The Clique Problem A clique is a subset of a graph's vertices such that every pair of vertices in the subset is connected by an edge. In simpler words, a cl...
The Clique Problem
A clique is a subset of a graph's vertices such that every pair of vertices in the subset is connected by an edge. In simpler words, a clique is a group of friends or colleagues who are all connected to each other.
Formal Definition:
A clique is a subset C of a graph G such that for all a, b, c in C, the distance between a and b is equal to the distance between a and c.
Examples:
A---B---C
A---B---C---D
Complexity:
The complexity of the clique problem is O(n^2), where n is the number of vertices in the graph. This is because the algorithm must check all pairs of vertices in the graph, which takes O(n^2) time.
Algorithms for Solving the Clique Problem:
There are two main algorithms for solving the clique problem:
Clifton's Algorithm
Kruskal's Algorithm
Clifton's Algorithm:
The Clifton algorithm works by randomly selecting a subset of vertices and checking if it forms a clique. If it does, the algorithm adds the subset to the output and repeats the process with the remaining vertices. The algorithm is simple but can be inefficient for large graphs.
Kruskal's Algorithm:
The Kruskal algorithm works by selecting the minimum spanning tree of the graph and adding the vertices in the order in which they are visited by the tree. The algorithm is more efficient than Clifton's algorithm but can be more complex to implement.
Time Complexity of Clique Problem:
The time complexity of the clique problem is O(n^2), where n is the number of vertices in the graph. This is because the algorithm must check all pairs of vertices in the graph, which takes O(n^2) time