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
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
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.
