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The geometry of binary search trees
 In Proceedings of the 20th ACMSIAM Symposium on Discrete Algorithms (SODA 2009
, 2009
"... We present a novel connection between binary search trees (BSTs) and points in the plane satisfying a simple property. Using this correspondence, we achieve the following results: 1. A surprisingly clean restatement in geometric terms of many results and conjectures relating to BSTs and dynamic opti ..."
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Cited by 8 (0 self)
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We present a novel connection between binary search trees (BSTs) and points in the plane satisfying a simple property. Using this correspondence, we achieve the following results: 1. A surprisingly clean restatement in geometric terms of many results and conjectures relating to BSTs and dynamic optimality. 2. A new lower bound for searching in the BST model, which subsumes the previous two known bounds of Wilber [FOCS’86]. 3. The first proposal for dynamic optimality not based on splay trees. A natural greedy but offline algorithm was presented by Lucas [1988], and independently by Munro [2000], and was conjectured to be an (additive) approximation of the best binary search tree. We show that there exists an equalcost online algorithm, transforming the conjecture of Lucas and Munro into the conjecture that the greedy algorithm is dynamically optimal. 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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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
Upper Bounds for Maximally Greedy Binary Search Trees
"... Abstract. At SODA 2009, Demaine et al. presented a novel connection between binary search trees (BSTs) and subsets of points on the plane. This connection was independently discovered by Derryberry et al. As part of their results, Demaine et al. considered GreedyFuture, an offline BST algorithm that ..."
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Abstract. At SODA 2009, Demaine et al. presented a novel connection between binary search trees (BSTs) and subsets of points on the plane. This connection was independently discovered by Derryberry et al. As part of their results, Demaine et al. considered GreedyFuture, an offline BST algorithm that greedily rearranges the search path to minimize the cost of future searches. They showed that GreedyFuture is actually an online algorithm in their geometric view, and that there is a way to turn GreedyFuture into an online BST algorithm with only a constant factor increase in total search cost. Demaine et al. conjectured this algorithm was dynamically optimal, but no upper bounds were given in their paper. We prove the first nontrivial upper bounds for the cost of search operations using GreedyFuture including giving an access lemma similar to that found in Sleator and Tarjan’s classic paper on splay trees. 1
Advanced Data Structures JanApr 2012 Lecturer: Venkatesh Raman
, 2012
"... In the last lecture we studied MovetoFront (MTF) Heuristic for a list and its competitive ratio. We also introduced Binary Search Trees (BST) and optimal BSTs. In today’s lecture, we will be analysing Splay Trees and see that they perform as well as an optimal static BST without maintaining extra ..."
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In the last lecture we studied MovetoFront (MTF) Heuristic for a list and its competitive ratio. We also introduced Binary Search Trees (BST) and optimal BSTs. In today’s lecture, we will be analysing Splay Trees and see that they perform as well as an optimal static BST without maintaining extra information for balancing the tree. We also discuss the scenario when we have the freedom to begin the search anywhere instead of always starting from the root node. 2 Statically Optimal Search Given a sequence of length m and having n distinct elements. We look at the elements one by one. We want to implement following operations: • insert (i) – If i has not been seen so far in the sequence, insert it in the tree. • access(i) – If i exists in the tree, return a pointer to it. Operation of insert(i) is O(n 2) atmost. We can ignore insertion and w.l.o.g assume we have a tree with all the keys initially and we are performing a sequence of access(i) operations on it. 2.1 Information Entropy and Search Given a set of keys S = [1... n], and frequency of access pi for i ∈ S. The information entropy H is, H = − ∑ piln(pi)
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"... For the past few lectures we have been studying self organizing binary search trees. In these two lectures we will study a novel connection between binary search trees(BSTs) and points satisfying a simple property from [1]. Using this correspondence, we will restate many results and conjectures rela ..."
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For the past few lectures we have been studying self organizing binary search trees. In these two lectures we will study a novel connection between binary search trees(BSTs) and points satisfying a simple property from [1]. Using this correspondence, we will restate many results and conjectures relating to BSTs and dynamic optimality. 2
OBST: A SelfAdjusting PeertoPeer Overlay Based on Multiple BSTs
"... Abstract—The design of scalable and robust overlay topologies has been a main research subject since the very origins of peertopeer (p2p) computing. Today, the corresponding optimization tradeoffs are fairly wellunderstood, at least in the static case and from a worstcase perspective. This paper ..."
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Abstract—The design of scalable and robust overlay topologies has been a main research subject since the very origins of peertopeer (p2p) computing. Today, the corresponding optimization tradeoffs are fairly wellunderstood, at least in the static case and from a worstcase perspective. This paper revisits the peertopeer topology design problem from a selforganization perspective. We initiate the study of topologies which are optimized to serve the communication demand, or even selfadjusting as demand changes. The appeal of this new paradigm lies in the opportunity to be able to go beyond the lower bounds and limitations imposed by a static, communicationoblivious, topology. For example, the goal of having short routing paths (in terms of hop count) does no longer conflict with the requirement of having low peer degrees. We propose a simple overlay topology OBST(k) which is composed of k (rooted and directed) Binary Search Trees (BSTs), where k is a parameter. We first prove some fundamental bounds on what can and cannot be achieved optimizing a topology towards a static communication pattern (a static OBST(k)). In particular, we show that the number of BSTs that constitute the overlay can have a large impact on the routing costs, and that a single additional BST may reduce the amortized communication costs from Ω(log n) to O(1), where n is the number of peers. Subsequently, we discuss a natural selfadjusting extension of OBST(k), in which frequently communicating partners are “splayed together”. I.