On Hardness of Jumbled Indexing

  • Amihood Amir
  • Timothy M. Chan
  • Moshe Lewenstein
  • Noa Lewenstein
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8572)


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|Σ|) time allowing quick queries afterwards, and there is another naive algorithm that requires no preprocessing but has O(nlog|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 − 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 \(\Omega(n^{2-\epsilon_r})\) preprocessing time or \(\Omega(n^{1-\delta_r})\) query time.


Query Time Input String Alphabet Size Naive Algorithm Text Size 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Amihood Amir
    • 1
    • 2
  • Timothy M. Chan
    • 3
  • Moshe Lewenstein
    • 1
  • Noa Lewenstein
    • 4
  1. 1.Bar-Ilan UniversityIsrael
  2. 2.Johns Hopkins UniversityUSA
  3. 3.University of WaterlooCanada
  4. 4.Netanya CollegeIsrael

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