Testing for forbidden order patterns in an array

Ilan Newman, Yuri Rabinovich, Deepak Rajendraprasad, Christian Sohler

Research output: Contribution to journal ā€ŗ Article ā€ŗ peer-review


A sequence f āˆ¶ [n] ā†’ R contains a pattern šœ‹ Ļ€āˆˆSk, that is, a permutations of [k], iff there are indices i1 < ā€¦ < ik, such that f(ix) > f(iy) whenever Ļ€(x) > Ļ€(y). Otherwise, f is Ļ€-free. We study the property testing problem of distinguishing, for a fixed Ļ€, between Ļ€-free sequences and the sequences which differ from any Ļ€-free sequence in more than Ļµ n places. Our main findings are as follows: (1) For monotone patterns, that is, Ļ€ = (k,k āˆ’ 1,ā€¦,1) and Ļ€ = (1,2,ā€¦,k), there exists a nonadaptive one-sided error Ļµ-test of (Ļµšœ–āˆ’1 log n)O(k2) query complexity. For any other Ļ€, any nonadaptive one-sided error test requires Ī©(āˆšn) queries. The latter lower-bound is tight for Ļ€ = (1,3,2). For specific Ļ€āˆˆSk it can be strengthened to Ī©(n1 āˆ’ 2/(k + 1)). The general case upper-bound is O(Ļµāˆ’1/kn1 āˆ’ 1/k). (2) For adaptive testing the situation is quite different. In particular, for any Ļ€āˆˆS3 there exists an adaptive Ļµ-tester of (Ļµšœ–āˆ’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

Publisher Copyright:
Ā© 2019 Wiley Periodicals, Inc.


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

ASJC Scopus subject areas

  • Software
  • General Mathematics
  • Computer Graphics and Computer-Aided Design
  • Applied Mathematics


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  • 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 proceeding ā€ŗ Conference contribution ā€ŗ peer-review

    Open Access

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