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#include "graph/shortest-path/dijkstra-fibonacchi-heap.hpp"
負辺のないグラフで単一始点全点間最短路を求めるアルゴリズム.
通常の dijkstra 法では std::priority_queue を使用していたが, これをフィボナッチヒープにすることで計算量を落とせる(実用上早くなるかは知らない).
std::priority_queue
dijkstra_fibonacchi_heap(g, s)
g
s
#pragma once #include "../graph-template.hpp" #include "../../structure/heap/fibonacchi-heap.hpp" /** * @brief Dijkstra-Fibonacchi-Heap(単一始点最短路) * @docs docs/dijkstra-fibonacchi-heap.md */ template< typename T > vector< T > dijkstra_fibonacchi_heap(Graph< T > &g, int s) { const auto INF = numeric_limits< T >::max(); using Heap = FibonacchiHeap< T, int >; using Node = typename Heap::Node; Heap heap; vector< Node * > keep(g.size(), nullptr); vector< T > dist; dist.assign(g.size(), INF); dist[s] = 0; keep[s] = heap.push(dist[s], s); while(!heap.empty()) { T cost; int idx; tie(cost, idx) = heap.pop(); if(dist[idx] < cost) continue; for(auto &e : g[idx]) { auto next_cost = cost + e.cost; if(dist[e.to] <= next_cost) continue; if(keep[e.to] == nullptr) { dist[e.to] = next_cost; keep[e.to] = heap.push(dist[e.to], e.to); } else { T d = dist[e.to] - next_cost; heap.decrease_key(keep[e.to], d); dist[e.to] -= d; } } } return dist; }
#line 2 "graph/shortest-path/dijkstra-fibonacchi-heap.hpp" #line 2 "graph/graph-template.hpp" /** * @brief Graph Template(グラフテンプレート) */ template< typename T = int > struct Edge { int from, to; T cost; int idx; Edge() = default; Edge(int from, int to, T cost = 1, int idx = -1) : from(from), to(to), cost(cost), idx(idx) {} operator int() const { return to; } }; template< typename T = int > struct Graph { vector< vector< Edge< T > > > g; int es; Graph() = default; explicit Graph(int n) : g(n), es(0) {} size_t size() const { return g.size(); } void add_directed_edge(int from, int to, T cost = 1) { g[from].emplace_back(from, to, cost, es++); } void add_edge(int from, int to, T cost = 1) { g[from].emplace_back(from, to, cost, es); g[to].emplace_back(to, from, cost, es++); } void read(int M, int padding = -1, bool weighted = false, bool directed = false) { for(int i = 0; i < M; i++) { int a, b; cin >> a >> b; a += padding; b += padding; T c = T(1); if(weighted) cin >> c; if(directed) add_directed_edge(a, b, c); else add_edge(a, b, c); } } inline vector< Edge< T > > &operator[](const int &k) { return g[k]; } inline const vector< Edge< T > > &operator[](const int &k) const { return g[k]; } }; template< typename T = int > using Edges = vector< Edge< T > >; #line 1 "structure/heap/fibonacchi-heap.hpp" /** * @brief Fibonacchi-Heap(フィボナッチヒープ) * @see https://www.cs.princeton.edu/~wayne/teaching/fibonacci-heap.pdf */ template< typename key_t, typename val_t > struct FibonacchiHeap { struct Node { key_t key; val_t val; Node *left, *right, *child, *par; int sz; bool mark; Node(const key_t &key, const val_t &val) : key(key), val(val), left(this), right(this), par(nullptr), child(nullptr), sz(0), mark(false) {} }; Node *root; size_t sz; vector< Node * > rank; FibonacchiHeap() : root(nullptr), sz(0) {} size_t size() const { return sz; } bool empty() const { return sz == 0; } void update_min(Node *t) { if(!root || t->key < root->key) { root = t; } } void concat(Node *&r, Node *t) { if(!r) { r = t; } else { t->left->right = r->right; r->right->left = t->left; t->left = r; r->right = t; } } void delete_node(Node *t) { t->left->right = t->right; t->right->left = t->left; t->left = t; t->right = t; } Node *push(const key_t &key, const val_t &val) { ++sz; auto node = new Node(key, val); concat(root, node); update_min(node); return node; } Node *consolidate(Node *s, Node *t) { if(root == s || s->key < t->key) { delete_node(t); ++s->sz; t->par = s; concat(s->child, t); return s; } else { delete_node(s); ++t->sz; s->par = t; concat(t->child, s); return t; } } pair< key_t, val_t > pop() { --sz; Node *rem = root; auto ret = make_pair(rem->key, rem->val); { root = root->left == root ? nullptr : root->left; delete_node(rem); } if(rem->child) { concat(root, rem->child); } if(root) { { Node *base = root, *cur = base; do { cur->par = nullptr; update_min(cur); cur = cur->right; } while(cur != base); } { Node *base = root; int last = -1; do { Node *nxt = base->right; while(base->sz < rank.size() && rank[base->sz]) { Node *u = rank[base->sz]; rank[base->sz] = nullptr; base = consolidate(u, base); } if(base->sz >= rank.size()) rank.resize(base->sz + 1); last = max(last, base->sz); rank[base->sz] = base; base = nxt; } while(base != root); for(int i = last; i >= 0; i--) rank[i] = nullptr; } } return ret; } inline void mark_dfs(Node *t) { if(!t->par) { t->mark = false; } else if(t->mark) { mark_dfs(t->par); t->par->child = t->left == t ? nullptr : t->left; delete_node(t); t->sz--; t->mark = false; t->par = nullptr; concat(root, t); } else { t->mark = true; t->sz--; } } void decrease_key(Node *t, const key_t &d) { t->key -= d; if(!t->par) { update_min(t); return; } if(t->par->key <= t->key) { return; } t->sz++; t->mark = true; mark_dfs(t); update_min(t); } }; #line 5 "graph/shortest-path/dijkstra-fibonacchi-heap.hpp" /** * @brief Dijkstra-Fibonacchi-Heap(単一始点最短路) * @docs docs/dijkstra-fibonacchi-heap.md */ template< typename T > vector< T > dijkstra_fibonacchi_heap(Graph< T > &g, int s) { const auto INF = numeric_limits< T >::max(); using Heap = FibonacchiHeap< T, int >; using Node = typename Heap::Node; Heap heap; vector< Node * > keep(g.size(), nullptr); vector< T > dist; dist.assign(g.size(), INF); dist[s] = 0; keep[s] = heap.push(dist[s], s); while(!heap.empty()) { T cost; int idx; tie(cost, idx) = heap.pop(); if(dist[idx] < cost) continue; for(auto &e : g[idx]) { auto next_cost = cost + e.cost; if(dist[e.to] <= next_cost) continue; if(keep[e.to] == nullptr) { dist[e.to] = next_cost; keep[e.to] = heap.push(dist[e.to], e.to); } else { T d = dist[e.to] - next_cost; heap.decrease_key(keep[e.to], d); dist[e.to] -= d; } } } return dist; }