Should Static Search Trees Ever Be Unbalanced?

TL;DR

This paper investigates whether static search trees should be unbalanced and introduces methods to restructure unbalanced trees into nearly optimal balanced trees with comparable average access times and improved worst-case performance.

Contribution

The paper presents novel methods to convert unbalanced static search trees into balanced trees with minimal increase in access time and near-optimal height.

Findings

Balanced trees have similar average access time as unbalanced trees.

Rebalanced trees have worst-case height within one of the optimal.

Rebalancing improves worst-case access time significantly.

Abstract

In this paper we study the question of whether or not a static search tree should ever be unbalanced. We present several methods to restructure an unbalanced k-ary search tree $T$ into a new tree $R$ that preserves many of the properties of $T$ while having a height of $\log_k n +1$ which is one unit off of the optimal height. More specifically, we show that it is possible to ensure that the depth of the elements in $R$ is no more than their depth in $T$ plus at most $\log_k \log_k n +2$. At the same time it is possible to guarantee that the average access time $P(R)$ in tree $R$ is no more than the average access time $P(T)$ in tree $T$ plus $O(\log_k P(T))$. This suggests that for most applications, a balanced tree is always a better option than an unbalanced one since the balanced tree has similar average access time and much better worst case access time.