# Faster algorithms for 1-mappability of a sequence

## TL;DR

This paper introduces faster algorithms for the 1-mappability problem in sequences, significantly improving efficiency over previous methods with worst-case and average-case solutions.

## Contribution

The authors develop two new algorithms with improved worst-case time complexities and an average-case algorithm for integer alphabets, advancing the state of the art in sequence mappability.

## Key findings

- Algorithms with worst-case time O(mn) and O(n log^2 n)
- Average-case algorithm with linear space and time for certain m
- Significant reduction in computational complexity for 1-mappability

## Abstract

In the k-mappability problem, we are given a string x of length n and integers m and k, and we are asked to count, for each length-m factor y of x, the number of other factors of length m of x that are at Hamming distance at most k from y. We focus here on the version of the problem where k = 1. The fastest known algorithm for k = 1 requires time O(mn log n/ log log n) and space O(n). We present two algorithms that require worst-case time O(mn) and O(n log^2 n), respectively, and space O(n), thus greatly improving the state of the art. Moreover, we present an algorithm that requires average-case time and space O(n) for integer alphabets if m = {\Omega}(log n/ log {\sigma}), where {\sigma} is the alphabet size.