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. 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 $r\ge 3$ 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.