Faster algorithms for 1-mappability of a sequence
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(mnlogn/loglogn) using space O(n). Here we present two new algorithms that require worst-case time O(mn) and O(nlognloglogn), 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=Ω(logσn). Notably, we show that this algorithm is generalizable for arbitrary k, requiring average-case time O(kn) and space O(n) if m=Ω(klogσn), 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.
|Theoretical Computer Science|
|Organisation||Centrum Wiskunde & Informatica, Amsterdam (CWI), The Netherlands|
Alzamel, M, Charalampopoulos, P, Iliopoulos, C.S, Pissis, S, Radoszewski, J, & Sung, W.-K. (2019). Faster algorithms for 1-mappability of a sequence. Theoretical Computer Science. doi:10.1016/j.tcs.2019.04.026