# Speeding Up Motif Finding solved by 1540

July 2, 2012, midnight by Rosalind Team

Topics: String Algorithms

In “Finding a Motif in DNA”, we discussed the problem of searching a genome for a known motif. Because of the large scale of eukaryotic genomes, we need to accomplish this computational task as efficiently as possible.

The standard method for locating one string $t$ as a substring of another string $s$ (and perhaps one you implemented in “Finding a Motif in DNA”) is to move a sliding window across the larger string, at each step starting at $s[k]$ and matching subsequent symbols of $t$ to symbols of $s$. After we have located a match or mismatch, we then shift the window backwards to begin searching at $s[k+1]$.

The potential weakness of this method is as follows: say we have matched 100 symbols of $t$ to $s$ before reaching a mismatch. The window-sliding method would then move back 99 symbols of $s$ and start comparing $t$ to $s$; can we avoid some of this sliding?

For example, say that we are looking for $t = \mathrm{ACGTACGT}$ in $s = \mathrm{TAGGTACGTACGGCATCACG}$. From $s[6]$ to $s[12]$, we have matched seven symbols of $t$, and yet $s[13]$ = G produces a mismatch with $t[8]$ = T. We don't need to go all the way back to $s[7]$ and start matching with $t$ because $s[7] = \mathrm{C}$, $s[8] = \mathrm{G}$, and $s[9] = \mathrm{T}$ are all different from $t[1] = \mathrm{A}$. What about $s[10]$? Because $t[1:4] = t[5:8] = \mathrm{ACGT}$, the previous mismatch of $s[13] = \mathrm{G}$ and $t[8] = \mathrm{T}$ guarantees the same mismatch with $s[13]$ and $t[4]$. Following this analysis, we may advance directly to $s[14]$ and continue sliding our window, without ever having to move it backward.

This method can be generalized to form the framework behind the Knuth-Morris-Pratt algorithm (KMP), which was published in 1977 and offers an efficiency boost for determining whether a given motif can be located within a larger string.

## Problem

A prefix of a length $n$ string $s$ is a substring $s[1:j]$; a suffix of $s$ is a substring $s[k:n]$.

The failure array of $s$ is an array $P$ of length $n$ for which $P[k]$ is the length of the longest substring $s[j:k]$ that is equal to some prefix $s[1:k-j+1]$, where $j$ cannot equal $1$ (otherwise, $P[k]$ would always equal $k$). By convention, $P[1] = 0$.

Given: A DNA string $s$ (of length at most 100 kbp) in FASTA format.

Return: The failure array of $s$.

## Sample Dataset

>Rosalind_87
CAGCATGGTATCACAGCAGAG


## Sample Output

0 0 0 1 2 0 0 0 0 0 0 1 2 1 2 3 4 5 3 0 0


## Extra Information

If you would like a more precise technical explanation of the Knuth-Morris-Pratt algorithm, please take a look at this site