Chain-splay trees, or, how to achieve and prove -competitiveness by splaying

Γεωργακόπουλος, Γεώργιος

doi:10.1016/j.ipl.2007.10.001

Cited by 17 publications

(12 citation statements)

References 23 publications

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“…For an overview of results related to dynamic optimality, we refer to the survey of Iacono [19]. Besides Tango trees, other O(log log n)-competitive BST algorithms have been proposed, similarly using Wilber's first bound (multi-splay [38] and chain-splay [17]). Using [13], all the known bounds can be simultaneously achieved by combining several BST algorithms.…”

Section: Overview Of Techniquesmentioning

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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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: Overview Of Techniquesmentioning

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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“…Conversely, prior to this work, even if a dynamically optimal BST algorithm had been found, it would not have been clear whether it satisfied the unified bound to within any factor that was o(lg n) since dynamic optimality by itself says nothing about actual formulaic bounds, and prior to this work no competitive factor better than O(lg n) was known for the cost of the optimal BST algorithm in comparison to the unified bound. See [7], [8], and [9] for progress on dynamic optimality in the BST model.…”

Section: Introduction and Related Workmentioning

confidence: 99%

Skip-Splay: Toward Achieving the Unified Bound in the BST Model

Derryberry

Sleator

2009

Lecture Notes in Computer Science

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“…This in turn implies that, for every input sequence X, the value of the WB-1 bound is within an O(log log n) factor from OPT(X). Follow-up work [WDS06,Geo08] improved several aspects of Tango Trees, but it did not improve the approximation factor. Additional lower bounds on OPT, that subsume both the WB-1 and the WB-2 bounds, were suggested in [DHI + 09, DSW05], but unfortunately it is not clear how to exploit them in algorithm design.…”

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confidence: 99%

Pinning Down the Strong Wilber 1 Bound for Binary Search Trees

Chalermsook,

Chuzhoy,

Saranurak

2019

Preprint

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The dynamic optimality conjecture, postulating the existence of an O(1)-competitive online algorithm for binary search trees (BSTs), is among the most fundamental open problems in dynamic data structures. Despite extensive work and some notable progress, including, for example, the Tango Trees (Demaine et al., FOCS 2004), that give the best currently known O(log log n)-competitive algorithm, the conjecture remains widely open. One of the main hurdles towards settling the conjecture is that we currently do not have approximation algorithms achieving better than an O(log log n)-approximation, even in the offline setting. All known nontrivial algorithms for BST's so far rely on comparing the algorithm's cost with the so-called Wilber's first bound (WB-1). Therefore, establishing the worst-case relationship between this bound and the optimal solution cost appears crucial for further progress, and it is an interesting open question in its own right.Our contribution is two-fold. First, we show that the gap between the WB-1 bound and the optimal solution value can be as large as Ω(log log n/ log log log n); in fact, the gap holds even for several stronger variants of the bound. Second, we provide a simple algorithm, that, given an integer D > 0, obtains an O(D)-approximation in time exp O n 1/2 Ω(D) log n . In particular, this gives a constant-factor approximation sub-exponential time algorithm. Moreover, we obtain a simpler and cleaner efficient O(log log n)-approximation algorithm that can be used in an online setting. Finally, we suggest a new bound, that we call Guillotine Bound, that is stronger than WB, while maintaining its algorithm-friendly nature, that we hope will lead to better algorithms. All our results use the geometric interpretation of the problem, leading to cleaner and simpler analysis.

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Chain-splay trees, or, how to achieve and prove -competitiveness by splaying

Cited by 17 publications

References 23 publications

Pattern-Avoiding Access in Binary Search Trees

Pattern-Avoiding Access in Binary Search Trees

Skip-Splay: Toward Achieving the Unified Bound in the BST Model

Pinning Down the Strong Wilber 1 Bound for Binary Search Trees

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