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(n(2)|Sigma|) time allowing quick queries afterwards, and there is another naive algorithm that requires no preprocessing but has O(n log |Sigma|) 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 omega(1) requires Omega(n(2-epsilon)) preprocessing time or Omega(n(1-delta)) query time for any epsilon, delta > 0. In fact, under a stronger 3SUM-hardness assumption, for any constant alphabet size r >= 3 there exist describable fixed constant epsilon(r) and delta(r) such that jumbled indexing requires Omega(n(2-epsilon r)) preprocessing time or Omega(n(1-delta r)) query time.