算法90----图的最短路径

一、单源最短路径：迪杰斯特拉【权值需非负】

目标：从某个起点出发，找到到各个点的最短距离。

思路：一个S集合存已经遍历的顶点，一个Q集合存未遍历的顶点。一个dist列表存从初始点到当前点的最短路径，即dist[i] 表示初始点到i所需的最短距离。

如果用堆优先队列来找dist中最小值，时间复杂度为O（E+VlogV)，否则时间复杂度为O（V2）,V 为顶点。

伪代码：

　　

代码：

MAX_value = 999999 def dijkstra(graph, s): dist = [MAX_value] * len(graph) dist[s] = 0 S = [] Q = [i for i in range(len(graph))] while Q: u_dist = min([d for i,d in enumerate(dist) if i in Q]) u = dist.index(u_dist) S.append(u) Q.remove(u) for v , d in enumerate(graph[u]): if 0 < d < MAX_value: dist[v] = min(dist[v] , dist[u] + d) return dist if __name__ == '__main__': graph_list = [[0, 9, MAX_value, MAX_value, MAX_value, 14, 15, MAX_value], [9, 0, 24, MAX_value, MAX_value, MAX_value, MAX_value, MAX_value], [MAX_value, 24, 0, 6, 2, 18, MAX_value, 19], [MAX_value, MAX_value, 6, 0, 11, MAX_value, MAX_value, 6], [MAX_value, MAX_value, 2, 11, 0, 30, 20, 16], [14, MAX_value, 18, MAX_value, 30, 0, 5, MAX_value], [15, MAX_value, MAX_value, MAX_value, 20, 5, 0, 44], [MAX_value, MAX_value, 19, 6, 16, MAX_value, 44, 0]] distance = dijkstra(graph_list, 0) print(distance)

 二、弗洛伊德算法【所有节点的最短路径】---动态规划

思路：时间复杂度O（n3)，

代码：DP相当于上面的Dis矩阵

MAX_value = 99999
def Floyd(graph):
    N = len(graph)
    DP = [[MAX_value] * N for i in range(N)]
    path = [[MAX_value] * N for i in range(N)]
    #####初始化DP和path矩阵。DP相当于动态规划的DP矩阵，path【i】【j】是记录i到j的最短路径的中间节点。
    for i in range(N):
        for j in range(N):
            DP[i][j] = graph[i][j]
            path[i][j] = -1
    ######动态规划过程
    for i in range(N):
        for j in range(N):
            for k in range(N):
                if DP[i][j] > DP[i][k] + DP[k][j]:
                    DP[i][j] = DP[i][k] + DP[k][j]
                    path[i][j] = k
    return DP,path
if __name__ == '__main__':
    graph_list = [[0, 9, MAX_value, MAX_value, MAX_value, 14, 15, MAX_value],
                  [9, 0, 24, MAX_value, MAX_value, MAX_value, MAX_value, MAX_value],
                  [MAX_value, 24, 0, 6, 2, 18, MAX_value, 19],
                  [MAX_value, MAX_value, 6, 0, 11, MAX_value, MAX_value, 6],
                  [MAX_value, MAX_value, 2, 11, 0, 30, 20, 16],
                  [14, MAX_value, 18, MAX_value, 30, 0, 5, MAX_value],
                  [15, MAX_value, MAX_value, MAX_value, 20, 5, 0, 44],
                  [MAX_value, MAX_value, 19, 6, 16, MAX_value, 44, 0]]
    print(Floyd(graph_list))