Theoretical insights and an experimental comparison of tango trees and multi-splay trees

TL;DR

This paper compares the theoretical and experimental performance of tango trees, multi-splay trees, and splay trees, revealing limitations of tango trees and insights into multi-splay behavior.

Contribution

It provides the first experimental implementation of tango trees, compares their performance with multi-splay and splay trees, and analyzes their practical efficiency and theoretical bounds.

Findings

Tango trees are less optimal on certain access sequences.

Multi-splay trees perform well on tested sequences, with insights into their working set behavior.

Tango trees have a running time of Θ(n log log n) on sequential access.

Abstract

The tango tree is the first proven $O(\lg \lg n)$-competitive binary search tree (BST). We present the first ever experimental implementation of tango trees and compare the running time of the tango tree with the multi-splay tree and the splay tree on a variety of families of access sequences. We construct access sequences that are intended to test specific properties of BSTs. The results of the other experiments demonstrate the optimality of the splay tree and multi-splay tree on these accesses, while simultaneously demonstrating the tango trees inability to achieve optimality. We prove that the running time of tango trees on the sequential access is $\Theta(n \lg \lg n)$, which provides insight into why the $\Theta(\lg \lg n)$ slow down exists on many access sequences. Motivated by experimental results, we conduct a deeper analysis of the working set access on multi-splay trees, leading to new insights about multi-splay tree behavior. Finally, all of the experiments also reveal insights about large constants and lower order terms in the multi-splay tree, which make it less practical than the splay tree, even though its proven competitive bound is tighter.