Tree, Back, Edge and Cross Edges in DFS of Graph
Consider a directed graph given in below, DFS of the below graph is 1 2 4 6 3 5 7 8. In below diagram if DFS is applied on this graph a tree is obtained which is connected using green edges.
Tree Edge: It is an edge which is present in the tree obtained after applying DFS on the graph. All the Green edges are tree edges.
Forward Edge: It is an edge (u, v) such that v is descendant but not part of the DFS tree. Edge from 1 to 8 is a forward edge.
Back edge: It is an edge (u, v) such that v is ancestor of node u but not part of DFS tree. Edge from 6 to 2 is a back edge. Presence of back edge indicates a cycle in directed graph.
Cross Edge: It is a edge which connects two node such that they do not have any ancestor and a descendant relationship between them. Edge from node 5 to 4 is cross edge.
Time Complexity(DFS)
Since all the nodes and vertices are visited, the average time complexity for DFS on a graph is O(V + E), where V is the number of vertices and E is the number of edges. In case of DFS on a tree, the time complexity is O(V), where V is the number of nodes.
Algorithm(DFS)
Pick any node. If it is unvisited, mark it as visited and recur on all its adjacent nodes.
Repeat until all the nodes are visited, or the node to be searched is found.
Example:
Implement DFS using adjacency list take a directed graph of size n=10, and randomly select number of edges in the graph varying from 9 to 45. Identify each edge as forward edge, tree edge, back edge and cross edge.
Python3
# codeimport randomclass 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 += 1if __name__ == "__main__": n = 10 g = Graph(n) g.create_random_graph() g.print_graph_list() g.print_graph_matrix() g.dfs() |
OUTPUT:
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