A survey of simple permutations

Brignall, Robert

doi:10.1017/cbo9780511902499.003

Cited by 33 publications

(44 citation statements)

References 37 publications

(63 reference statements)

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“…(1,6) and the gray circle at (3,5). Therefore the augmentation of the black square adds the gray square at (3,6). The topwing of R(2, 2) consists of the gray circle at (4,4) and hence the gray square at (4,6) is added.…”

Section: Decomposition Theorem and Applicationsmentioning

confidence: 99%

“…For a vast majority of access sequences, the optimal access cost by any dynamic BST (online or offline) is 2 Θ(m log n) and any static balanced tree can achieve this bound. 3 Hence, dynamic optimality is almost trivial on most sequences. However, when an input sequence has complexity o(m log n), a candidate BST must "learn" and "exploit" the specific structure of the sequence, in order to match the optimum.…”

Section: Introductionmentioning

confidence: 99%

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Pattern-Avoiding Access in Binary Search Trees

Chalermsook

Goswami

Kozma

et al. 2015

2015 IEEE 56th Annual Symposium on Foundations of Computer Science

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The dynamic optimality conjecture is perhaps the most fundamental open question about binary search trees (BST). It postulates the existence of an asymptotically optimal online BST, i.e. one that is constant factor competitive with any BST on any input access sequence. The two main candidates for dynamic optimality in the literature are splay trees [Sleator and Tarjan, 1985], and Greedy [Lucas, 1988;Munro, 2000;Demaine et al. 2009]. Despite BSTs being among the simplest data structures in computer science, and despite extensive effort over the past three decades, the conjecture remains elusive. Dynamic optimality is trivial for almost all sequences: the optimum access cost of most length-n sequences is Θ(n log n), achievable by any balanced BST. Thus, the obvious missing step towards the conjecture is an understanding of the "easy" access sequences, and indeed the most fruitful research direction so far has been the study of specific sequences, whose "easiness" is captured by a parameter of interest. For instance, splay provably achieves the bound of O(nd) when d roughly measures the distances between consecutive accesses (dynamic finger), the average entropy (static optimality), or the delays between multiple accesses of an element (working set). The difficulty of proving dynamic optimality is witnessed by other highly restricted special cases that remain unresolved; one prominent example is the traversal conjecture [Sleator and Tarjan, 1985], which states that preorder sequences (whose optimum is linear) are linear-time accessed by splay trees; no online BST is known to satisfy this conjecture.In this paper, we prove two different relaxations of the traversal conjecture for Greedy: (i) Greedy is almost linear for preorder traversal, (ii) if a linear-time preprocessing 1 is allowed, Greedy is in fact linear. These statements are corollaries of our more general results that express the complexity of access sequences in terms of a pattern avoidance parameter k. Pattern avoidance is a well-established concept in combinatorics, and the classes of input sequences thus defined are rich, e.g. the k = 3 case includes preorder sequences. For any sequence X with parameter k, our most general result shows that Greedy achieves the cost n2 α(n) O(k) where α is the inverse Ackermann function. Furthermore, a broad subclass of parameter-k sequences has a natural combinatorial interpretation as k-decomposable sequences. For this class of inputs, we obtain an n2 O(k 2 ) bound for Greedy when preprocessing is allowed. For k = 3, these results imply (i) and (ii). To our knowledge, these are the first upper bounds for Greedy that are not known to hold for any other online BST. To obtain these results we identify an input-revealing property of Greedy. Informally, this means that the execution log partially reveals the structure of the access sequence. This property facilitates the use of rich technical tools from forbidden submatrix theory.Further studying the intrinsic complexity of k-decomposable sequences, we make several observ...

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Section: Decomposition Theorem and Applicationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Pattern-Avoiding Access in Binary Search Trees

Chalermsook

Goswami

Kozma

et al. 2015

2015 IEEE 56th Annual Symposium on Foundations of Computer Science

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“…Every permutation π of [n] has intervals of length 0, 1, and n; π is said to be simple if it has no other intervals. For an extensive study of simple permutations, we refer the reader to Brignall's survey [13]. Going in the other direction, given a permutation σ of length m and nonempty permutations α 1 , .…”

Section: Inflations Simple Permutations Alternations and Substitutmentioning

confidence: 99%

Small permutation classes

Vatter

2011

Proceedings of the London Mathematical Society

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We establish a phase transition for permutation classes (downsets of permutations under the permutation containment order): there is an algebraic number κ, approximately 2.20557, for which there are only countably many permutation classes of growth rate (a.k.a. Stanley-Wilf limit) less than κ but uncountably many permutation classes of growth rate κ, answering a question of Klazar. We go on to completely characterize the possible sub-κ growth rates of permutation classes, answering a question of Kaiser and Klazar. Central to our proofs are the concepts of generalized grid classes (introduced herein), partial well-order, the substitution decomposition, and atomicity (a.k.a. the joint embedding property).

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“…In the word case, an infinite antichain is given by any infinite collection of words with no consecutive occurrences of any letter, for example (ab) n , n ∈ N. Likewise, in the permutation case, any infinite set of permutations without non-trivial intervals (also known as simple permutations) would suffice; there are many such examples in the literature (see [1] and [2]). …”

Section: Theorem 315 the Class Of Equivalence Relations Is Pwo Withmentioning

confidence: 99%

Homomorphic Image Orders on Combinatorial Structures

Huczynska

Ruškuc

2014

Order

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Combinatorial structures have been considered under various orders, including substructure order and homomorphism order. In this paper, we investigate the homomorphic image order, corresponding to the existence of a surjective homomorphism between two structures. We distinguish between strong and induced forms of the order and explore how they behave in the context of different common combinatorial structures. We focus on three aspects: antichains and partial well-order, the joint preimage property and the dual amalgamation property. The two latter properties are natural analogues of the well-known joint embedding property and amalgamation property, and are investigated here for the first time.

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A survey of simple permutations

Cited by 33 publications

References 37 publications

Pattern-Avoiding Access in Binary Search Trees

Pattern-Avoiding Access in Binary Search Trees

Small permutation classes

Homomorphic Image Orders on Combinatorial Structures

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