Binary search trees and rectangulations

TL;DR

This paper explores novel geometric and combinatorial connections between binary search trees and rectangulations, offering new insights and potential tools for longstanding open problems like the dynamic optimality conjecture.

Contribution

It establishes a new correspondence between BST operations and rectangulation flips, providing a fresh geometric perspective on BST search processes.

Findings

BST execution traces correspond to rectangulation flip sequences

Reinterpretation of Small Manhattan Network in terms of rectangulations

New simplified geometric view of BST operations resembling shortest path algorithms

Abstract

We revisit the classical problem of searching in a binary search tree (BST) using rotations, and present novel connections of this problem to a number of geometric and combinatorial structures. In particular, we show that the execution trace of a BST that serves a sequence of queries is in close correspondence with the flip-sequence between two rectangulations. (Rectangulations are well-studied combinatorial objects also known as mosaic floorplans.) We also reinterpret Small Manhattan Network, a problem with known connections to the BST problem, in terms of flips in rectangulations. We apply further transformations to the obtained geometric model, to arrive at a particularly simple view of the BST problem that resembles sequences of edge-relaxations in a shortest path algorithm. Our connections yield new results and observations for all structures concerned. In this draft we present some preliminary findings. BSTs with rotations are among the most fundamental and most thoroughly studied objects in computer science, nonetheless they pose long-standing open questions, such as the dynamic optimality conjecture of Sleator and Tarjan (STOC 1983). Our hope is that the correspondences presented in this paper provide a new perspective on this old problem and bring new tools to the study of dynamic optimality.