Tilings

doi:10.1201/b18255-15

Cited by 27 publications

(34 citation statements)

References 153 publications

(235 reference statements)

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“…Whether {C n (1324)} is P-recursive remains a long-standing open problem in the area (see e.g. [Ste,V2]). In fact, there seems to be some recent strong experimental evidence against sequence {C n (1324)} being P-recursive (cf.…”

Section: Final Remarks and Open Problems 61mentioning

confidence: 99%

Permutation patterns are hard to count

Garrabrant¹,

Pak²

2015

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

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Section: Final Remarks and Open Problems 61mentioning

confidence: 99%

Permutation patterns are hard to count

Garrabrant¹,

Pak²

2015

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

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“…For two permutations π 1 of size n 1 and π 2 of size n 2 , the direct sum of π 1 and π 2 is defined by [15]. The direct sum operation consists to putting the elements of π 2 on the right and above the elements of π 1 .…”

Section: [J] If π[J] < π[I] (Resp π[I] < π[J])mentioning

confidence: 99%

“…binary) separating trees are not well-suited to handle this task. We use here the notion of compact separating tree (also known as decomposition tree [15]). Informally, in compact separating tree, we strive for every node to have as many children as possible.…”

Section: Tmentioning

confidence: 99%

“…To mention just a few examples, they are the permutations whose permutation graphs are cographs (i.e. P 4 -free graphs); equivalently, a separable permutation is a permutation that can be obtained from the trivial permutation 1 by direct sums and skew sums [15]. While the term separable permutation dates only to the work of Bose, Buss, and Lubiw [5], these permutations first arose in Avis and Newborns work on pop stacks [3].…”

Section: Introductionmentioning

confidence: 99%

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Pattern Matching for Separable Permutations

Neou

Rizzi

2016

String Processing and Information Retrieval

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Abstract. Given a permutation π (called the text) of size n and another permutation σ (called the pattern) of size k, the NP-complete pattern containment problem asks whether σ is contained in π as an order-isomorphic subsequence. In this paper, we focus on separable permutations (those permutations that avoid both 2413 and 3142, or, equivalently, that admit a separating tree). The main contributions presented in this paper are as follows.-We simplify the algorithm of Ibarra to detect an occurrence of a separable permutation in a permutation and show how to reduce the space complexity from O(n 3 k) to O(n 3 log k). -In case both the text and the pattern are separable permutations, we give a more practicable alternative O(n 2 k) time and O(nk) space algorithm. Furtheremore, we show how to use this approach to decide in O(nk 3 2 ) time whether a separable permutation is a disjoint union of two given permutations of size k and -We give a O(n 6 k) time and O(n 4 log k) space algorithm to compute the longest common pattern of two permutations of size at most n (provided that at least one of these permutations is separable). This improves upon the existing O(n 8 ) time algorithm. -Finally, we give a O(n 6 k) time and O(kn 4 ) space algorithm to detect an occurrence of a bivincular separable permutation in a permutation. (Bivincular patterns generalize classical permutations by requiring that positions and values involved in an occurrence may be forced to be adjacent).

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“…On the other hand, π does not contain the pattern 123 since it contains no increasing subsequence of length three. Since 1985, when the first systematic study of Restricted Permutations [34] was published by Simion and Schmidt, the area of permutation patterns has become a rapidly growing field of discrete mathematics, more specifically of combinatorics [5,25,36]. Many applications of permutation patterns have been discovered: their relation to stack and deque sorting, genome sequences in computational biology, statistical mechanics and in general their numerous connections to other combinatorial objects [25].…”

Section: Introductionmentioning

confidence: 99%

A Fast Algorithm for Permutation Pattern Matching Based on Alternating Runs

Bruner

Lackner

2015

Algorithmica

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The NP-complete Permutation Pattern Matching problem asks whether a k-permutation P is contained in a n-permutation T as a pattern. This is the case if there exists an order-preserving embedding of P into T . In this paper, we present a fixed-parameter algorithm solving this problem with a worst-case runtime of O(1.79 run(T ) · n · k), where run(T ) denotes the number of alternating runs of T . This algorithm is particularly well-suited for instances where T has few runs, i.e., few ups and downs. Moreover, since run(T ) < n, this can be seen as a O(1.79 n · n · k) algorithm which is the first to beat the exponential 2 n runtime of brute-force search. Furthermore, we prove that under standard complexity theoretic assumptions such a fixedparameter tractability result is not possible for run(P ).

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Tilings

Cited by 27 publications

References 153 publications

Permutation patterns are hard to count

Permutation patterns are hard to count

Pattern Matching for Separable Permutations

A Fast Algorithm for Permutation Pattern Matching Based on Alternating Runs

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