Floyd算法
Floyd算法又称为Floyd-Warshell算法,其实Warshell算法是离散数学中求传递闭包的算法,两者的思想是一致的。Floyd算法是求解多源最短路时通常选用的算法,经过一次算法即可求出任意两点之间的最短距离,并且可以处理有负权边的情况(但无法处理负权环),算法的时间复杂度是 O ( n 3 ) O(n^3) O(n3),空间复杂度是 O ( n 2 ) O(n^2) O(n2)。
python
import numpy as np
def floyd(adjacent_matrix, source, target):
"""
:param adjacent_matrix: 图邻接矩阵
:param source: 起点
:param target: 终点
:return: shortest_path
"""
num_node = len(adjacent_matrix)
# 计算
"""
矩阵D记录顶点间的最小路径
例如D[0][3]= 10,说明顶点0 到 3 的最短路径为10;
矩阵P记录顶点间最小路径中的中转点
例如P[0][3]= 1 说明,0 到 3的最短路径轨迹为:0 -> 1 -> 3。
"""
distance = np.zeros(shape=(num_node, num_node), dtype=np.int_)
path = np.zeros(shape=(num_node, num_node), dtype=np.int_)
for v in range(num_node):
for w in range(num_node):
distance[v][w] = adjacent_matrix[v][w]
path[v][w] = w
# 弗洛伊德算法的核心部分
for k in range(num_node): # k为中间点
for v in range(num_node): # v 为起点
for w in range(num_node): # w为起点
if distance[v][w] > (distance[v][k] + distance[k][w]):
distance[v][w] = distance[v][k] + distance[k][w]
path[v][w] = path[v][k]
print(np.asarray(path))
shortest_path = [source]
k = path[source][target]
while k != target:
shortest_path.append(k)
k = path[k][target]
shortest_path.append(target)
return shortest_path
if __name__ == "__main__":
M = 1e6
adjacent_matrix = [
[0, 12, M, M, M, 16, 14],
[12, 0, 10, M, M, 7, M],
[M, 10, 0, 3, 5, 6, M],
[M, M, 3, 0, 4, M, M],
[M, M, 5, 4, 0, 2, 8],
[16, 7, 6, M, 2, 0, 9],
[14, M, M, M, 8, 9, 0],
]
shortest_path = floyd(adjacent_matrix, 0, 3)
print(shortest_path)
# [0, 6, 3, M, M, M],
# [6, 0, 2, 5, M, M],
# [3, 2, 0, 3, 4, M],
# [M, 5, 3, 0, 5, 3],
# [M, M, 4, 5, 0, 5],
# [M, M, M, 3, 5, 0]
适应场景
Floyd-Warshall算法由于其 O ( n 3 ) O(n^3) O(n3)的时间复杂度,适用于节点数比较少且图比较稠密的情况。对于边数较少的稀疏图,使用基于边的算法(如Dijkstra或Bellman-Ford)通常会更高效。