$\newcommand{\O}{\mathrm{O}}$
Dijkstra 法を Fibonacci Heap を用いて高速化したアルゴリズム.
すでにフィボナッチヒープ内にある頂点についての最短距離の更新はキー値減算を用いることで $1$ 回あたりならし $\O (1)$ で行えるため, 新たに要素を insert する必要はない.
結果 insert および delete は $n$ 回となるので計算量は $\O (n \log n + m)$ となる.
元論文は "Fibonacci Heaps and Their Uses in Improved Network
Optimization Algorithms" [Fredman, Tarjan 1984]
時間計算量: $\O (n \log n + m)$
template<typename _Key, typename _Tp> class FibonacciHeap { public: using data_type = pair<_Key, _Tp>; class node { public: data_type _data; unsigned short int _child; bool _mark; node *_par, *_prev, *_next, *_ch_last; node(data_type&& data) : _data(move(data)), _child(0), _mark(false), _par(nullptr), _prev(nullptr), _next(nullptr), _ch_last(nullptr){} inline const _Key& get_key() const noexcept { return _data.first; } void insert(node *cur){ if(_ch_last) insert_impl(cur, _ch_last); else _ch_last = cur, _ch_last->_prev = _ch_last->_next = _ch_last; ++_child, cur->_par = this; } void erase(node *cur){ if(cur == cur->_prev){ _ch_last = nullptr; }else{ erase_impl(cur); if(cur == _ch_last) _ch_last = cur->_prev; } --_child, cur->_par = nullptr; } }; private: size_t _size; node *_minimum; vector<node*> rank; static void insert_impl(node *cur, node *next){ cur->_prev = next->_prev, cur->_next = next; cur->_prev->_next = cur, next->_prev = cur; } static void erase_impl(node *cur){ cur->_prev->_next = cur->_next, cur->_next->_prev = cur->_prev; } void root_insert(node *cur){ if(_minimum){ insert_impl(cur, _minimum); if(cur->get_key() < _minimum->get_key()) _minimum = cur; }else{ _minimum = cur, _minimum->_prev = _minimum->_next = _minimum; } } void root_erase(node *cur){ if(cur == cur->_prev) _minimum = nullptr; else erase_impl(cur); } void _delete(node *cur){ root_erase(cur); delete cur; } template<typename Key, typename Data> node *_push(Key&& key, Data&& data){ ++_size; data_type new_data(forward<Key>(key), forward<Data>(data)); node* new_node = new node(move(new_data)); root_insert(new_node); return new_node; } void _pop(){ assert(_size > 0); --_size; if(_size == 0){ _delete(_minimum); return; } if(_minimum->_ch_last){ for(node *cur = _minimum->_ch_last->_next;;){ node *next = cur->_next; _minimum->erase(cur), root_insert(cur); if(!_minimum->_ch_last) break; cur = next; } } node *next_minimum = _minimum->_next; for(node*& cur : rank) cur = nullptr; for(node *cur = next_minimum; cur != _minimum;){ if(cur->get_key() < next_minimum->get_key()) next_minimum = cur; node *next = cur->_next; unsigned int deg = cur->_child; if(rank.size() <= deg) rank.resize(deg + 1, nullptr); while(rank[deg]){ if(cur->get_key() < rank[deg]->get_key() || cur == next_minimum){ root_erase(rank[deg]), cur->insert(rank[deg]); }else{ root_erase(cur), rank[deg]->insert(cur); cur = rank[deg]; } rank[deg++] = nullptr; if(rank.size() <= deg) rank.resize(deg + 1, nullptr); } rank[deg] = cur; cur = next; } _delete(_minimum); _minimum = next_minimum; } void _decrease_key(node *cur, const _Key& key){ assert(!(key < (_Key)0)); node *change = ((cur->_data.first -= key) < _minimum->get_key()) ? cur : nullptr; if(!cur->_par || !(cur->get_key() < cur->_par->get_key())){ if(change) _minimum = change; return; } while(true){ node *next = cur->_par; next->erase(cur), root_insert(cur); cur->_mark = false, cur = next; if(!cur->_par) break; if(!cur->_mark){ cur->_mark = true; break; } } if(change) _minimum = change; } void clear_dfs(node *cur){ if(cur->_ch_last){ for(node *_cur = cur->_ch_last->_next;;){ node *next = _cur->_next; if(_cur == cur->_ch_last){ clear_dfs(_cur); break; }else{ clear_dfs(_cur); } _cur = next; } } delete cur; return; } void _clear(){ if(!_minimum) return; for(node *cur = _minimum->_next;;){ node *next = cur->_next; if(cur == _minimum){ clear_dfs(cur); break; }else{ clear_dfs(cur); } cur = next; } } public: FibonacciHeap() noexcept : _size(0u), _minimum(nullptr){} // ~FibonacciHeap(){ _clear(); } inline bool empty() const noexcept { return (_size == 0); } inline size_t size() const noexcept { return _size; } inline const data_type& top() const noexcept { return _minimum->_data; } template<typename Key, typename Data> node *push(Key&& key, Data&& data){ return _push(forward<Key>(key), forward<Data>(data)); } void pop(){ _pop(); } void decrease_key(node *cur, const _Key& key){ _decrease_key(cur, key); } void clear(){ _clear(); _size = 0; rank.~vector<node*>(); } }; template<typename T> class Dijkstra { public: struct edge{ int to; T cost; }; const int V; vector<vector<edge> > G; vector<T> d; FibonacciHeap<T, int> fheap; typename FibonacciHeap<T, int>::node** keep; Dijkstra(int node_size) : V(node_size), G(V), d(V, numeric_limits<T>::max()), keep(new typename FibonacciHeap<T, int>::node*[V]){} // ~Dijkstra(){ // delete[] keep; // } // 有向グラフの場合 void add_edge(int u,int v,T cost){ G[u].pb((edge){v, cost}); } void solve(int s){ d[s] = 0; keep[s] = fheap.push(0, s); while(!fheap.empty()){ int v = fheap.top().second; fheap.pop(); for(autoamp; w : G[v]){ if(d[w.to] > d[v] + w.cost){ if(d[w.to] == numeric_limits<T>::max()){ keep[w.to] = fheap.push(d[v] + w.cost, w.to); }else{ fheap.decrease_key(keep[w.to], d[w.to] - (d[v] + w.cost)); } d[w.to] = d[v] + w.cost; } } } } };