Comparing Elastic-Degenerate Strings: Algorithms, Lower Bounds, and Applications

An elastic-degenerate (ED) string T is a sequence of n sets T[1],…,T[n] containing m strings in total whose cumulative length is N. We call n, m, and N the length, the cardinality and the size of T, respectively. The language of T is defined as ℒ(T) = {S_1 ⋯ S_n : S_i ∈ T[i] for all i ∈ [1,n]}. ED strings have been introduced to represent a set of closely-related DNA sequences, also known as a pangenome. The basic question we investigate here is: Given two ED strings, how fast can we check whether the two languages they represent have a nonempty intersection? We call the underlying problem the ED String Intersection (EDSI) problem. For two ED strings T₁ and T₂ of lengths n₁ and n₂, cardinalities m₁ and m₂, and sizes N₁ and N₂, respectively, we show the following: - There is no 𝒪((N₁N₂)^{1-ε})-time algorithm, thus no 𝒪((N₁m₂+N₂m₁)^{1-ε})-time algorithm and no 𝒪((N₁n₂+N₂n₁)^{1-ε})-time algorithm, for any constant ε > 0, for EDSI even when T₁ and T₂ are over a binary alphabet, unless the Strong Exponential-Time Hypothesis is false. - There is no combinatorial 𝒪((N₁+N₂)^{1.2-ε}f(n₁,n₂))-time algorithm, for any constant ε > 0 and any function f, for EDSI even when T₁ and T₂ are over a binary alphabet, unless the Boolean Matrix Multiplication conjecture is false. - An 𝒪(N₁log N₁log n₁+N₂log N₂log n₂)-time algorithm for outputting a compact (RLE) representation of the intersection language of two unary ED strings. In the case when T₁ and T₂ are given in a compact representation, we show that the problem is NP-complete. - An 𝒪(N₁m₂+N₂m₁)-time algorithm for EDSI. - An Õ(N₁^{ω-1}n₂+N₂^{ω-1}n₁)-time algorithm for EDSI, where ω is the exponent of matrix multiplication; the Õ notation suppresses factors that are polylogarithmic in the input size. We also show that the techniques we develop have applications outside of ED string comparison.