- Design and Analysis of Algorithms
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- Basics of Algorithms
- DAA - Introduction
- DAA - Analysis of Algorithms
- DAA - Methodology of Analysis
- Asymptotic Notations & Apriori Analysis
- Time Complexity
- Master’s Theorem
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- Divide & Conquer
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- DAA - Greedy Method
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- Dynamic Programming
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- Matrix Chain Multiplication
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- Longest Common Subsequence
- Travelling Salesman Problem | Dynamic Programming
- Randomized Algorithms
- Randomized Algorithms
- Randomized Quick Sort
- Karger’s Minimum Cut
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- Approximation Algorithms
- Approximation Algorithms
- Vertex Cover Problem
- Set Cover Problem
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- Graph Theory
- DAA - Spanning Tree
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- Optimal Cost Binary Search Trees
- Heap Algorithms
- DAA - Binary Heap
- DAA - Insert Method
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- Sorting Techniques
- DAA - Bubble Sort
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- DAA - Selection Sort
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- Searching Techniques
- Searching Techniques Introduction
- DAA - Linear Search
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- DAA - Interpolation Search
- DAA - Jump Search
- DAA - Exponential Search
- DAA - Fibonacci Search
- DAA - Sublist Search
- Complexity Theory
- Deterministic vs. Nondeterministic Computations
- DAA - Max Cliques
- DAA - Vertex Cover
- DAA - P and NP Class
- DAA - Cook’s Theorem
- NP Hard & NP-Complete Classes
- DAA - Hill Climbing Algorithm
- DAA Useful Resources
- DAA - Quick Guide
- DAA - Useful Resources
- DAA - Discussion
Design and Analysis Max Cliques
In an undirected graph, a clique is a complete sub-graph of the given graph. Complete sub-graph means, all the vertices of this sub-graph is connected to all other vertices of this sub-graph.
The Max-Clique problem is the computational problem of finding maximum clique of the graph. Max clique is used in many real-world problems.
Let us consider a social networking application, where vertices represent people’s profile and the edges represent mutual acquaintance in a graph. In this graph, a clique represents a subset of people who all know each other.
To find a maximum clique, one can systematically inspect all subsets, but this sort of brute-force search is too time-consuming for networks comprising more than a few dozen vertices.
Max-Clique Algorithm
The algorithm to find the maximum clique of a graph is relatively simple. The steps to the procedure are given below −
Step 1: Take a graph as an input to the algorithm with a non-empty set of vertices and edges.
Step 2: Create an output set and add the edges into it if they form a clique of the graph.
Step 3: Repeat Step 2 iteratively until all the vertices of the graph are checked, and the list does not form a clique further.
Step 4: Then the output set is backtracked to check which clique has the maximum edges in it.
Pseudocode
Algorithm: Max-Clique (G, n, k)
S := ф
for i = 1 to k do
t := choice (1…n)
if t є S then
return failure
S := S U t
for all pairs (i, j) such that i є S and j є S and i ≠ j do
if (i, j) is not a edge of the graph then
return failure
return success
Analysis
Max-Clique problem is a non-deterministic algorithm. In this algorithm, first we try to determine a set of k distinct vertices and then we try to test whether these vertices form a complete graph.
There is no polynomial time deterministic algorithm to solve this problem. This problem is NP-Complete.
Example
Take a look at the following graph. Here, the sub-graph containing vertices 2, 3, 4 and 6 forms a complete graph. Hence, this sub-graph is a clique. As this is the maximum complete sub-graph of the provided graph, it’s a 4-Clique.
