Arithmetic Binary Search Trees: Static Optimality in the Matching Model

TL;DR

This paper introduces Arithmetic BST (A-BST), a new dynamic binary search tree algorithm optimized for the matching model with reconfiguration costs, achieving static optimality for long sequences.

Contribution

We propose A-BST, a simple dynamic BST algorithm based on Shannon-Fano-Elias coding, tailored for the matching model with reconfiguration costs, and prove its static optimality.

Findings

A-BST is statically optimal for sequences of length Ω(n α log α).

A-BST effectively balances link traversal and reconfiguration costs.

The algorithm is simple and based on information coding principles.

Abstract

Motivated by recent developments in optical switching and reconfigurable network design, we study dynamic binary search trees (BSTs) in the matching model. In the classical dynamic BST model, the cost of both link traversal and basic reconfiguration (rotation) is $O(1)$. However, in the matching model, the BST is defined by two optical switches (that represent two matchings in an abstract way), and each switch (or matching) reconfiguration cost is $\alpha$ while a link traversal cost is still $O(1)$. In this work, we propose Arithmetic BST (A-BST), a simple dynamic BST algorithm that is based on dynamic Shannon-Fano-Elias coding, and show that A-BST is statically optimal for sequences of length $\Omega(n \alpha \log \alpha)$ where $n$ is the number of nodes (keys) in the tree.