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#include "graph/directed_mst.hpp"
#include "graph/base.hpp" #include "ds/unionfind/unionfind.hpp" template <typename GT, int NODES> struct Directed_MST_Solver { using T = typename GT::cost_type; GT &G; Directed_MST_Solver(GT &G) : G(G), pid(0) { pool = new Node[NODES]; assert(G.N + G.M <= NODES); } vc<int> calc(int root) { int N = G.N, M = G.M; vc<np> que(N); for (auto &e: G.edges) { que[e.to] = meld(que[e.to], new_node(e.frm, e.cost, e.id)); } vc<char> used(N + M); used[root] = 2; vc<Edge> best_edge(N + M); vc<int> par(N + M, -1); // merge 過程の木 vc<int> rt(N + M); FOR(i, N) rt[i] = i; UnionFind uf(N + M); int nxt = N; for (int s = 0; s < N; ++s) { if (used[s] != 0) continue; vc<int> path = {s}; while (1) { int a = path.back(); assert(used[a] == 0); used[a] = 1; if (!que[a]) { return {}; } best_edge[a] = pop(que[a]); int to = rt[uf[best_edge[a].to]]; if (used[to] == 0) { path.eb(to); continue; } if (used[to] == 2) break; // cycle 発見 int v = nxt++; que.eb(nullptr); while (1) { int w = POP(path); T sub = best_edge[w].cost; que[v] = meld(que[v], add(que[w], -sub)); uf.merge(v, w), par[w] = v; used[w] = 2; if (w == to) break; } rt[uf[v]] = v; path.eb(v); } for (auto &v: path) used[v] = 2; } vc<int> res; vc<bool> done(nxt); done[root] = 1; FOR_R(v, nxt) { if (done[v]) continue; int id = best_edge[v].id; res.eb(id); int x = G.edges[id].to; while (x != -1 && !done[x]) { done[x] = 1, x = par[x]; } } return res; } private: struct Edge { int to, id; T cost; }; struct Node { Node *l, *r; Edge e; T lazy; int s; }; Node *pool; using np = Node *; int pid; np new_node(int to, T cost, int id) { pool[pid].l = pool[pid].r = nullptr; pool[pid].s = 1; pool[pid].e = Edge{to, id, cost}; pool[pid].lazy = 0; return &(pool[pid++]); } np add(np a, T x) { if (a) a->e.cost += x, a->lazy += x; return a; } np meld(np a, np b) { if (!a) return b; if (!b) return a; if ((a->e.cost) > (b->e.cost)) swap(a, b); b = add(b, -(a->lazy)); a->r = (a->r ? meld(a->r, b) : b); if (!(a->l) || (a->l->s < a->r->s)) swap(a->l, a->r); a->s = (a->r ? a->r->s : 0) + 1; return a; } Edge pop(np &a) { Edge e = a->e; a = meld(add(a->l, a->lazy), add(a->r, a->lazy)); return e; } }; template <typename GT, int MAX_N> pair<typename GT::cost_type, vc<int>> directed_mst(GT &G, int root) { Directed_MST_Solver<GT, 2 * MAX_N> D(G); using T = typename GT::cost_type; auto I = D.calc(root); T cost = 0; for (auto &i: I) cost += G.edges[i].cost; return {cost, I}; };
#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); 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; } 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 3 "graph/directed_mst.hpp" template <typename GT, int NODES> struct Directed_MST_Solver { using T = typename GT::cost_type; GT &G; Directed_MST_Solver(GT &G) : G(G), pid(0) { pool = new Node[NODES]; assert(G.N + G.M <= NODES); } vc<int> calc(int root) { int N = G.N, M = G.M; vc<np> que(N); for (auto &e: G.edges) { que[e.to] = meld(que[e.to], new_node(e.frm, e.cost, e.id)); } vc<char> used(N + M); used[root] = 2; vc<Edge> best_edge(N + M); vc<int> par(N + M, -1); // merge 過程の木 vc<int> rt(N + M); FOR(i, N) rt[i] = i; UnionFind uf(N + M); int nxt = N; for (int s = 0; s < N; ++s) { if (used[s] != 0) continue; vc<int> path = {s}; while (1) { int a = path.back(); assert(used[a] == 0); used[a] = 1; if (!que[a]) { return {}; } best_edge[a] = pop(que[a]); int to = rt[uf[best_edge[a].to]]; if (used[to] == 0) { path.eb(to); continue; } if (used[to] == 2) break; // cycle 発見 int v = nxt++; que.eb(nullptr); while (1) { int w = POP(path); T sub = best_edge[w].cost; que[v] = meld(que[v], add(que[w], -sub)); uf.merge(v, w), par[w] = v; used[w] = 2; if (w == to) break; } rt[uf[v]] = v; path.eb(v); } for (auto &v: path) used[v] = 2; } vc<int> res; vc<bool> done(nxt); done[root] = 1; FOR_R(v, nxt) { if (done[v]) continue; int id = best_edge[v].id; res.eb(id); int x = G.edges[id].to; while (x != -1 && !done[x]) { done[x] = 1, x = par[x]; } } return res; } private: struct Edge { int to, id; T cost; }; struct Node { Node *l, *r; Edge e; T lazy; int s; }; Node *pool; using np = Node *; int pid; np new_node(int to, T cost, int id) { pool[pid].l = pool[pid].r = nullptr; pool[pid].s = 1; pool[pid].e = Edge{to, id, cost}; pool[pid].lazy = 0; return &(pool[pid++]); } np add(np a, T x) { if (a) a->e.cost += x, a->lazy += x; return a; } np meld(np a, np b) { if (!a) return b; if (!b) return a; if ((a->e.cost) > (b->e.cost)) swap(a, b); b = add(b, -(a->lazy)); a->r = (a->r ? meld(a->r, b) : b); if (!(a->l) || (a->l->s < a->r->s)) swap(a->l, a->r); a->s = (a->r ? a->r->s : 0) + 1; return a; } Edge pop(np &a) { Edge e = a->e; a = meld(add(a->l, a->lazy), add(a->r, a->lazy)); return e; } }; template <typename GT, int MAX_N> pair<typename GT::cost_type, vc<int>> directed_mst(GT &G, int root) { Directed_MST_Solver<GT, 2 * MAX_N> D(G); using T = typename GT::cost_type; auto I = D.calc(root); T cost = 0; for (auto &i: I) cost += G.edges[i].cost; return {cost, I}; };