Efficient Computation of Sequence Mappability

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.