On Dynamic Optimality for Binary Search Trees

Does there exist O(1)-competitive (self-adjusting) binary search tree (BST) algorithms? This is a well-studied problem. A simple offline BST algorithm GreedyFuture was proposed independently by Lucas and Munro, and they conjectured it to be O(1)-competitive. Recently, Demaine et al. gave a geometric view of the BST problem. This view allowed them to give an online algorithm GreedyArb with the same cost as GreedyFuture. However, no o(n)-competitive ratio was known for GreedyArb. In this paper we make progress towards proving O(1)-competitive ratio for GreedyArb by showing that it is O(\log n)-competitive

The Fresh-Finger Property

The unified property roughly states that searching for an element is fast when the current access is close to a recent access. Here, "close" refers to rank distance measured among all elements stored by the dictionary. We show that distance need not be measured this way: in fact, it is only necessary to consider a small working-set of elements to measure this rank distance. This results in a data structure with access time that is an improvement upon those offered by the unified property for many query sequences

In pursuit of the dynamic optimality conjecture

In 1985, Sleator and Tarjan introduced the splay tree, a self-adjusting binary search tree algorithm. Splay trees were conjectured to perform within a constant factor as any offline rotation-based search tree algorithm on every sufficiently long sequence---any binary search tree algorithm that has this property is said to be dynamically optimal. However, currently neither splay trees nor any other tree algorithm is known to be dynamically optimal. Here we survey the progress that has been made in the almost thirty years since the conjecture was first formulated, and present a binary search tree algorithm that is dynamically optimal if any binary search tree algorithm is dynamically optimal.Comment: Preliminary version of paper to appear in the Conference on Space Efficient Data Structures, Streams and Algorithms to be held in August 2013 in honor of Ian Munro's 66th birthda

Weighted dynamic finger in binary search trees

It is shown that the online binary search tree data structure GreedyASS performs asymptotically as well on a sufficiently long sequence of searches as any static binary search tree where each search begins from the previous search (rather than the root). This bound is known to be equivalent to assigning each item $i$ in the search tree a positive weight $w_i$ and bounding the search cost of an item in the search sequence $s_1,\ldots,s_m$ by $$O\left(1+ \log \frac{\displaystyle \sum_{\min(s_{i-1},s_i) \leq x \leq \max(s_{i-1},s_i)}w_x}{\displaystyle \min(w_{s_i},w_{s_{i-1}})} \right)$$ amortized. This result is the strongest finger-type bound to be proven for binary search trees. By setting the weights to be equal, one observes that our bound implies the dynamic finger bound. Compared to the previous proof of the dynamic finger bound for Splay trees, our result is significantly shorter, stronger, simpler, and has reasonable constants.Comment: An earlier version of this work appeared in the Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithm

Locally Self-Adjusting Skip Graphs

We present a distributed self-adjusting algorithm for skip graphs that minimizes the average routing costs between arbitrary communication pairs by performing topological adaptation to the communication pattern. Our algorithm is fully decentralized, conforms to the $\mathcal{CONGEST}$ model (i.e. uses $O(\log n)$ bit messages), and requires $O(\log n)$ bits of memory for each node, where $n$ is the total number of nodes. Upon each communication request, our algorithm first establishes communication by using the standard skip graph routing, and then locally and partially reconstructs the skip graph topology to perform topological adaptation. We propose a computational model for such algorithms, as well as a yardstick (working set property) to evaluate them. Our working set property can also be used to evaluate self-adjusting algorithms for other graph classes where multiple tree-like subgraphs overlap (e.g. hypercube networks). We derive a lower bound of the amortized routing cost for any algorithm that follows our model and serves an unknown sequence of communication requests. We show that the routing cost of our algorithm is at most a constant factor more than the amortized routing cost of any algorithm conforming to our computational model. We also show that the expected transformation cost for our algorithm is at most a logarithmic factor more than the amortized routing cost of any algorithm conforming to our computational model