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Longest Common Subsequence (LCS)

Longest Common Subsequence (LCS)

Find the longest subsequence common to two sequences using dynamic programming.

Family: Dynamic Programming Status: ✅ Complete

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Overview

The Longest Common Subsequence (LCS) problem is a fundamental dynamic programming problem

that finds the longest subsequence present in both given sequences. A subsequence is a sequence that appears in the same relative order, but not necessarily contiguous.

This algorithm has wide applications in bioinformatics (DNA sequence alignment), version control systems (diff algorithms), and text processing (plagiarism detection).

Mathematical Formulation

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Recurrence Relation

\[ F(n) = \begin{cases} 0 & \text{if } n = 0 \\ 1 & \text{if } n = 1 \\ F(n-1) + F(n-2) & \text{if } n > 1 \end{cases} \]

Problem Definition

Given two sequences X = [x₁, x₂, ..., xₘ] and Y = [y₁, y₂, ..., yₙ], find the longest

subsequence that appears in both sequences.

Key Properties

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  • Optimal Substructure

    LCS of prefixes is part of LCS of full sequences

  • Overlapping Subproblems

    Same subproblems solved multiple times in recursive approach

  • Subsequence Property

    Elements must appear in same relative order

Implementation Approaches

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Direct recursive implementation of recurrence relation

Complexity:

  • Time: O(2^(m+n))
  • Space: O(m+n)

Advantages

  • Simple to understand and implement

  • Direct translation of mathematical formulation

Disadvantages

  • Exponential time complexity

  • Redundant computation of same subproblems

Recursive approach with memoization to avoid redundant calculations

Complexity:

  • Time: O(m × n)
  • Space: O(m × n)

Advantages

  • Natural recursive structure

  • Only computes needed subproblems

  • Easy to understand

Disadvantages

  • Recursion stack overhead

  • May cause stack overflow for large inputs

Iterative approach building solution from base cases

Complexity:

  • Time: O(m × n)
  • Space: O(m × n)

Advantages

  • No recursion stack issues

  • Better space optimization possible

  • More predictable memory usage

Disadvantages

  • Computes all subproblems

  • Less intuitive than recursive approach

Uses only two rows of DP table instead of full 2D table

Complexity:

  • Time: O(m × n)
  • Space: O(min(m,n))

Advantages

  • Significant space savings

  • Same time complexity as standard DP

Disadvantages

  • Cannot reconstruct actual LCS sequence

  • Slightly more complex implementation

Complete Implementation

The full implementation with error handling, comprehensive testing, and additional variants is available in the source code:

Use Cases & Applications

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Application Categories

Bioinformatics

  • DNA sequence alignment: DNA sequence alignment

  • Protein sequence comparison: Protein sequence comparison

  • Phylogenetic analysis: Phylogenetic analysis

Version Control

  • Git diff algorithms: Git diff algorithms

  • File comparison: File comparison

  • Merge conflict resolution: Merge conflict resolution

Text Processing

  • Plagiarism detection: Plagiarism detection

  • Document similarity: Document similarity

  • Spell checking: Spell checking

Data Analysis

  • Time series analysis: Time series analysis

  • Pattern matching: Pattern matching

  • Anomaly detection: Anomaly detection

Educational Value

  • Understand dynamic programming principles: Understand dynamic programming principles

  • Learn to identify optimal substructure: Learn to identify optimal substructure

  • Practice with 2D DP problems: Practice with 2D DP problems

  • Understand space-time tradeoffs: Understand space-time tradeoffs

References & Further Reading

:material-book: Core Textbooks

:material-file-document:

:material-web: Online Resources

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Interactive Learning

Try implementing the different approaches yourself! This progression will give you deep insight into the algorithm's principles and applications.

Pro Tip: Start with the simplest implementation and gradually work your way up to more complex variants.

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Related Algorithms in Dynamic Programming:

  • Coin Change Problem - Find the minimum number of coins needed to make a given amount using dynamic programming.

  • 0/1 Knapsack Problem - Optimize value within weight constraints using dynamic programming for resource allocation.

  • Fibonacci Sequence - Classic sequence where each number is the sum of the two preceding ones, demonstrating recursion, memoization, and dynamic programming.

  • Edit Distance (Levenshtein Distance) - Calculate minimum operations to transform one string into another using dynamic programming.

  • Matrix Chain Multiplication - Find optimal parenthesization for matrix chain multiplication to minimize operations using dynamic programming.