Longest Increasing Subsequence under Persistent Comparison Errors

We study the problem of computing a longest increasing subsequence in a sequence S of n distinct elements in the presence of persistent comparison errors. In this model, Braverman and Mossel ( Noisy sorting without resampling , SODA 2008, pages 268–276, 2008) every comparison between two elements ca...

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Vydáno v:	Theory of computing systems Ročník 64; číslo 4; s. 662 - 680
Hlavní autor:	Geissmann, Barbara
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York Springer US 01.05.2020 Springer Nature B.V
Témata:	Algorithms Approximation Computation Computer Science Lower bounds Mathematical analysis Resampling Run time (computers) Special Issue on Approximation and Online Algorithms 2018 Theory of Computation Probabilistic persistent comparison errors Lower bounds Approximation algorithm Longest increasing subsequence
ISSN:	1432-4350, 1433-0490
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Abstract	We study the problem of computing a longest increasing subsequence in a sequence S of n distinct elements in the presence of persistent comparison errors. In this model, Braverman and Mossel ( Noisy sorting without resampling , SODA 2008, pages 268–276, 2008) every comparison between two elements can return the wrong result with some fixed (small) probability p , and comparisons cannot be repeated. Computing the longest increasing subsequence exactly is impossible in this model, therefore, the objective is to identify a subsequence that (i) is indeed increasing and (ii) has a length that approximates the length of the longest increasing subsequence. We present asymptotically tight upper and lower bounds on both the approximation factor and the running time. In particular, we present an algorithm that computes an O ( log n ) -approximation in O ( n log n ) time, with high probability. This approximation relies on the fact that we can approximately sort (Geissmann et al. Optimal Sorting with Persistent Comparison Errors , ArXiv e-prints 1804.07575, 2018) n elements in O ( n log n ) time such that the maximum dislocation of an element is O ( log n ) . For the lower bounds, we prove that (i) there is a set of sequences, such that on a sequence picked randomly from this set every algorithm must return an Ω ( log n ) -approximation with high probability, and (ii) any log n -approximation algorithm for longest increasing subsequence requires Ω ( n log n ) comparisons, even in the absence of errors.
AbstractList	We study the problem of computing a longest increasing subsequence in a sequence S of n distinct elements in the presence of persistent comparison errors. In this model, Braverman and Mossel (Noisy sorting without resampling, SODA 2008, pages 268–276, 2008) every comparison between two elements can return the wrong result with some fixed (small) probability p, and comparisons cannot be repeated. Computing the longest increasing subsequence exactly is impossible in this model, therefore, the objective is to identify a subsequence that (i) is indeed increasing and (ii) has a length that approximates the length of the longest increasing subsequence. We present asymptotically tight upper and lower bounds on both the approximation factor and the running time. In particular, we present an algorithm that computes an O(logn)-approximation in O(nlogn) time, with high probability. This approximation relies on the fact that we can approximately sort (Geissmann et al. Optimal Sorting with Persistent Comparison Errors, ArXiv e-prints 1804.07575, 2018) n elements in O(nlogn) time such that the maximum dislocation of an element is O(logn). For the lower bounds, we prove that (i) there is a set of sequences, such that on a sequence picked randomly from this set every algorithm must return an Ω(logn)-approximation with high probability, and (ii) any logn-approximation algorithm for longest increasing subsequence requires Ω(nlogn) comparisons, even in the absence of errors. We study the problem of computing a longest increasing subsequence in a sequence S of n distinct elements in the presence of persistent comparison errors. In this model, Braverman and Mossel ( Noisy sorting without resampling , SODA 2008, pages 268–276, 2008) every comparison between two elements can return the wrong result with some fixed (small) probability p , and comparisons cannot be repeated. Computing the longest increasing subsequence exactly is impossible in this model, therefore, the objective is to identify a subsequence that (i) is indeed increasing and (ii) has a length that approximates the length of the longest increasing subsequence. We present asymptotically tight upper and lower bounds on both the approximation factor and the running time. In particular, we present an algorithm that computes an O ( log n ) -approximation in O ( n log n ) time, with high probability. This approximation relies on the fact that we can approximately sort (Geissmann et al. Optimal Sorting with Persistent Comparison Errors , ArXiv e-prints 1804.07575, 2018) n elements in O ( n log n ) time such that the maximum dislocation of an element is O ( log n ) . For the lower bounds, we prove that (i) there is a set of sequences, such that on a sequence picked randomly from this set every algorithm must return an Ω ( log n ) -approximation with high probability, and (ii) any log n -approximation algorithm for longest increasing subsequence requires Ω ( n log n ) comparisons, even in the absence of errors.
Author	Geissmann, Barbara
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Cites_doi	10.1016/S0020-0190(00)00124-1 10.1016/0020-0190(77)90031-X 10.1093/nar/27.11.2369 10.1090/S0894-0347-99-00307-0 10.1016/0022-0000(80)90002-1 10.1007/s10878-006-7125-x 10.1017/S0963548304006297 10.1561/0400000002 10.1145/2701427 10.1016/j.ic.2010.04.003 10.1016/j.comgeo.2004.12.007 10.1016/j.dam.2011.05.010 10.1016/j.ipl.2004.10.014 10.1016/0012-365X(75)90103-X 10.1007/BF01683268 10.1093/bioinformatics/btg168 10.1090/S0273-0979-99-00796-X 10.1137/S0097539791195877 10.1007/978-3-319-44543-4_31 10.1145/2588555.2593662 10.1007/978-3-642-23719-5_62
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Snippet	We study the problem of computing a longest increasing subsequence in a sequence S of n distinct elements in the presence of persistent comparison errors. In... We study the problem of computing a longest increasing subsequence in a sequence S of n distinct elements in the presence of persistent comparison errors. In...
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SubjectTerms	Algorithms Approximation Computation Computer Science Lower bounds Mathematical analysis Resampling Run time (computers) Special Issue on Approximation and Online Algorithms 2018 Theory of Computation
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Title	Longest Increasing Subsequence under Persistent Comparison Errors
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