求最短路径的三种算法: Ford, Dijkstra和Floyd

Bellman-Ford算法

Bellman-Ford是一种容易理解的单源最短路径算法, Bellman-Ford算法需要两个数组进行辅助:

• dis[i]: 存储顶点i到源点已知最短路径
• path[i]: 存储顶点i到源点已知最短路径上, i的前一个顶点.

若图有n个顶点, 则图中最长简单路径长度不超过n-1, 因此Ford算法进行n-1次迭代确保获得最短路径.

Ford算法的每次迭代遍历所有边, 并对边进行松弛(relax)操作. 对边e进行松弛是指: 若从源点通过e.start到达e.stop的路径长小于已知最短路径, 则更新已知最短路径.

为了便于描述, 本文采用python实现算法. 首先实现两个工具函数:

INF = 1e6

def make_mat(m, n, fill=None):
    mat = []
    for i in range(m):
        mat.append([fill] * n)
    return mat

def get_edges(graph):
    n = len(graph)
    edges = []
    for i in range(n):
        for j in range(n):
            if graph[i][j] != 0:
                edges.append((i, j, graph[i][j]))
    return edges

make_mat用于初始化二维数组, get_edges用于将图由邻接矩阵表示变换为边的列表.

接下来就可以实现Bellman-Ford算法了:

def ford(graph, v0):
    n = len(graph)
    edges = get_edges(graph)
    dis = [INF] * n
    dis[v0] = 0
    path = [0] * n

    for k in range(n-1):
        for edge in edges:
            # relax
            if dis[edge[0]] + edge[2] < dis[edge[1]]:
                dis[edge[1]] = dis[edge[0]] + edge[2]
                path[edge[1]] = edge[0]
   return dis, path

初始化后执行迭代和松弛操作, 非常简单.

由path[i]获得最短路径的前驱顶点, 逐次迭代得到从顶点i到源点的最短路径. 倒序即可得源点到i的最短路径.

def show(path, start, stop):
    i = stop
    tmp = [stop]
    while i != start:
        i = path[i]
        tmp.append(i)
    return list(reversed(tmp))

Ford算法允许路径的权值为负, 但是若路径中存在总权值为负的环的话, 每次经过该环最短路径长就会减少. 因此, 图中的部分点不存在最短路径(最短路径长为负无穷).

若路径中不存在负环, 则进行n-1次迭代后不存在可以进行松弛的边. 因此再遍历一次边, 若存在可松弛的边说明图中存在负环.

这样改进得到可以检测负环的Ford算法:

def ford(graph, v0):
    n = len(graph)
    edges = get_edges(graph)
    dis = [INF] * n
    dis[v0] = 0
    path = [0] * n

    for k in range(n-1):
        for edge in edges:
            # relax
            if dis[edge[0]] + edge[2] < dis[edge[1]]:
                dis[edge[1]] = dis[edge[0]] + edge[2]
                path[edge[1]] = edge[0]

    # check negative loop
    flag = False
    for edge in edges:
        # try to relax
        if dis[edge[0]] + edge[2] < dis[edge[1]]:
            flag = True
            break
    if flag:
        return False
    return dis, path

• 完整python代码

Dijkstra算法

Dijkstra算法是一种贪心算法, 但可以保证求得全局最优解. Dijkstra算法需要和Ford算法同样的两个辅助数组:

• dis[i]: 存储顶点i到源点已知最短路径
• path[i]: 存储顶点i到源点已知最短路径上, i的前一个顶点.

Dijkstra算法的核心仍然是松弛操作, 但是选择松弛的边的方法不同. Dijkstra算法使用一个小顶堆存储所有未被访问过的边, 然后每次选择其中最小的进行松弛.

def dijkstra(graph, v0):
    n = len(graph)
    dis = [INF] * n
    dis[v0] = 0
    path = [0] * n

    unvisited = get_edges(graph)
    heapq.heapify(unvisited)

    while len(unvisited):
        u = heapq.heappop(unvisited)[1]
        for v in range(len(graph[u])):
            w = graph[u][v]
            if dis[u] + w < dis[v]:
                dis[v] = dis[u] + w
                path[v] = u

    return dis, path

• 完整python代码

Floyd

floyd算法是采用动态规划思想的多源最短路径算法. 它同样需要两个辅助数组, 但作为多源最短路径算法, 其结构不同:

• dis[i][j]: 保存从顶点i到顶点j的已知最短路径, 初始化为直接连接
• path[i][j]: 保存从顶点i到顶点j的已知最短路径上下一个顶点, 初始化为j

def floyd(graph):
	# init
    m = len(graph)
    dis = make_mat(m, m, fill=0)
    path = make_mat(m, m, fill=0)
    for i in range(m):
        for j in range(m):
            dis[i][j] = graph[i][j]
            path[i][j] = j

    for k in range(m):
        for i in range(m):
            for j in range(m):
                # relax
                if dis[i][k] + dis[k][j] < dis[i][j]:
                    dis[i][j] = dis[i][k] + dis[k][j]
                    path[i][j] = path[i][k]

    return dis, path

算法核心是遍历顶点k, i, j. 若从顶点i经过顶点k到达顶点j的路径, 比已知从i到j的最短路径短, 则更新已知最短路径.

• 完整python代码