Algorithm Implementations

A collection of classic algorithm implementations including brute force approaches, dynamic programming solutions, and matrix operations. Each implementation includes analysis of time complexity and algorithmic approach.

Implementations

1. Brute Force Algorithm

• Implementation of exhaustive search approach
• Applications:
  • String matching
  • Pattern finding
  • Combinatorial problems
• Time Complexity Analysis
• Time Complexity: O(n!) for permutation problems
• Space Complexity: O(n) for recursion depth
• Test cases demonstrating various scenarios

2. Knapsack Problem

Different approaches to solving the 0/1 Knapsack problem:

• Dynamic Programming Solution
  • Time Complexity: O(nW)
  • Space Complexity: O(nW)
• Approximation Algorithms
  • Greedy Approach O(n log n) time for sorting
  • FPTAS O(n/ε) time where ε is the approximation factor (Fully Polynomial Time Approximation Scheme)
• Performance comparison between approaches

3. Longest Common Subsequence (LCS)

Dynamic Programming implementation of LCS:

• Matrix-based solution
• Backtracking for sequence reconstruction
• Time Complexity: O(mn)
• Space Complexity: O(mn)
• Example cases with visualizations

4. Strassen's Matrix Multiplication

• Implementation of Strassen's algorithm for matrix multiplication
• Comparison with conventional matrix multiplication
• Time Complexity: O(n^2.807)
• Optimizations and threshold implementations
• Performance analysis with varying matrix sizes