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This approach uses breadth-first search (BFS) to try and color the graph using two colors, validating its bipartiteness. By starting at any unvisited node and attempting to color it with one color, you color all its neighbors with an alternating color. If at any point a neighbor is found to already have the same color, then the graph cannot be bipartite.
The time complexity is O(V + E) where V is the number of vertices and E is the number of edges, as each edge and vertex is visited once in BFS. The space complexity is O(V) due to the storage needed for the color array and the BFS queue.
1from collections import deque
2
3def bfsCheck(start, graph, colors):
4 queue = deque([start])
5 colors[start] = 0
6
7 while queue:
8 node = queue.popleft()
9 for neighbor in graph[node]:
10 if colors[neighbor] == -1: # Not colored
11 colors[neighbor] = 1 - colors[node]
12 queue.append(neighbor)
13 elif colors[neighbor] == colors[node]:
14 return False
15 return True
16
17def isBipartite(graph):
18 n = len(graph)
19 colors = [-1] * n
20
21 for i in range(n):
22 if colors[i] == -1:
23 if not bfsCheck(i, graph, colors):
24 return False
25 return True
26
27# Example usage:
28graph = [[1, 3], [0, 2], [1, 3], [0, 2]]
29print(isBipartite(graph))The Python version employs BFS checking with the collections.deque structure for efficient queue operations. Nodes are colored sequentially, and any contradiction in coloring through adjacent vertices leads to a false result.
The depth-first search (DFS) approach is an alternative to BFS for examining bipartiteness. You use DFS recursively to color the graph and check for conflicts. The method still involves checking whether adjacent nodes have opposing colors, iterating through components one by one as needed.
The time complexity is O(V + E), and the space complexity is O(V) due to recursive depth stack space utilization.
1
JavaScript imparts the recursive DFS method facilitating the coloring and adjacency check, inducing appropriate backtracking where a non-bipartite configuration occurs. As nodes get revisited, the initial states' integrity is validated against symptoms of unviable coloration.