On hardness of jumbled indexing

Jumbled indexing is the problem of indexing a text T for queries that ask whether there is a substring of T matching a pattern represented as a Parikh vector, i.e., the vector of frequency counts for each character. Jumbled indexing has garnered a lot of interest in the last four years; for a partial list see [2,6,13,16,17,20,22,24,26,30,35,36]. There is a naive algorithm that preprocesses all answers in O(n²|∑|) time allowing quick queries afterwards, and there is another naive algorithm that requires no preprocessing but has O(nlog|∑|) query time. Despite a tremendous amount of effort there has been little improvement over these running times. In this paper we provide good reason for this. We show that, under a 3SUM-hardness assumption, jumbled indexing for alphabets of size ω(1) requires Ω(n^2-ε) preprocessing time or Ω(n^1-δ) query time for any ε,δ > 0. In fact, under a stronger 3SUM-hardness assumption, for any constant alphabet size r ≥ 3 there exist describable fixed constant ε_r and δ_r such that jumbled indexing requires Ω(n^2-εr) preprocessing time or Ω(n ^1-δr) query time.