The document discusses the Longest Common Subsequence (LCS) concept, which is a significant application of dynamic programming used to determine common patterns in sequences. It outlines the process for calculating LCS using brute force and dynamic programming techniques, highlighting its relevance in areas like bioinformatics and software tools for comparison. Key steps include characterizing the LCS, defining the recursive relationship, and computing its length effectively.

1. Longest Common
Subsequence(LCS)
An application of Dynamic Programming Approach
Tania Sahito

2. Subsequence
• X is a subsequence of Y if X is obtained by dropping elements of Y
without changing order of elements which we did not drop
• Note: string and sequence are used interchangeably because
“string is also a sequence of characters”.
• Example
• “ricm” is a subsequence of “Curriculum”

3. • Common subsequence
• Given 2 sequences X and Y , we say Z is a common subsequence of X and Y if
Z is present in X and Y.
• Longest Common Subsequence
• A common subsequence of the maximal length among all the
subsequences(There may be many LCS)
• Remember: Length of LCS(A,B)=Similarity between A And B
• Higher the Length(LCS(A,B))=more similarity

4. Motivation for LCS
• Bioinformatics: LCS is used in Molecular Biology for DNA
sequences
• Unix File Comparison(Program named as “Diff”)
• Screen reDisplay(in Emacs text editor)
• Misspelt Words Matching

5. We already know…
• Dynamic Programming involves following steps
• Characterize optimal substructure
• Recursively define the value of an optimal solution
• Compute the value bottom up
• If needed construct an optimal solution

6. Step#01: Characterizing the LCS
LCS by Brute Force Algorithm Design
• Two sequences
• X=x1,x2...xm
• Y=y1,y2y3…yn
Steps:
Generate all possible subsequences of A
Check which are also subsequences of B
Return the longest
Time complexity: O(n2M)
2M
Cost of
comparison=O(n)

7. Step#01: Characterize the LCS(DP)
• There may exist following cases

8. Step#02:Recursive Relation

9. Step#03: Computing the length of LCS

10. Step#04: Constructing an LCS

11. Example:What do "spanking" and "amputation" have in
common?

12. Modifications in LCS Algorithm
Some Questions:
• Do we need B?
• Do we need whole C (for finding longest length)?
• As we can see
• Use Linked List instead of Table
• Time complexity=O(MN. Max(M,N))