Structural Properties of Search Trees with 2-way Comparisons

TL;DR

This paper investigates the structural properties of 2-way comparison search trees, revealing limitations of existing optimization techniques and providing new bounds that inform the choice of comparison types at the root.

Contribution

It introduces new threshold bounds for key weights in 2WCST's and demonstrates that Monge property-based optimization techniques do not extend to 2WCST's.

Findings

Monge property does not apply to 2WCST's

New threshold bounds for key weights in 2WCST's

Optimal 2WCST algorithms remain slower than 3WCST algorithms

Abstract

Optimal 3-way comparison search trees (3WCST's) can be computed using standard dynamic programming in time O(n^3), and this can be further improved to O(n^2) by taking advantage of the Monge property. In contrast, the fastest algorithm in the literature for computing optimal 2-way comparison search trees (2WCST's) runs in time O(n^4). To shed light on this discrepancy, we study structure properties of 2WCST's. On one hand, we show some new threshold bounds involving key weights that can be helpful in deciding which type of comparison should be at the root of the optimal tree. On the other hand, we also show that the standard techniques for speeding up dynamic programming (the Monge property / quadrangle inequality) do not apply to 2WCST's.