Weighted dynamic finger in binary search trees

TL;DR

This paper proves a strong, simplified, and shorter bound for binary search trees that matches the dynamic finger bound, improving understanding of their efficiency in search sequences.

Contribution

It introduces a new, stronger, and simpler proof of the dynamic finger bound for binary search trees, using weighted bounds and the GreedyASS structure.

Findings

GreedyASS performs asymptotically as well as static trees on long sequences.

The bound generalizes to a weighted, finger-type bound.

The new proof is shorter, stronger, and has reasonable constants.

Abstract

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.