from collections import deque from collections.abc import Iterator from dataclasses import dataclass from typing import Optional, Union """ Finding the shortest path in 0-1-graph in O(E + V) which is faster than dijkstra. 0-1-graph is the weighted graph with the weights equal to 0 or 1. Link: https://codeforces.com/blog/entry/22276 """ @dataclass class Edge: """Weighted directed graph edge.""" destination_vertex: int weight: int class AdjacencyList: """Graph adjacency list.""" def __init__(self, size: int): self._graph: list[list[Edge]] = [[] for _ in range(size)] self._size = size def __getitem__(self, vertex: int) -> Iterator[Edge]: """Get all the vertices adjacent to the given one.""" return iter(self._graph[vertex]) @property def size(self): return self._size def add_edge(self, from_vertex: int, to_vertex: int, weight: int): """ >>> g = AdjacencyList(2) >>> g.add_edge(0, 1, 0) >>> g.add_edge(1, 0, 1) >>> list(g[0]) [Edge(destination_vertex=1, weight=0)] >>> list(g[1]) [Edge(destination_vertex=0, weight=1)] >>> g.add_edge(0, 1, 2) Traceback (most recent call last): ... ValueError: Edge weight must be either 0 or 1. >>> g.add_edge(0, 2, 1) Traceback (most recent call last): ... ValueError: Vertex indexes must be in [0; size). """ if weight not in (0, 1): raise ValueError("Edge weight must be either 0 or 1.") if to_vertex < 0 or to_vertex >= self.size: raise ValueError("Vertex indexes must be in [0; size).") self._graph[from_vertex].append(Edge(to_vertex, weight)) def get_shortest_path(self, start_vertex: int, finish_vertex: int) -> Optional[int]: """ Return the shortest distance from start_vertex to finish_vertex in 0-1-graph. 1 1 1 0--------->3 6--------7>------->8 | ^ ^ ^ |1 | | | |0 v 0| |0 1| 9-------->10 | | | ^ 1 v | | |0 1--------->2<-------4------->5 0 1 1 >>> g = AdjacencyList(11) >>> g.add_edge(0, 1, 0) >>> g.add_edge(0, 3, 1) >>> g.add_edge(1, 2, 0) >>> g.add_edge(2, 3, 0) >>> g.add_edge(4, 2, 1) >>> g.add_edge(4, 5, 1) >>> g.add_edge(4, 6, 1) >>> g.add_edge(5, 9, 0) >>> g.add_edge(6, 7, 1) >>> g.add_edge(7, 8, 1) >>> g.add_edge(8, 10, 1) >>> g.add_edge(9, 7, 0) >>> g.add_edge(9, 10, 1) >>> g.add_edge(1, 2, 2) Traceback (most recent call last): ... ValueError: Edge weight must be either 0 or 1. >>> g.get_shortest_path(0, 3) 0 >>> g.get_shortest_path(0, 4) Traceback (most recent call last): ... ValueError: No path from start_vertex to finish_vertex. >>> g.get_shortest_path(4, 10) 2 >>> g.get_shortest_path(4, 8) 2 >>> g.get_shortest_path(0, 1) 0 >>> g.get_shortest_path(1, 0) Traceback (most recent call last): ... ValueError: No path from start_vertex to finish_vertex. """ queue = deque([start_vertex]) distances: list[Union[int, None]] = [None] * self.size distances[start_vertex] = 0 while queue: current_vertex = queue.popleft() current_distance = distances[current_vertex] if current_distance is None: continue for edge in self[current_vertex]: new_distance = current_distance + edge.weight dest_vertex_distance = distances[edge.destination_vertex] if ( isinstance(dest_vertex_distance, int) and new_distance >= dest_vertex_distance ): continue distances[edge.destination_vertex] = new_distance if edge.weight == 0: queue.appendleft(edge.destination_vertex) else: queue.append(edge.destination_vertex) if distances[finish_vertex] is None: raise ValueError("No path from start_vertex to finish_vertex.") return distances[finish_vertex] if __name__ == "__main__": import doctest doctest.testmod()