What is a bipartite graph?
A bipartite graph is a graph whose vertex set can be divided into two mutually exclusive subsets. In other words, it is a division such that all edges of the graph only exist between vertices of the two different sets.
The most common example of a bipartite graph is the “matching” problem. For instance, when matching students to classes, students and classes can be entered as each of the respective sets.
Bipartite graphs have the property that they can always be colored with two colors.
Problem Description
Write a function to determine if the given undirected graph is bipartite.
The given graph is presented in the form of an adjacency list, and the vertices are connected from 0 to n-1.
The function should return true if the graph is bipartite, and false otherwise.
Input Example
n = 4
edges = [[0, 1], [0, 3], [1, 2], [2, 3]]
Output Example
false
Problem Solving Process
-
Understanding the Structure of the Graph
The given graph consists of nodes and edges, with each node connected to other nodes.
We will represent the graph in an undirected linked list format.
Many programming languages, including Java and Swift, can implement this structure using arrays or hashmaps. -
Properties of Bipartite Graphs and Search Methods
A bipartite graph can be divided into two sets of vertices,
where all adjacent vertices must belong to different sets. Utilizing this property, we can employ depth-first search (DFS) or breadth-first search (BFS)
as an approach to color the graph. -
Exploring the Graph Using DFS or BFS
We start exploring the graph by coloring each vertex.
Two colors are used (e.g., 1 and -1), and if we revisit a node that is already colored,
we can determine that it is not a bipartite graph if the colors match.
Code Implementation
We will now implement an algorithm to determine bipartiteness in Swift.
class Solution {
func isBipartite(_ graph: [[Int]]) -> Bool {
let n = graph.count
var color = Array(repeating: -1, count: n) // -1 means uncolored
for i in 0..<n {="" if="" color[i]="" !="-1" continue="" }="" var="" queue:="" [int]="[i]" while="" !queue.isempty="" let="" node="queue.removeFirst()" for="" neighbor="" in="" graph[node]="" color[neighbor]="=" -1="" -="" color[node]="" queue.append(neighbor)="" else="" return="" false="" true="" <="" code=""></n>
<h2>Examples and Explanation</h2>
<p>The above code traverses the given graph, coloring the nodes and checking for re-visits to determine bipartiteness.<br>
In the example above, the graph takes the following form:</p>
<pre> 0 -- 1
| |
3 -- 2
</pre>
<p>In this case, nodes 0 and 1 have different colors, 1 and 2 have different colors, and 2 and 3 have different colors.<br>
However, nodes 0 and 3 have the same color, indicating that it is not a bipartite graph.<br>
This can be verified through BFS exploration.</p>
<h2>Conclusion</h2>
<p>In this article, we explained the concept of bipartite graphs and the process of their determination,<br>
and we explored how to implement this in Swift.<br>
This problem is useful for laying the foundations of algorithms and data structures,<br>
and it can be applied in various interview questions.<br>
Therefore, understanding bipartite graphs and coloring algorithms deeply through this problem is crucial.</p>
<h2>References</h2>
<ul>
<li><a href="https://en.wikipedia.org/wiki/Bipartite_graph">Wikipedia - Bipartite Graph</a></li>
<li><a href="https://leetcode.com/problems/is-graph-bipartite/">LeetCode: Is Graph Bipartite?</a></li>
</ul>
<p></p>
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