Faster Algorithms for 1-Mappability of a Sequence

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 $\mathcal {O}(mn \log n/\log \log n)$ and space $\mathcal {O}(n)$. We present two new algorithms that require worst-case time $\mathcal {O}(mn)$ and $\mathcal {O}(n \log n \log \log n)$, respectively, and space $\mathcal {O}(n)$, thus greatly improving the state of the art. Moreover, we present another algorithm that requires average-case time and space $\mathcal {O}(n)$ for integer alphabets of size $\sigma$ if $m=\varOmega (\log _\sigma n)$. Notably, we show that this algorithm is generalizable for arbitrary k, requiring average-case time $\mathcal {O}(kn)$ and space $\mathcal {O}(n)$ if $m=\varOmega (k\log _\sigma n)$.

M. Alzamel and C.S. Iliopoulos—Partially supported by the Onassis Foundation.

J. Radoszewski—Supported by the “Algorithms for text processing with errors and uncertainties” project carried out within the HOMING programme of the Foundation for Polish Science co-financed by the European Union under the European Regional Development Fund.