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. There is a naive algorithm that preprocesses all answers in O(n~2|Σ|) time allowing quick queries afterwards, and there is another naive algorithm that requires no preprocessing but has O(n log |Σ|) 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-∈)) preprocessing time or Ω(n~(1-δ)) query time for any ∈, δ ＞ 0. In fact, under a stronger 3SUM-hardness assumption, for any constant alphabet size r > 3 there exist describable fixed constant ∈_r and δ_r such that jumbled indexing requires Ω(n~(2-∈_r)) preprocessing time or Ω(n~(1-δ-r)) query time.
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