}); relabel(n, es); return es; } }; // namespace Random /** * @brief Random */ #line 3 "Graph/bimatching.hpp" vector<int> BiMatching(int n, int m, vector<vector<int>> &g) { rep(v, 0, n) Random::shuffle(ALL(g[v])); vector<int> L(n, -1), R(m, -1); for (;;) { vector<int> par(n, -1), root(n, -1); queue<int> que; rep(i, 0, n) if (L[i] == -1) { que.push(i); root[i] = i; } bool upd = 0; while (!que.empty()) { int v = que.front(); que.pop(); if (L[root[v]] != -1) continue; for (auto u : g[v]) { if (R[u] == -1) { while (u != -1) { R[u] = v; swap(L[v], u); v = par[v]; } upd = 1; break; } int to = R[u]; if (par[to] == -1) { root[to] = root[v]; par[to] = v; que.push(to); } } } if (!upd) break; } return L; } /** * @brief Bipartite Matching */ #line 2 "Graph/scc.hpp" struct SCC{ int n,m,cur; vector<vector<int>> g; vector<int> low,ord,id; SCC(int _n=0):n(_n),m(0),cur(0),g(_n),low(_n),ord(_n,-1),id(_n){} void resize(int _n){ n=_n; g.resize(n); low.resize(n); ord.resize(n,-1); id.resize(n); } void add_edge(int u,int v){g[u].emplace_back(v);} void dfs(int v,vector<int>& used){ ord[v]=low[v]=cur++; used.emplace_back(v); for(auto& nxt:g[v]){ if(ord[nxt]==-1){ dfs(nxt,used); chmin(low[v],low[nxt]); } else{ chmin(low[v],ord[nxt]); } } if(ord[v]==low[v]){ while(1){ int add=used.back(); used.pop_back(); ord[add]=n; id[add]=m; if(v==add)break; } m++; } } void run(){ vector<int> used; rep(v,0,n)if(ord[v]==-1)dfs(v,used); for(auto& x:id)x=m-1-x; } }; /** * @brief Strongly Connected Components */ #line 4 "Graph/dmdecomp.hpp" vector<vector<int>> DMdecomposition(int n, int m, vector<vector<int>> &g, vector<int> match) { if (match.empty()) match = BiMatching(n, m, g); vector G(n + m, vector<int>()), REV(n + m, vector<int>()); rep(i, 0, n) for (auto &j : g[i]) { G[i].push_back(j + n); REV[j + n].push_back(i); } vector<int> R(m, -1); rep(i, 0, n) if (match[i] != -1) { G[match[i] + n].push_back(i); REV[i].push_back(match[i] + n); R[match[i]] = i; } vector<int> V0, Vinf; queue<int> que; vector<int> used(n + m); rep(i, 0, n) if (match[i] == -1) { used[i] = 1; que.push(i); } while (!que.empty()) { int v = que.front(); que.pop(); Vinf.push_back(v); for (auto &to : G[v]) if (!used[to]) { used[to] = 1; que.push(to); } } rep(i, 0, m) if (R[i] == -1) { used[i + n] = 1; que.push(i + n); } while (!que.empty()) { int v = que.front(); que.pop(); V0.push_back(v); for (auto &to : REV[v]) if (!used[to]) { used[to] = 1; que.push(to); } } SCC scc(n + m); rep(i, 0, n + m) for (auto &to : G[i]) { if (!used[i] and !used[to]) scc.add_edge(i, to); } scc.run(); vector group(scc.m, vector<int>()); rep(i, 0, n + m) if (!used[i]) group[scc.id[i]].push_back(i); vector<vector<int>> ret; ret.push_back(V0); for (auto &v : group) ret.push_back(v); ret.push_back(Vinf); return ret; } /** * @brief DM decomposition */