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. There exists an algorithm to solve this problem for k = 1 requiring time O(mn log n/ log log n) using space O(n). Here we present two new algorithms that require worst-case time O(mn) and O(n log n log log n), respectively, and space O(n), thus greatly improving the previous result. Moreover, we present another algorithm that requires average-case time and space O(n) for integer alphabets of size σ if m = (log σ n). Notably, we show that this algorithm is generalizable for arbitrary k, requiring average-case time O(kn) and space O(n) if m = (k log σ n), assuming that the letters are independent and uniformly distributed random variables. Finally, we provide an experimental evaluation of our average-case algorithm demonstrating its competitiveness to the state-of-the-art implementation.

Efficient Computation of Sequence Mappability

“…All algorithms use O(n) space solutions of Alzamel et al [3] work also on strings over integer alphabets {1, . .…”

“…Overview of the algorithm Intuitively, the algorithm proceeds by efficiently simulating transformations of a compact trie of modified substrings, initially containing all substrings T m i . 3 The elementary transformations are guided by the smaller-to-larger principle, and each of them consists in copying one subtree unto its sibling, with an appropriate modification introduced to each copied substring in order to match the label of the edge leading to the sibling. This process effectively results in registering one mismatch for a large batch of substrings at once, and therefore lays a foundation to solve the main problem in the aforementioned time.…”

“…In this section, we generalize the O(nm)-time algorithm for k = 1 and integer alphabets from [3]. To this end, we make use of an approach from [6].…”

“…Through standard deamortization, we could achieve, with high probability, O(1)-time operations after O(n)-time initialization. However, this would not benefit the main results of this paper 3. The true course of the algorithm will not actually perform much of its operations on a compact trie, but the intuition is best conveyed by visualizing them this way.…”

The k-mappability problem revisited

“…For a text with constant alphabet Σ and k ∈ O(1), they present an algorithm with linear space and O(n log k+1 n) time. For the case in which k = 1 and a constant size alphabet, a faster algorithm with linear space and O(n log(n) log log(n)) time was presented in [2].In this work, we enhance the techniques of [2] to obtain an algorithm with linear space and O(n log(n)) time for k = 1. Our algorithm removes the constraint of the alphabet being of constant size.…”