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12
Splay trees, DavenportSchinzel sequences, and the deque conjecture
, 2007
"... We introduce a new technique to bound the asymptotic performance of splay trees. The basic idea is to transcribe, in an indirect fashion, the rotations performed by the splay tree as a DavenportSchinzel sequence S, none of whose subsequences are isomorphic to fixed forbidden subsequence. We direct ..."
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Cited by 15 (5 self)
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We introduce a new technique to bound the asymptotic performance of splay trees. The basic idea is to transcribe, in an indirect fashion, the rotations performed by the splay tree as a DavenportSchinzel sequence S, none of whose subsequences are isomorphic to fixed forbidden subsequence. We direct this technique towards Tarjan’s deque conjecture and prove that n deque operations require O(nα ∗ (n)) time, where α ∗ (n) is the minimum number of applications of the inverseAckermann function mapping n to a constant. We are optimistic that this approach could be directed towards other open conjectures on splay trees such as the traversal and split conjectures.
Dynamic Optimality–Almost
 Proc. 45th Annu. IEEE Sympos. Foundations Comput. Sci
"... We present an O(lg lg n)competitive online binary search tree, improving upon the best previous (trivial) competitive ratio of O(lg n). This is the first major progress on Sleator and Tarjan’s dynamic optimality conjecture of 1985 that O(1)competitive binary search trees exist. 1. ..."
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Cited by 11 (1 self)
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We present an O(lg lg n)competitive online binary search tree, improving upon the best previous (trivial) competitive ratio of O(lg n). This is the first major progress on Sleator and Tarjan’s dynamic optimality conjecture of 1985 that O(1)competitive binary search trees exist. 1.
Dynamic Optimality for Skip Lists and BTrees
, 2008
"... Sleator and Tarjan [39] conjectured that splay trees are dynamically optimal binary search trees (BST). In this context, we study the skip list data structure introduced by Pugh [35]. We prove that for a class of skip lists that satisfy a weak balancing property, the workingset bound is a lower bou ..."
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Cited by 5 (1 self)
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Sleator and Tarjan [39] conjectured that splay trees are dynamically optimal binary search trees (BST). In this context, we study the skip list data structure introduced by Pugh [35]. We prove that for a class of skip lists that satisfy a weak balancing property, the workingset bound is a lower bound on the time to access any sequence. Furthermore, we develop a deterministic selfadjusting skip list whose running time matches the workingset bound, thereby achieving dynamic optimality in this class. Finally, we highlight the implications our bounds for skip lists have on multiway branching search trees such as Btrees, (ab)trees, and other variants as well as their binary tree representations. In particular, we show a selfadjusting Btree that is dynamically optimal both in internal and external memory.
SkipSplay: Toward Achieving the Unified Bound in the BST Model
"... Abstract. We present skipsplay, the first binary search tree algorithm known to have a running time that nearly achieves the unified bound. Skipsplay trees require only O(m lg lg n + UB(σ)) time to execute a query sequence σ = σ1...σm. The skipsplay algorithm is simple and similar to the splay al ..."
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Cited by 4 (2 self)
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Abstract. We present skipsplay, the first binary search tree algorithm known to have a running time that nearly achieves the unified bound. Skipsplay trees require only O(m lg lg n + UB(σ)) time to execute a query sequence σ = σ1...σm. The skipsplay algorithm is simple and similar to the splay algorithm. 1 Introduction and Related Work Although the worstcase access cost for comparisonbased dictionaries is Ω(lg n), many sequences of operations are highly nonrandom, allowing tighter, instancespecific running time bounds to be achieved by algorithms that adapt to the input sequence. Splay trees [1] are an example of such an adaptive algorithm
An O(log log n)competitive binary search tree with optimal worstcase access times. Obtained on December 7, 2009 from: http://cgm.cs.mcgill.ca/ vida/pubs/papers/ZipperTrees.pdf
"... We present the zipper tree, the first O(log log n)competitive online binary search tree that performs each access in O(log n) worstcase time. This shows that for binary search trees, optimal worstcase access time and nearoptimal amortized access time can be guaranteed simultaneously. 1 ..."
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Cited by 2 (0 self)
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We present the zipper tree, the first O(log log n)competitive online binary search tree that performs each access in O(log n) worstcase time. This shows that for binary search trees, optimal worstcase access time and nearoptimal amortized access time can be guaranteed simultaneously. 1
Adaptive Binary Search Trees
, 2009
"... A ubiquitous problem in the field of algorithms and data structures is that of searching for an element from an ordered universe. The simple yet powerful binary search tree (BST) model provides a rich family of solutions to this problem. Although BSTs require Ω(lg n) time per operation in the wors ..."
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Cited by 1 (0 self)
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A ubiquitous problem in the field of algorithms and data structures is that of searching for an element from an ordered universe. The simple yet powerful binary search tree (BST) model provides a rich family of solutions to this problem. Although BSTs require Ω(lg n) time per operation in the worst case, various adaptive BST algorithms are capable of exploiting patterns in the sequence of queries to achieve tighter, inputsensitive, bounds that can be o(lg n) in many cases. This thesis furthers our understanding of what is achievable in the BST model along two directions. First, we make progress in improving instancespecific lower bounds in the BST model. In particular, we introduce a framework for generating lower bounds on the cost that any BST algorithm must pay to execute a query sequence,
Finger Search on Balanced Search Trees
, 2006
"... This thesis introduces the concept of a heterogeneous decomposition of a balanced search tree and apply it to the following problems: • How can finger search be implemented without changing the representation of a RedBlack Tree, such as introducing extra storage to the nodes? (Answer: Any degreeba ..."
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This thesis introduces the concept of a heterogeneous decomposition of a balanced search tree and apply it to the following problems: • How can finger search be implemented without changing the representation of a RedBlack Tree, such as introducing extra storage to the nodes? (Answer: Any degreebalanced search tree can support finger search without modification in its representation by maintaining an auxiliary data structure of logarithmic size and suitably modifying the search algorithm to make use of this auxiliary data structure.) • Do MultiSplay Trees, which is known to be O(log log n)competitive to the optimal binary search trees, have the Dynamic Finger property? (Answer: This is work in progress. We believe the answer is yes.)
Université Libre de Bruxelles
"... Abstract. We present a general transformation for combining a constant number of binary search tree data structures (BSTs) into a single BST whose running time is within a constant factor of the minimum of any “wellbehaved ” bound on the running time of the given BSTs, for any online access sequenc ..."
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Abstract. We present a general transformation for combining a constant number of binary search tree data structures (BSTs) into a single BST whose running time is within a constant factor of the minimum of any “wellbehaved ” bound on the running time of the given BSTs, for any online access sequence. (A BST has a wellbehaved bound with f(n) overhead if it spends at most O(f(n)) time per access and its bound satisfies a weak sense of closure under subsequences.) In particular, we obtain a BST data structure that is O(log log n) competitive, satisfies the working set bound (and thus satisfies the static finger bound and the static optimality bound), satisfies the dynamic finger bound, satisfies the unified bound with an additive O(log log n) factor, and performs each access in worstcase O(log n) time. 1