This very small ppt presentation describes Longest Common Sub-sequence. This is one of the topics of Analysis of Algorithms

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))