# The Stony Brook Algorithm Repository

## INPUT OUTPUT

Input Description: A set S of strings S_1,...,S_n.

Problem: What is the longest string c such that for each S_i, 1 \leq i \leq n, the characters of c appear as a scattered subsequence of S_i?

Excerpt from The Algorithm Design Manual: The problem of longest common subsequence arises whenever we search for similarities across multiple texts. A particularly important application is in finding a consensus among DNA sequences. The genes for building particular proteins evolve with time, but the functional regions must remain consistent in order to work correctly. By finding the longest common subsequence of the same gene in different species, we learn what has been conserved over time.

The longest common substring problem is a special case of edit distance, when substitutions are forbidden and only exact character match, insert, and delete are allowable edit operations. Under these conditions, the edit distance between p and t is n+m-2 |lcs(p,t)|, since we can delete the missing characters from p to the lcs(p,t) and insert the missing characters from $t$ to transform p to t. This is particularly interesting because the longest common subsequence can be faster to compute than edit distance.

## Recommended Books

 Algorithms on Strings, Trees, and Sequences by Dan Gusfield Introduction to Computational Biology by M. Waterman Handbook of Algorithms and Data Structures by G. Gonnet and R. Baeza-Yates Computational Discrete Mathematics: Combinatorics and Graph Theory with Mathematica by S. Pemmaraju and S. Skiena Introduction to Algorithms by T. Cormen and C. Leiserson and R. Rivest and C. Stein Introduction to Algorithms by U. Manber Data Structures and Algorithms by A. Aho and J. Hopcroft and J. Ullman

## Related Problems

   Approximate String Matching   Shortest Common Superstring   Suffix Trees and Arrays