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#include "graph/shortest_path/K_shortest_path.hpp"
#include "graph/base.hpp" // (cost, vs, es) template <typename T, typename GT> vc<tuple<T, vc<int>, vc<int>>> K_shortest_path(GT& G, int s, int t, int K) { assert(GT::is_directed); const int N = G.N; // (cost, vs, es) vc<tuple<T, vc<int>, vc<int>>> res; // 探索の起点 (es, ng_edges) vc<pair<vc<int>, vc<int>>> nodes; // 答の候補 (cost, es, ng_es, n), (ng_es, n):その path を見つけたときの条件 vc<tuple<T, vc<int>, vc<int>, int>> paths; nodes.eb(vc<int>(), vc<int>()); vc<T> dist(N, infty<T>); vc<bool> ng_v(N); vc<bool> ng_e(G.M); vc<int> par(N, -1); while (len(res) < K) { for (auto&& [es, ng_es]: nodes) { fill(all(par), -1); fill(all(ng_v), 0); fill(all(ng_e), 0); fill(all(dist), infty<T>); T pref_cost = 0; for (auto&& x: es) pref_cost += G.edges[x].cost; for (auto&& x: es) ng_v[G.edges[x].frm] = 1, ng_e[x] = 1; for (auto&& x: ng_es) ng_e[x] = 1; // dijkstra pqg<pair<T, int>> que; auto add = [&](int v, T d, int p) -> void { if (chmin(dist[v], d)) { que.emplace(d, v); par[v] = p; } }; int s0 = (es.empty() ? s : G.edges[es.back()].to); add(s0, pref_cost, -1); while (len(que)) { auto [dv, v] = POP(que); if (dv != dist[v]) continue; if (v == t) break; for (auto&& e: G[v]) { if (ng_e[e.id] || ng_v[e.to]) continue; add(e.to, dv + e.cost, e.id); } } // failed if (par[t] == -1) continue; // restore path vc<int> add_e; { int v = t; while (v != s0) { add_e.eb(par[v]); v = G.edges[par[v]].frm; } } reverse(all(add_e)); int n = len(es); // find a path es.insert(es.end(), all(add_e)); paths.eb(dist[t], es, ng_es, n); } // choose best path if (len(paths) == 0) break; pair<int, T> best = {-1, infty<T>}; FOR(i, len(paths)) { T cost = get<0>(paths[i]); if (chmin(best.se, cost)) best.fi = i; } int idx = best.fi; swap(paths[idx], paths[len(paths) - 1]); auto [cost, es, ng_es, n] = POP(paths); vc<int> vs = {s}; for (auto&& x: es) vs.eb(G.edges[x].to); res.eb(cost, vs, es); nodes.clear(); FOR(k, n, len(es)) { // k 番目の辺までを固定して vc<int> new_es = {es.begin(), es.begin() + k}; vc<int> new_ng = ng_es; new_ng.eb(es[k]); nodes.eb(new_es, new_ng); } } return res; }
#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 "graph/shortest_path/K_shortest_path.hpp" // (cost, vs, es) template <typename T, typename GT> vc<tuple<T, vc<int>, vc<int>>> K_shortest_path(GT& G, int s, int t, int K) { assert(GT::is_directed); const int N = G.N; // (cost, vs, es) vc<tuple<T, vc<int>, vc<int>>> res; // 探索の起点 (es, ng_edges) vc<pair<vc<int>, vc<int>>> nodes; // 答の候補 (cost, es, ng_es, n), (ng_es, n):その path を見つけたときの条件 vc<tuple<T, vc<int>, vc<int>, int>> paths; nodes.eb(vc<int>(), vc<int>()); vc<T> dist(N, infty<T>); vc<bool> ng_v(N); vc<bool> ng_e(G.M); vc<int> par(N, -1); while (len(res) < K) { for (auto&& [es, ng_es]: nodes) { fill(all(par), -1); fill(all(ng_v), 0); fill(all(ng_e), 0); fill(all(dist), infty<T>); T pref_cost = 0; for (auto&& x: es) pref_cost += G.edges[x].cost; for (auto&& x: es) ng_v[G.edges[x].frm] = 1, ng_e[x] = 1; for (auto&& x: ng_es) ng_e[x] = 1; // dijkstra pqg<pair<T, int>> que; auto add = [&](int v, T d, int p) -> void { if (chmin(dist[v], d)) { que.emplace(d, v); par[v] = p; } }; int s0 = (es.empty() ? s : G.edges[es.back()].to); add(s0, pref_cost, -1); while (len(que)) { auto [dv, v] = POP(que); if (dv != dist[v]) continue; if (v == t) break; for (auto&& e: G[v]) { if (ng_e[e.id] || ng_v[e.to]) continue; add(e.to, dv + e.cost, e.id); } } // failed if (par[t] == -1) continue; // restore path vc<int> add_e; { int v = t; while (v != s0) { add_e.eb(par[v]); v = G.edges[par[v]].frm; } } reverse(all(add_e)); int n = len(es); // find a path es.insert(es.end(), all(add_e)); paths.eb(dist[t], es, ng_es, n); } // choose best path if (len(paths) == 0) break; pair<int, T> best = {-1, infty<T>}; FOR(i, len(paths)) { T cost = get<0>(paths[i]); if (chmin(best.se, cost)) best.fi = i; } int idx = best.fi; swap(paths[idx], paths[len(paths) - 1]); auto [cost, es, ng_es, n] = POP(paths); vc<int> vs = {s}; for (auto&& x: es) vs.eb(G.edges[x].to); res.eb(cost, vs, es); nodes.clear(); FOR(k, n, len(es)) { // k 番目の辺までを固定して vc<int> new_es = {es.begin(), es.begin() + k}; vc<int> new_ng = ng_es; new_ng.eb(es[k]); nodes.eb(new_es, new_ng); } } return res; }