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 equal-cost online algorithm, transforming the conjecture of Lucas and Munro into the conjecture that the greedy algorithm is dynamically optimal.

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The geometry of binary search trees

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.

Dynamic optimality - almost [competitive online binary search tree]

This paper initiates the study of locally self-adjusting networks: networks whose topology adapts dynamically and in a decentralized manner, to the communication pattern $\sigma$ . Our vision can be seen as a distributed generalization of the self-adjusting datastructures introduced by Sleator and Tarjan, 1985: In contrast to their splay trees which dynamically optimize the lookup costs from a single node (namely the tree root), we seek to minimize the routing cost between arbitrary communication pairs in the network. As a first step, we study distributed binary search trees (BSTs), which are attractive for their support of greedy routing. We introduce a simple model which captures the fundamental tradeoff between the benefits and costs of self-adjusting networks. We present the SplayNet algorithm and formally analyze its performance, and prove its optimality in specific case studies. We also introduce lower bound techniques based on interval cuts and edge expansion, to study the limitations of any demand-optimized network. Finally, we extend our study to multi-tree networks, and highlight an intriguing difference between classic and distributed splay trees.

http://net.t-labs.tu-berlin.de/~stefan/ton15splay.pdf

SplayNet: towards locally self-adjusting networks

The Dynamic Optimality Conjecture [ST85] states that splay trees are competitive (within a constant competitive factor) among the class of all binary search tree (BST) algorithms. Despite 20 years of research this conjecture is still unresolved. Recently, Demaine et al. [DHIP04] suggested searching for alternative algorithms which have small but non-constant competitive factors. They proposed Tango, a BST algorithm which is nearly dynamically optimal - its competitive ratio is O(log log n) instead of a constant. Unfortunately, for many access patterns, such as random and sequential, Tango is worse than other BST algorithms by a factor of log log n.In this paper, we introduce the multi-splay tree (MST) data structure, which is the first O(log log n)-competitive BST to simultaneously achieve O(log n) amortized cost and O(log2n) worst-case cost per query. We also prove the sequential access lemma for MSTs, which states that sequentially accessing all keys takes linear time. Thus, MSTs are O(log log n)-competitive like Tango but, unlike Tango, require only O(log n) amortized time per access in an arbitrary sequence and only O(1) amortized time per access during a sequential access sequence.Furthermore, we generalize the standard framework for competitive analysis of BST algorithms to include updates (insertions and deletions) in addition to queries. In doing so, we extend the lower bound of Wilber [Wil89] and Demaine et al. [DHIP04] to handle these update operations. We show how MSTs can be modified to support these update operations and be O(log log n)-competitive in the new framework while maintaining the rest of the properties above.

http://www.cs.cmu.edu/~chengwen/paper/MST.pdf

O(log log n)-competitive dynamic binary search trees

The physical topology is emerging as the next frontier in an ongoing effort to render communication networks more flexible While first empirical results indicate that these flexibilities can be exploited to reconfigure and optimize the network toward the workload it serves and, eg, providing the same bandwidth at lower infrastructure cost, only little is known today about the fundamental algorithmic problems underlying the design of reconfigurable networks This paper initiates the study of the theory of demand-aware, self-adjusting networks Our main position is that self-adjusting networks should be seen through the lense of self-adjusting datastructures Accordingly, we present a taxonomy classifying the different algorithmic models of demand-oblivious, fixed demand-aware, and reconfigurable demand-aware networks, introduce a formal model, and identify objectives and evaluation metrics We also demonstrate, by examples, the inherent advantage of demand-aware networks over state-of-the-art demand-oblivious, fixed networks (such as expanders) We conclude by observing that the usefulness of self-adjusting networks depends on the spatial and temporal locality of the demand; as relevant data is scarce, we call for community action

Toward demand-aware networking: a theory for self-adjusting networks

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.

/pdf/dynamic-optimality-almost-30s88gr3nh.pdf

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Papers

Dynamic Optimality—Almost