Results 1  10
of
18
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 ..."
Abstract

Cited by 16 (5 self)
 Add to MetaCart
(Show Context)
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. ..."
Abstract

Cited by 11 (1 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
(Show Context)
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 ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
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,
A Selfadjusting Data Structure for Multidimensional Point Sets ⋆
"... Abstract. A data structure is said to be selfadjusting if it dynamically reorganizes itself to adapt to the pattern of accesses. Efficiency is typically measured in terms of amortized complexity, that is, the average running time of an access over an arbitrary sequence of accesses. The best known e ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. A data structure is said to be selfadjusting if it dynamically reorganizes itself to adapt to the pattern of accesses. Efficiency is typically measured in terms of amortized complexity, that is, the average running time of an access over an arbitrary sequence of accesses. The best known example of such a data structure is Sleator and Tarjan’s splay tree. In this paper, we introduce a selfadjusting data structure for storing multidimensional point data. The data structure is based on a quadtreelike subdivision of space. Like a quadtree, the data structure implicitly encodes a subdivision of space into cells of constant combinatorial complexity. Each cell is either a quadtree box or the settheoretic difference of two such boxes. Similar to the traditional splay tree, accesses are based on an splaying operation that restructures the tree in order to bring an arbitrary internal node to the root of the tree. We show that many of the properties enjoyed by traditional splay trees can be generalized to this multidimensional version.
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 ..."
Abstract
 Add to MetaCart
(Show Context)
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
unknown title
"... of sequence of searches in a splay tree is within a constant factor of the cost of the same sequence of searches in any dynamic binary search tree, even if the competing search algorithm knows the entire sequence in advance. Equivalently, in the language of online algorithms, Sleator and Tarjan conj ..."
Abstract
 Add to MetaCart
(Show Context)
of sequence of searches in a splay tree is within a constant factor of the cost of the same sequence of searches in any dynamic binary search tree, even if the competing search algorithm knows the entire sequence in advance. Equivalently, in the language of online algorithms, Sleator and Tarjan conjectured that splay trees are O(1)competitive against an optimal offline adversary. After more than 20 years, this dynamic optimality conjecture remains one of the most frustrating and important open problems in the field of data structures. In fact, it is not known whether any O(1)competitive offline binary search tree exists, nor whether computing the optimal offline strategy for a given access sequence is NPhard. The depth of our ignorance is humbling. One potential way to prove the dynamic optimality conjecture is to develop tighter lower bounds on the cost of the optimal offline search strategy. Three nontrivial lower bounds are currently known: the interleave and alternation bounds proved by Robert Wilber * [5], and the much more recent rectangle cover bound proved by Jonathan Derryberry*, Danny Sleator, and Chengwen Wang * [2]. Conversely, one might approach the problem by developing new dynamic binary search trees that more closely approach the known lower bounds. Essentially only one such structure is known: the tango trees developed by Erik Demaine, Dion Harmon*, John Iacono, and Mihai Pătra¸scu * [1].