Testing for forbidden order patterns in an array

Ilan Newman, Yuri Rabinovich, Deepak Rajendraprasad, Christian Sohler

Research output: Contribution to journalArticlepeer-review


A sequence f ∶ [n] → R contains a pattern 𝜋 π∈Sk, that is, a permutations of [k], iff there are indices i1 < … < ik, such that f(ix) > f(iy) whenever π(x) > π(y). Otherwise, f is π-free. We study the property testing problem of distinguishing, for a fixed π, between π-free sequences and the sequences which differ from any π-free sequence in more than ϵ n places. Our main findings are as follows: (1) For monotone patterns, that is, π = (k,k − 1,…,1) and π = (1,2,…,k), there exists a nonadaptive one-sided error ϵ-test of (ϵ𝜖−1 log n)O(k2) query complexity. For any other π, any nonadaptive one-sided error test requires Ω(√n) queries. The latter lower-bound is tight for π = (1,3,2). For specific π∈Sk it can be strengthened to Ω(n1 − 2/(k + 1)). The general case upper-bound is O(ϵ−1/kn1 − 1/k). (2) For adaptive testing the situation is quite different. In particular, for any π∈S3 there exists an adaptive ϵ-tester of (ϵ𝜖−1 log n)O(1) query complexity.

Original languageEnglish
Pages (from-to)402-426
Number of pages25
JournalRandom Structures and Algorithms
Issue number2
StatePublished - Sep 2019

Bibliographical note

Funding Information:
information: This research was supported by the Israel Science Foundation; 497/17. ERC Grant; 307696.This research was supported by The Israel Science Foundation, number 497/17 [I.N.]. This work was done while visiting the Caesarea Rothschild Institute, University of Haifa [D.R.]. The author acknowledges the support of ERC grant 307696 [C.S.].

Publisher Copyright:
© 2019 Wiley Periodicals, Inc.


  • Adaptive Testing
  • Monotonicity Testing
  • Non-adaptive Testing
  • Property Testing

ASJC Scopus subject areas

  • Software
  • Mathematics (all)
  • Computer Graphics and Computer-Aided Design
  • Applied Mathematics


Dive into the research topics of 'Testing for forbidden order patterns in an array'. Together they form a unique fingerprint.
  • Testing for forbidden order patterns in an array

    Newman, I., Rabinovich, Y., Rajendraprasad, D. & Sohler, C., 2017, 28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017. Klein, P. N. (ed.). Association for Computing Machinery, p. 1582-1597 16 p. (Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms; vol. 0).

    Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

    Open Access

Cite this