TY - GEN

T1 - On hardness of jumbled indexing

AU - Amir, Amihood

AU - Chan, Timothy M.

AU - Lewenstein, Moshe

AU - Lewenstein, Noa

PY - 2014

Y1 - 2014

N2 - 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(n2-ε) preprocessing time or Ω(n1-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 εr and δr such that jumbled indexing requires Ω(n2-εr) preprocessing time or Ω(n 1-δr) query time.

AB - 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(n2-ε) preprocessing time or Ω(n1-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 εr and δr such that jumbled indexing requires Ω(n2-εr) preprocessing time or Ω(n 1-δr) query time.

UR - http://www.scopus.com/inward/record.url?scp=84904174181&partnerID=8YFLogxK

U2 - 10.1007/978-3-662-43948-7_10

DO - 10.1007/978-3-662-43948-7_10

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AN - SCOPUS:84904174181

SN - 9783662439470

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 114

EP - 125

BT - Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Proceedings

PB - Springer Verlag

T2 - 41st International Colloquium on Automata, Languages, and Programming, ICALP 2014

Y2 - 8 July 2014 through 11 July 2014

ER -