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#include "graph/shortest-path/dijkstra.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. 各地点でもっとも近い頂点から距離が確定していく. 距離順でソートされたヒープを用いると, 効率よく距離を確定していくことができる.
dijkstra(g, s)
: 重み付きグラフ g
で, 頂点 s
から全点間の最短コストを求める. dist
には最短コスト(到達できないとき型の最大値), from
にはその頂点の直前に訪れた頂点(頂点 s
または到達できない頂点は $-1$), id
はその頂点の直前に使った辺番号が格納される.#pragma once
#include "../graph-template.hpp"
/**
* @brief Dijkstra(単一始点最短路)
*
*/
template< typename T >
struct ShortestPath {
vector< T > dist;
vector< int > from, id;
};
template< typename T >
ShortestPath< T > dijkstra(const Graph< T > &g, int s) {
const auto INF = numeric_limits< T >::max();
vector< T > dist(g.size(), INF);
vector< int > from(g.size(), -1), id(g.size(), -1);
using Pi = pair< T, int >;
priority_queue< Pi, vector< Pi >, greater<> > que;
dist[s] = 0;
que.emplace(dist[s], s);
while(!que.empty()) {
T cost;
int idx;
tie(cost, idx) = que.top();
que.pop();
if(dist[idx] < cost) continue;
for(auto &e : g[idx]) {
auto next_cost = cost + e.cost;
if(dist[e.to] <= next_cost) continue;
dist[e.to] = next_cost;
from[e.to] = idx;
id[e.to] = e.idx;
que.emplace(dist[e.to], e.to);
}
}
return {dist, from, id};
}
#line 2 "graph/shortest-path/dijkstra.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 4 "graph/shortest-path/dijkstra.hpp"
/**
* @brief Dijkstra(単一始点最短路)
*
*/
template< typename T >
struct ShortestPath {
vector< T > dist;
vector< int > from, id;
};
template< typename T >
ShortestPath< T > dijkstra(const Graph< T > &g, int s) {
const auto INF = numeric_limits< T >::max();
vector< T > dist(g.size(), INF);
vector< int > from(g.size(), -1), id(g.size(), -1);
using Pi = pair< T, int >;
priority_queue< Pi, vector< Pi >, greater<> > que;
dist[s] = 0;
que.emplace(dist[s], s);
while(!que.empty()) {
T cost;
int idx;
tie(cost, idx) = que.top();
que.pop();
if(dist[idx] < cost) continue;
for(auto &e : g[idx]) {
auto next_cost = cost + e.cost;
if(dist[e.to] <= next_cost) continue;
dist[e.to] = next_cost;
from[e.to] = idx;
id[e.to] = e.idx;
que.emplace(dist[e.to], e.to);
}
}
return {dist, from, id};
}