Longest Common Subsequence (Dynamic Programming)
Find the longest subsequence common to two sequences using dynamic programming
Longest Common Subsequence (LCS)
LCS finds the longest subsequence that appears in the same relative order in both sequences, but not necessarily contiguous.
Theory & Concepts
Definition:
A subsequence is a sequence that can be derived from another sequence by deleting some or no elements without changing the order of remaining elements.
Key Points:
- Optimal substructure: LCS(X[1..m], Y[1..n]) depends on LCS of smaller subproblems
- Overlapping subproblems: same subproblems appear multiple times
- Bottom-up approach builds solution table systematically
- Time complexity O(mn), space complexity O(mn)
Applications:
- DNA sequence analysis in bioinformatics
- Version control systems (diff algorithms)
- Spell checkers and text similarity
- Data compression and file comparison
University Guidelines
Exam Weightage
6-8 marks MarksSyllabus Coverage:
- Longest Common Subsequence problem
- Dynamic programming formulation
- Recurrence relation development
- Table filling approach
- Backtracking to find actual LCS
Common Questions:
- Find LCS of two given strings
- Construct DP table step by step
- Trace back to find the actual subsequence
- Analyze time and space complexity
Exam Tips:
Draw the DP table clearly
Show the recurrence relation
Trace back from bottom-right corner
Highlight matching characters
Algorithm Details
O(m × n) where m, n are string lengths
Time Complexity
O(m × n) for DP table
Space Complexity
Algorithm Steps:
- 1Create DP table of size (m+1) × (n+1)
- 2Initialize first row and column with zeros
- 3For each character pair:
- 4If characters match: dp[i][j] = dp[i-1][j-1] + 1
- 5Else: dp[i][j] = max(dp[i-1][j], dp[i][j-1])
- 6Backtrack from dp[m][n] to construct LCS
Configure LCS Problem
String 1:
A
B
C
D
G
H
String 2:
A
E
D
F
H
R
LCS Results
🔤
Enter two strings and click "Find LCS" to see the dynamic programming solution