图的 DFS 中的树、后、边和交叉边
考虑下面给出的有向图，下图的 DFS 为 1 2 4 6 3 5 7 8。在下图中，如果在该图中应用 DFS，则会获得使用绿色边缘连接的树。
树边：它是在对图应用 DFS 后获得的树中存在的边。所有的绿色边都是树边。
前向边：它是一条边 (u, v)，使得 v 是后代但不是 DFS 树的一部分。从1 到 8的边是前边。
后边：它是一条边 (u, v)，使得 v 是节点 u 的祖先，但不是 DFS 树的一部分。从6 到 2的边是后边。后边缘的存在表示有向图中的循环。
交叉边：它是连接两个节点的边，它们之间没有任何祖先和后代关系。从节点5 到 4的边是交叉边。
时间复杂度（DFS）
由于访问了所有节点和顶点，图上 DFS 的平均时间复杂度为 O(V + E)，其中 V 是顶点数，E 是边数。对于树上的 DFS，时间复杂度为 O(V)，其中 V 是节点数。
算法(DFS)
选择任何节点。如果它未被访问，则将其标记为已访问并在其所有相邻节点上重复。
重复直到所有节点都被访问过，或者找到要搜索的节点。
例子：
使用邻接表实现 DFS 取一个大小为 n=10 的有向图，并在图中随机选择 9 到 45 条边。将每条边标识为前边、树边、后边和交叉边。
Python3
# code
import random
class Graph:
# instance variables
def __init__(self, v):
# v is the number of nodes/vertices
self.time = 0
self.traversal_array = []
self.v = v
# e is the number of edge (randomly chosen between 9 to 45)
self.e = random.randint(9, 45)
# adj. list for graph
self.graph_list = [[] for _ in range(v)]
# adj. matrix for graph
self.graph_matrix = [[0 for _ in range(v)] for _ in range(v)]
# function to create random graph
def create_random_graph(self):
# add edges upto e
for i in range(self.e):
# choose src and dest of each edge randomly
src = random.randrange(0, self.v)
dest = random.randrange(0, self.v)
# re-choose if src and dest are same or src and dest already has an edge
while src == dest and self.graph_matrix[src][dest] == 1:
src = random.randrange(0, self.v)
dest = random.randrange(0, self.v)
# add the edge to graph
self.graph_list[src].append(dest)
self.graph_matrix[src][dest] = 1
# function to print adj list
def print_graph_list(self):
print("Adjacency List Representation:")
for i in range(self.v):
print(i, "-->", *self.graph_list[i])
print()
# function to print adj matrix
def print_graph_matrix(self):
print("Adjacency Matrix Representation:")
for i in self.graph_matrix:
print(i)
print()
# function the get number of edges
def number_of_edges(self):
return self.e
# function for dfs
def dfs(self):
self.visited = [False]*self.v
self.start_time = [0]*self.v
self.end_time = [0]*self.v
for node in range(self.v):
if not self.visited[node]:
self.traverse_dfs(node)
print()
print("DFS Traversal: ", self.traversal_array)
print()
def traverse_dfs(self, node):
# mark the node visited
self.visited[node] = True
# add the node to traversal
self.traversal_array.append(node)
# get the starting time
self.start_time[node] = self.time
# increment the time by 1
self.time += 1
# traverse through the neighbours
for neighbour in self.graph_list[node]:
# if a node is not visited
if not self.visited[neighbour]:
# marks the edge as tree edge
print('Tree Edge:', str(node)+'-->'+str(neighbour))
# dfs from that node
self.traverse_dfs(neighbour)
else:
# when the parent node is traversed after the neighbour node
if self.start_time[node] > self.start_time[neighbour] and self.end_time[node] < self.end_time[neighbour]:
print('Back Edge:', str(node)+'-->'+str(neighbour))
# when the neighbour node is a descendant but not a part of tree
elif self.start_time[node] < self.start_time[neighbour] and self.end_time[node] > self.end_time[neighbour]:
print('Forward Edge:', str(node)+'-->'+str(neighbour))
# when parent and neighbour node do not have any ancestor and a descendant relationship between them
elif self.start_time[node] > self.start_time[neighbour] and self.end_time[node] > self.end_time[neighbour]:
print('Cross Edge:', str(node)+'-->'+str(neighbour))
self.end_time[node] = self.time
self.time += 1
if __name__ == "__main__":
n = 10
g = Graph(n)
g.create_random_graph()
g.print_graph_list()
g.print_graph_matrix()
g.dfs()输出：
