Add Longest Common Subsequence (LCS) dynamic programming algorithm [HACKTOBERFEST 2025]

[piyushkumar0707]

🚀 Overview

This PR implements the Longest Common Subsequence (LCS) algorithm - a fundamental dynamic programming solution used in bioinformatics, version control, and text analysis.

✨ Features

• ✅ Complete DP implementation - O(m*n) time with full DP table
• ✅ Space-optimized version - O(min(m,n)) space for large inputs
• ✅ Path reconstruction - Returns actual LCS string, not just length
• ✅ Multiple sequence types - Works with strings and arrays
• ✅ All solutions finder - Finds multiple LCS when they exist
• ✅ File diff simulation - Real-world version control application

🎯 Why This Matters

LCS algorithm is essential for:

• Bioinformatics - DNA/protein sequence alignment and analysis
• Version control - Git diff functionality and merge conflict resolution
• Plagiarism detection - Finding common text segments
• Data compression - Identifying repeated subsequences
• Text analysis - Document similarity and comparison
• Edit distance - Foundation for Levenshtein distance calculations

📚 Implementation Details

• Time Complexity: O(m × n) where m, n are sequence lengths
• Space Complexity: O(m × n) standard, O(min(m, n)) optimized
• Algorithm: Dynamic programming with optimal substructure
• Output: Both LCS length and actual subsequence

🔧 Advanced Functions

• lcs_string() - Complete solution with path reconstruction
• lcs_length_optimized() - Space-efficient O(min(m,n)) version
• find_all_lcs() - Finds all possible LCS solutions
• lcs_array() - Works with integer arrays, not just strings
• print_dp_table() - Educational DP table visualization

🧬 Bioinformatics Application

DNA Sequence Analysis:

dna1 <- "ATCGATCGATCG"
dna2 <- "ATGCGATGCATG"
result <- lcs_string(dna1, dna2)
# Finds common genetic patterns
similarity <- result$length / max(nchar(dna1), nchar(dna2)) * 100

[Copilot]

Pull Request Overview

This PR implements the Longest Common Subsequence (LCS) dynamic programming algorithm with multiple variants and optimizations. The implementation includes both basic and advanced features for finding common subsequences between strings and arrays.

• Complete LCS implementation with O(m*n) time complexity and path reconstruction
• Space-optimized version using O(min(m,n)) space for memory efficiency
• Multiple LCS finder and array support for different data types

Reviewed Changes

Copilot reviewed 2 out of 2 changed files in this pull request and generated 4 comments.

File	Description
dynamic_programming/longest_common_subsequence.r	Complete LCS algorithm implementation with multiple variants, optimizations, and comprehensive test cases
DIRECTORY.md	Added entry for the new dynamic programming section and LCS algorithm

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