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#include "graph/eulerwalk.hpp"
#include "ds/unionfind/unionfind.hpp" #include "graph/base.hpp" #include "graph/vs_to_es.hpp" // (vs, es) or empty template <typename GT> pair<vc<int>, vc<int>> euler_walk(GT& G, int s = -1) { const int N = G.N, M = G.M; assert(G.is_prepared()); assert(N > 0); if (s == -1) { vc<int> deg(N); for (auto&& e: G.edges) { if constexpr (GT::is_directed) { deg[e.frm]++, deg[e.to]--; } else { deg[e.frm]++, deg[e.to]++; } } if constexpr (GT::is_directed) { s = max_element(all(deg)) - deg.begin(); if (deg[s] == 0) s = (M == 0 ? 0 : G.edges[0].frm); } else { s = [&]() -> int { FOR(v, N) if (deg[v] & 1) return v; return (M == 0 ? 0 : G.edges[0].frm); }(); } } if (M == 0) return {{s}, {}}; vc<int> D(N), its(N), eu(M), vs, st = {s}; FOR(v, N) its[v] = G.indptr[v]; ++D[s]; while (!st.empty()) { int x = st.back(), y, e, &it = its[x], end = G.indptr[x + 1]; if (it == end) { vs.eb(x); st.pop_back(); continue; } auto& ee = G.csr_edges[it++]; y = ee.to, e = ee.id; if (!eu[e]) { D[x]--, D[y]++; eu[e] = 1; st.eb(y); } } for (auto&& x: D) if (x < 0) return {{}, {}}; if (len(vs) != M + 1) return {{}, {}}; reverse(all(vs)); auto es = vs_to_es(G, vs, false); return {vs, es}; } template <typename GT> bool has_euler_walk(GT& G, int s = -1) { int N = G.N, M = G.M; if (M == 0) return true; if constexpr (!GT::is_directed) { vc<int> odd(N); for (auto& e: G.edges) odd[e.frm] ^= 1, odd[e.to] ^= 1; int n_odd = 0; for (auto x: odd) n_odd += x; if (n_odd >= 4) return false; if (s != -1 && n_odd == 2 && !odd[s]) return false; UnionFind uf(N); for (auto& e: G.edges) uf.merge(e.frm, e.to); vector<int> cnt_edge(N); for (auto& e: G.edges) cnt_edge[uf[e.frm]]++; if (s != -1 && cnt_edge[uf[s]] == 0) return false; // 辺がある成分を数える int nc = 0; for (int v = 0; v < N; ++v) { if (uf[v] == v && cnt_edge[v] >= 1) ++nc; } return nc <= 1; } else { int N = G.N; vc<int> in(N), out(N); for (auto& e: G.edges) out[e.frm]++, in[e.to]++; int ng = 0; FOR(v, N) ng += abs(out[v] - in[v]); if (ng >= 4) return false; if (s != -1 && ng == 2 && out[s] != in[s] + 1) return false; UnionFind uf(N); for (auto& e: G.edges) uf.merge(e.frm, e.to); vector<int> cnt_edge(N); for (auto& e: G.edges) cnt_edge[uf[e.frm]]++; if (s != -1 && cnt_edge[uf[s]] == 0) return false; // 辺がある成分を数える int nc = 0; for (int v = 0; v < N; ++v) { if (uf[v] == v && cnt_edge[v] >= 1) ++nc; } return nc <= 1; } }
#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/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} // 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; } 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 "graph/vs_to_es.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 4 "graph/vs_to_es.hpp" template <typename GT> vc<int> vs_to_es(GT& G, vc<int>& vs, bool allow_use_twice = false) { assert(!vs.empty()); HashMap<int> MP(G.M); vc<int> nxt(G.M, -1); auto get = [&](ll a, ll b) -> u64 { if (!GT::is_directed && a > b) swap(a, b); return a * G.N + b; }; FOR(eid, G.M) { u64 k = get(G.edges[eid].frm, G.edges[eid].to); int x = MP.get(k, -1); nxt[eid] = x, MP[k] = eid; } int n = len(vs); vc<int> es(n - 1); FOR(i, n - 1) { u64 k = get(vs[i], vs[i + 1]); int eid = MP.get(k, -1); assert(eid != -1); es[i] = eid; if (!allow_use_twice) { MP[k] = nxt[eid]; } } return es; } #line 4 "graph/eulerwalk.hpp" // (vs, es) or empty template <typename GT> pair<vc<int>, vc<int>> euler_walk(GT& G, int s = -1) { const int N = G.N, M = G.M; assert(G.is_prepared()); assert(N > 0); if (s == -1) { vc<int> deg(N); for (auto&& e: G.edges) { if constexpr (GT::is_directed) { deg[e.frm]++, deg[e.to]--; } else { deg[e.frm]++, deg[e.to]++; } } if constexpr (GT::is_directed) { s = max_element(all(deg)) - deg.begin(); if (deg[s] == 0) s = (M == 0 ? 0 : G.edges[0].frm); } else { s = [&]() -> int { FOR(v, N) if (deg[v] & 1) return v; return (M == 0 ? 0 : G.edges[0].frm); }(); } } if (M == 0) return {{s}, {}}; vc<int> D(N), its(N), eu(M), vs, st = {s}; FOR(v, N) its[v] = G.indptr[v]; ++D[s]; while (!st.empty()) { int x = st.back(), y, e, &it = its[x], end = G.indptr[x + 1]; if (it == end) { vs.eb(x); st.pop_back(); continue; } auto& ee = G.csr_edges[it++]; y = ee.to, e = ee.id; if (!eu[e]) { D[x]--, D[y]++; eu[e] = 1; st.eb(y); } } for (auto&& x: D) if (x < 0) return {{}, {}}; if (len(vs) != M + 1) return {{}, {}}; reverse(all(vs)); auto es = vs_to_es(G, vs, false); return {vs, es}; } template <typename GT> bool has_euler_walk(GT& G, int s = -1) { int N = G.N, M = G.M; if (M == 0) return true; if constexpr (!GT::is_directed) { vc<int> odd(N); for (auto& e: G.edges) odd[e.frm] ^= 1, odd[e.to] ^= 1; int n_odd = 0; for (auto x: odd) n_odd += x; if (n_odd >= 4) return false; if (s != -1 && n_odd == 2 && !odd[s]) return false; UnionFind uf(N); for (auto& e: G.edges) uf.merge(e.frm, e.to); vector<int> cnt_edge(N); for (auto& e: G.edges) cnt_edge[uf[e.frm]]++; if (s != -1 && cnt_edge[uf[s]] == 0) return false; // 辺がある成分を数える int nc = 0; for (int v = 0; v < N; ++v) { if (uf[v] == v && cnt_edge[v] >= 1) ++nc; } return nc <= 1; } else { int N = G.N; vc<int> in(N), out(N); for (auto& e: G.edges) out[e.frm]++, in[e.to]++; int ng = 0; FOR(v, N) ng += abs(out[v] - in[v]); if (ng >= 4) return false; if (s != -1 && ng == 2 && out[s] != in[s] + 1) return false; UnionFind uf(N); for (auto& e: G.edges) uf.merge(e.frm, e.to); vector<int> cnt_edge(N); for (auto& e: G.edges) cnt_edge[uf[e.frm]]++; if (s != -1 && cnt_edge[uf[s]] == 0) return false; // 辺がある成分を数える int nc = 0; for (int v = 0; v < N; ++v) { if (uf[v] == v && cnt_edge[v] >= 1) ++nc; } return nc <= 1; } }