Efficient Computation of Sequence Mappability

Abstract

Sequence mappability is an important task in genome re-sequencing. In the (k, m)-mappability problem, for a given sequence T of length n, our goal is to compute a table whose ith entry is the number of indices $j \ne i$ such that length-m substrings of T starting at positions i and j have at most k mismatches. Previous works on this problem focused on heuristic approaches to compute a rough approximation of the result or on the case of $k=1$. We present several efficient algorithms for the general case of the problem. Our main result is an algorithm that works in $\mathcal {O}(n \min \{m^k,\log ^{k+1} n\})$ time and $\mathcal {O}(n)$ space for $k=\mathcal {O}(1)$. It requires a careful adaptation of the technique of Cole et al. [STOC 2004] to avoid multiple counting of pairs of substrings. We also show $\mathcal {O}(n^2)$-time algorithms to compute all results for a fixed m and all $k=0,\ldots ,m$ or a fixed k and all $m=k,\ldots ,n-1$. Finally we show that the (k, m)-mappability problem cannot be solved in strongly subquadratic time for $k,m = \varTheta (\log n)$ unless the Strong Exponential Time Hypothesis fails.

J. Radoszewski and J. Straszyński—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.