Pattern Matching for k-Track Permutations

Bulteau, Laurent; Rizzi, Roméo

doi:10.1007/978-3-319-94667-2_9

“…Further related work The complexity of the PPM problem has also been studied under the stronger restriction that the text is pattern-avoiding. The problem is polynomial-time solvable if is monotone [22] or 2-monotone [4,6,20,35,46], but NP-hard if is 3-monotone [38]. A broader characterization is missing.…”

Section: Theoremmentioning

confidence: 99%

Finding and Counting Permutations via CSPs

Berendsohn

¹

,

Kozma

²

,

Marx

³

2021

View full text Add to dashboard Cite

Permutation patterns and pattern avoidance have been intensively studied in combinatorics and computer science, going back at least to the seminal work of Knuth on stack-sorting (1968). Perhaps the most natural algorithmic question in this area is deciding whether a given permutation of length n contains a given pattern of length k. In this work we give two new algorithms for this well-studied problem, one whose running time is $$n^{k/4 + o(k)}$$ n k / 4 + o ( k ) , and a polynomial-space algorithm whose running time is the better of $$O(1.6181^n)$$ O ( 1 . 6181 n ) and $$O(n^{k/2 + 1})$$ O ( n k / 2 + 1 ) . These results improve the earlier best bounds of $$n^{0.47k + o(k)}$$ n 0.47 k + o ( k ) and $$O(1.79^n)$$ O ( 1 . 79 n ) due to Ahal and Rabinovich (2000) resp. Bruner and Lackner (2012) and are the fastest algorithms for the problem when $$k \in \varOmega (\log {n})$$ k ∈ Ω ( log n ) . We show that both our new algorithms and the previous exponential-time algorithms in the literature can be viewed through the unifying lens of constraint-satisfaction. Our algorithms can also count, within the same running time, the number of occurrences of a pattern. We show that this result is close to optimal: solving the counting problem in time $$f(k) \cdot n^{o(k/\log {k})}$$ f ( k ) · n o ( k / log k ) would contradict the exponential-time hypothesis (ETH). For some special classes of patterns we obtain improved running times. We further prove that 3-increasing (4321-avoiding) and 3-decreasing (1234-avoiding) permutations can, in some sense, embed arbitrary permutations of almost linear length, which indicates that a sub-exponential running time is unlikely with the current techniques, even for patterns from these restricted classes.

show abstract

“…Examples include separable permutations [11,27,47,3], i.e. avoiding (2, 4, 1, 3) and (3, 1, 4, 2), k-increasing or k-decreasing patterns [17], k-monotone text [24,15], patterns of length 3 or 4 [3], patterns avoiding (3, 2, 1) [25], text or pattern avoiding (2, 1, 3) and (2, 3, 1) [37], linear permutations [1], permutations with few runs [13].…”

Section: Theorem 2 Algorithm S Solves Permutation Pattern Matching Us...mentioning

confidence: 99%

Faster and simpler algorithms for finding large patterns in permutations

Kozma¹

2019

Preprint

0

View full text Add to dashboard Cite

Permutation patterns and pattern avoidance have been intensively studied in combinatorics and computer science, going back at least to the seminal work of Knuth on stack-sorting (1968). Perhaps the most natural algorithmic question in this area is deciding whether a given permutation of length n contains a given pattern of length k.In this work we give two new algorithms for this well-studied problem, one whose running time is n 0.44k+o(k) , and one whose running time is the better of O(1.6181 n ) and n k/2+o (k) . These results improve the earlier best bounds of Ahal and Rabinovich (2000), and Bruner and Lackner (2012), and are the fastest algorithms for the problem when k = Ω(log n). When k = o(log n), the parameterized algorithm of Guillemot and Marx (2013) dominates.Our second algorithm uses polynomial space and is significantly simpler than all previous approaches with comparable running times, including an n k/2+o(k) algorithm proposed by Guillemot and Marx. Our approach can be summarized as follows: "for every matching of the even-valued entries of the pattern, try to match all odd-valued entries left-to-right". For the special case of patterns that are Jordan-permutations, we show an improved, subexponential running time.

show abstract

“…The complexity of the PPM problem has also been studied under the stronger restriction that the text τ is pattern-avoiding. The problem is polynomialtime solvable if τ is monotone [21] or 2-monotone [19,34,6,4,45], but NP-hard if τ is 3-monotone [37]. A broader characterization is missing.…”

Section: Theorem 5 the Treewidth Of The Incidence Graph Ofmentioning

confidence: 99%

Finding and counting permutations via CSPs

Berendsohn¹,

Kozma²,

Marx³

2019

Preprint

0

View full text Add to dashboard Cite

Permutation patterns and pattern avoidance have been intensively studied in combinatorics and computer science, going back at least to the seminal work of Knuth on stack-sorting (1968). Perhaps the most natural algorithmic question in this area is deciding whether a given permutation of length n contains a given pattern of length k.In this work we give two new algorithms for this well-studied problem, one whose running time is n k/4+o (k) , and a polynomial-space algorithm whose running time is the better of O(1.6181 n ) and O(n k/2+1 ). These results improve the earlier best bounds of n 0.47k+o(k) and O(1.79 n ) due to Ahal and Rabinovich (2000) resp. Bruner and Lackner (2012) and are the fastest algorithms for the problem when k ∈ Ω(log n). We show that both our new algorithms and the previous exponential-time algorithms in the literature can be viewed through the unifying lens of constraint-satisfaction.Our algorithms can also count, within the same running time, the number of occurrences of a pattern. We show that this result is close to optimal: solving the counting problem in time f (k) • n o(k/ log k) would contradict the exponential-time hypothesis (ETH). For some special classes of patterns we obtain improved running times. We further prove that 3-increasing and 3-decreasing permutations can, in some sense, embed arbitrary permutations of almost linear length, which indicates that an algorithm with sub-exponential running time is unlikely, even for patterns from these restricted classes.

show abstract

Pattern Matching for k-Track Permutations

Cited by 4 publications

References 17 publications

Finding and Counting Permutations via CSPs

Finding and Counting Permutations via CSPs

Faster and simpler algorithms for finding large patterns in permutations

Finding and counting permutations via CSPs

Contact Info

Product

Resources

About