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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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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.
Locally SelfAdjusting Tree Networks
"... Abstract—This paper initiates the study of selfadjusting networks (or distributed data structures) whose topologies dynamically adapt to a communication pattern σ. We present a fully decentralized selfadjusting solution called SplayNet. A SplayNet is a distributed generalization of the classic spl ..."
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Abstract—This paper initiates the study of selfadjusting networks (or distributed data structures) whose topologies dynamically adapt to a communication pattern σ. We present a fully decentralized selfadjusting solution called SplayNet. A SplayNet is a distributed generalization of the classic splay tree concept. It ensures short paths (which can be found using localgreedy routing) between communication partners while minimizing topological rearrangements. We derive an upper bound for the amortized communication cost of a SplayNet based on empirical entropies of σ, and show that SplayNets have several interesting convergence properties. For instance, SplayNets features a provable online optimality under special requests scenarios. We also investigate the optimal static network and prove different lower bounds for the average communication cost based on graph cuts and on the empirical entropy of the communication pattern σ. From these lower bounds it follows, e.g., that SplayNets are optimal in scenarios where the requests follow a product distribution as well. Finally, this paper shows that in contrast to the Minimum Linear Arrangement problem which is generally NPhard, the optimal static tree network can be computed in polynomial time for any guest graph, despite the exponentially large graph family. We complement our formal analysis with a small simulation study on a Facebook graph. I.
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.)
Lecturer: Venkatesh Raman
, 2012
"... In the last lecture we saw Splay trees and the different bounds proved on them like the working set bound, the static finger bound, the sequential access bound and the dynamic finger bound. These bounds show that Splay trees execute certain classes of access sequences in o(mlgn) time, but they all p ..."
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In the last lecture we saw Splay trees and the different bounds proved on them like the working set bound, the static finger bound, the sequential access bound and the dynamic finger bound. These bounds show that Splay trees execute certain classes of access sequences in o(mlgn) time, but they all provide O(mlgn) upper bounds on access sequences that sometimes take Θ(m) time to execute on Splay trees. Splay trees are conjectured to be O(1)competitive for all access sequences. In this lecture we discuss Tango tree, an online BST data structure that is O(lglgn)competitive against the optimal offline BST data structure on every access sequence. This reduces the competitive gap from the previously known O(lgn) to O(lglgn). Tango Tree originates in a paper by Demaine, Harmon, Lacono and Patrascu [1]. We also discuss a variation of the lower bound of Wilber [2], called the interleave bound (IB) on which today’s results are based. THEOREM (Wilber ’89): COSTOPT (σ) = Ω(IB(σ)) Today we will prove the following theorems:THEOREM 1. IB(σ)/2 − n is a lower bound on COSTOPT (σ), the cost of the optimal offline BST that serves access sequence σ. Using theorem 1 we shall prove the following main result on Tango trees: