String Attractors: Verification and Optimization
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String attractors [STOC 2018] are combinatorial objects recently introduced to unify all known dictionary compression techniques in a single theory. A set Γ⊆ [1..n] is a k-attractor for a string S∈[1..σ]^n if and only if every distinct substring of S of length at most k has an occurrence straddling at least one of the positions in Γ. Finding the smallest k-attractor is NP-hard for k≥3, but polylogarithmic approximations can be found using reductions from dictionary compressors. It is easy to reduce the k-attractor problem to a set-cover instance where string's positions are interpreted as sets of substrings. The main result of this paper is a much more powerful reduction based on the truncated suffix tree. Our new characterization of the problem leads to more efficient algorithms for string attractors: we show how to check the validity and minimality of a k-attractor in near-optimal time and how to quickly compute exact and approximate solutions. For example, we prove that a minimum 3-attractor can be found in optimal O(n) time when σ∈ O(√( n)) for any constant ϵ>0, and 2.45-approximation can be computed in O(n) time on general alphabets. To conclude, we introduce and study the complexity of the closely-related sharp-k-attractor problem: to find the smallest set of positions capturing all distinct substrings of length exactly k. We show that the problem is in P for k=1,2 and is NP-complete for constant k≥ 3.
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