The diameter of caterpillar associahedra

TL;DR

This paper investigates the diameter of caterpillar associahedra, revealing it depends on the number of vertices, leaves, and leaf distribution entropy, and establishes a connection to binary search tree searching complexities.

Contribution

It introduces a precise asymptotic bound for the diameter of caterpillar associahedra and links it to binary search tree search problems.

Findings

Diameter is $ heta(n + m imes (H+1))$.

Lower bound uses Wilber's first lower bound for dynamic BSTs.

Upper bound reduces to static BST search.

Abstract

The caterpillar associahedron $\mathcal{A}(G)$ is a polytope arising from the rotation graph of search trees on a caterpillar tree $G$, generalizing the rotation graph of binary search trees (BSTs) and thus the conventional associahedron. We show that the diameter of $\mathcal{A}(G)$ is $\Theta(n + m \cdot (H+1))$, where $n$ is the number of vertices, $m$ is the number of leaves, and $H$ is the entropy of the leaf distribution of $G$. Our proofs reveal a strong connection between caterpillar associahedra and searching in BSTs. We prove the lower bound using Wilber's first lower bound for dynamic BSTs, and the upper bound by reducing the problem to searching in static BSTs.