Approximate Pattern Matching on Elastic-Degenerate Text

An elastic-degenerate string is a sequence of $n$ sets of strings of total length $N$. It has been introduced to represent a multiple alignment of several closely-related sequences (eg pan-genome) compactly. In this representation, substrings of these sequences that match exactly are collapsed, while in positions where the sequences differ, all possible variants observed at that location are listed. The natural problem that arises is finding all matches of a deterministic pattern of length $m$ in an elastic-degenerate text. There exists a non-combinatorial $cO(nm^{1.381}+N)$-time algorithm to solve this problem on-line~cite{icalp2019}. In this paper, we study the same problem under the edit distance model and present an $cO(k^2mG+kN)$-time and $cO(m)$-space algorithm, where $G$ is the total number of strings in the elastic-degenerate text and $k$ is the maximum edit distance allowed. We also present a simple $cO(kmG+kN)$-time and $cO(m)$-space algorithm for solving the problem under Hamming distance.