Closest string

In theoretical computer science, closest string is the name of an NP-hard computational problem, which tries to find the geometrical center of a set of input strings.

To understand the word "center" it is necessary to define a distance between two strings. Usually, this problem is studied with the Hamming distance in mind.

Formal definition

More formally, given n length-m strings $s_{1},s_{2},\dots ,s_{n}$ and a radius k, the closest string problem seeks for a new length-m string s such that $d(s,s_{i})\leq k\ \forall s_{i}$ where d denotes the Hamming distance.

Motivation

Closest string is an intensively studied facet of the problem of finding signals in DNA.

Simplifications and data reductions

Instances of closest string may contain information that is not essential to the problem. In some sense, the usual input of closest string contains information, that does not contribute to the hardness of the problem. For example, if some strings contain the character a, but none contains the character z, we could replace all as with zs and this modified instance would essentially be equivalent, that is: given a solution of the modified instance, we could tell the original solution and vice versa.

Normalizing the input

When all input strings that share the same length are written on top of each other, they form a matrix. Now, we can see that certain row types have essentially the same implications to the solution. Say, given a column with three entries, (a,a,b), then we might change the solution string, but not the solvability, by replacing this column with another column (x,x,y), because they express the same structure. This structure being that the first two entries are the same while the third is different.

We can, in each column, replace the character that occurs the most often with a, the character that occurs the second most often, and so forth. An input instance with this property is called normalized.

If we have a solution to the modified instance, we can find the original instance by remapping the characters of the solution to its original version in every column.

The order of the columns does not contribute to the hardness of the problem. That means, if we permute all input strings according to a certain permutation $\pi$ and obtain a solution string s to that modified instance, then $\pi ^{-1}(s)$ will be a solution to the original instance.

Example

Given an instance with three input strings 'uvwx', 'xuwv', 'xvwu'. This could be written as a matrix like this:

${\begin{matrix}u&v&w&x\\x&u&w&v\\x&v&w&u\end{matrix}}$

The first column has the values (u,x,x). As x is the character that appears the most often, we replace it by a, and we replace u, the second most often character, by b, obtaining the new first column (b,a,a).

Doing the same with all columns gives the new normalized instance

${\begin{matrix}b&a&a&a\\a&b&a&b\\a&a&a&c\end{matrix}}$

Data reduction obtained from normalization

Normalizing the input reduces the alphabet size to at most the number of input strings. This can be useful for algorithms whose running times depend on the alphabet size.

Approximability

Li et al. evolved a polynomial-time approximation scheme which is practically unusable because of the large hidden constants.

Fixed-parameter tractability

Closest String can be solved in $O(kL+kd\cdot d^{d})$, where k is the number of input strings, L is the length of all strings and d is the desired maximum distance from the solution string to any input string.

Relations to other problems

Closest string is a special case of the more general closest substring problem, which is strictly more difficult. While closest string turns out to be fixed-parameter tractable in a number of ways, closest substring is W[1]-hard with regard to these parameters.