Testing permutation properties through subpermutations

Hoppen, Carlos; Kohayakawa, Yoshiharu; Moreira, Carlos Gustavo; Sampaio, Rudini Menezes

doi:10.1016/j.tcs.2011.03.002

Cited by 31 publications

(33 citation statements)

References 19 publications

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“…Works on string properties related to forbidden or occurring patterns in labeled posets include the papers on testing sortedness [9,21] and the lower bound of Fischer [22], and many others, see for example, [3,7,24,37], and citations therein. The problem of testing hereditary properties of permutations has been studied before by Hoppen and coworkers [31] under the rectangular distance (discrepancy of intervals) and by Klimošová and Král [36] under Kendall's tau-distance (the normalized number of transpositions). Unlike the edit distance used in this paper, in both the distance measures discussed above, local changes do not contribute much to the distance: For example, the sequence (2, 1, 4, 3, … , n, n − 1) is close to being monotone with respect to these two distances, while according to the edit distance it is not.…”

Section: Other Related Workmentioning

confidence: 99%

Testing for forbidden order patterns in an array

Newman

Rabinovich

Rajendraprasad

et al. 2019

Random Struct Algorithms

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A sequence f:false[nfalse]→double-struckR contains a pattern π∈scriptSk, 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 false(ϵ−1normallognfalse)Ofalse(k2false) query complexity. For any other π, any nonadaptive one‐sided error test requires normalΩfalse(nfalse) queries. The latter lower‐bound is tight for π = (1,3,2). For specific π∈scriptSk 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∈scriptS3 there exists an adaptive ϵ‐tester of false(ϵ−1normallog1emnfalse)Ofalse(1false) query complexity.

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Section: Other Related Workmentioning

confidence: 99%

Testing for forbidden order patterns in an array

Newman

Rabinovich

Rajendraprasad

et al. 2019

Random Struct Algorithms

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“…Works on string properties related to forbidden or occurring patterns in labeled posets include the papers on testing sortedness [13,6] and the lower bound of Fischer [14], and many others, see e.g., [4,26,2,16], and others. The problem of testing hereditary properties of permutations has been studied before by Hoppen et al [21] under the rectangular distance (discrepancy of intervals) and by Klimošová and Král' [25] under Kendall's tau-distance (the normalized number of transpositions). Unlike the edit distance used in this paper, in both the distance measures discussed above, local changes do not contribute much to the distance: For example, the sequence (2, 1, 4, 3, .…”

Section: Delete Distance Vs Hamming Distancementioning

confidence: 99%

Testing for Forbidden Order Patterns in an Array

Newman¹,

Rabinovich²,

Rajendraprasad³

et al. 2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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In this paper, we study testing of sequence properties that are defined by forbidden order patterns. A sequence f : {1, . . . , n} → R of length n contains a pattern π ∈ S k (S k is the group of permutations of k elements), iff there are indices i 1 < i 2 < · · · < i k , such that f (i x ) > f (i y ) whenever π(x) > π(y). If f does not contain π, we say f is π-free. For example, for π = (2, 1), the property of being π-free is equivalent to being non-decreasing, i.e. monotone. The property of being (k, k − 1, . . . , 1)-free is equivalent to the property of having a partition into at most k − 1 non-decreasing subsequences.Let π ∈ S k , k constant, be a (forbidden) pattern. Assuming f is stored in an array, we consider the property testing problem of distinguishing the case that f is π-free from the case that f differs in more than n places from any π-free sequence. We show the following results: There is a clear dichotomy between the monotone patterns and the non-monotone ones:• For monotone patterns of length k, i.e., (k, k − 1, . . . , 1) and (1, 2, . . . , k), we design non-adaptive one-sided error -tests of (complexity.• For non-monotone patterns, we show that for any size-k non-monotone π, any non-adaptive one-sided error -test requires at least Ω( √ n) queries. This general lower bound can be further strengthened for specific non-monotone k-length patterns to Ω(n 1−2/(k+1) ).On the other hand, there always exists a nonadaptive one-sided error -test for π ∈ S k with O( −1/k n 1−1/k ) query complexity. Again, this general upper bound can be further strengthened for specific non-monotone patterns. E.g., for π = * Department of Computer Science, University of Haifa, Israel. (1, 3, 2), we describe an -test with (almost tight) query complexity of O( √ n).Finally, we show that adaptivity can make a big difference in testing non-monotone patterns, and develop an adaptive algorithm that for any π ∈ S 3 , tests π-freeness by making ( −1 log n) O(1) queries. For all algorithms presented here, the running times are linear in their query complexity.

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“…From this we derive an analogous statement which quantifies how small d (π , σ (k, π)) is. This will prove to be useful in the characterization of testability in [15].…”

Section: Rectangular Distancementioning

confidence: 99%