The Power and Limitations of Static Binary Search Trees with Lazy Finger

TL;DR

This paper investigates static binary search trees with lazy finger search, revealing their limitations compared to dynamic trees and providing a new runtime expression and a method to compute optimal lazy finger trees.

Contribution

It introduces a non-entropy-based runtime expression for optimal lazy finger trees and presents a dynamic programming approach to compute them.

Findings

Lazy finger trees outperform classic root-finger trees but are less powerful than dynamic trees.

The runtime of optimal lazy finger trees is not entropy-based, unlike root-finger trees.

A dynamic programming method is proposed to compute the optimal lazy finger trees.

Abstract

A static binary search tree where every search starts from where the previous one ends (lazy finger) is considered. Such a search method is more powerful than that of the classic optimal static trees, where every search starts from the root (root finger), and less powerful than when rotations are allowed---where finding the best rotation based tree is the topic of the dynamic optimality conjecture of Sleator and Tarjan. The runtime of the classic root-finger tree can be expressed in terms of the entropy of the distribution of the searches, but we show that this is not the case for the optimal lazy finger tree. A non-entropy based asymptotically-tight expression for the runtime of the optimal lazy finger trees is derived, and a dynamic programming-based method is presented to compute the optimal tree.