The rotation distance of brooms

TL;DR

This paper studies the rotation distance between search trees called brooms on a specific class of graphs, providing a quasi-quadratic time algorithm for computing these distances, which relates to a major open problem in the field.

Contribution

It introduces an efficient quasi-quadratic time algorithm for computing rotation distances between brooms on complete split graphs, bridging a gap in understanding for this class.

Findings

Rotation distance between brooms can be computed in quasi-quadratic time.

The associahedron of a complete split graph interpolates between the stellohedron and permutohedron.

The problem relates to the open problem of computing rotation distances in binary search trees.

Abstract

The associahedron $\mathcal{A}(G)$ of a graph $G$ has the property that its vertices can be thought of as the search trees on $G$ and its edges as the rotations between two search trees. If $G$ is a simple path, then $\mathcal{A}(G)$ is the usual associahedron and the search trees on $G$ are binary search trees. Computing distances in the graph of $\mathcal{A}(G)$, or equivalently, the rotation distance between two binary search trees, is a major open problem. Here, we consider the different case when $G$ is a complete split graph. In that case, $\mathcal{A}(G)$ interpolates between the stellohedron and the permutohedron, and all the search trees on $G$ are brooms. We show that the rotation distance between any two such brooms and therefore the distance between any two vertices in the graph of the associahedron of $G$ can be computed in quasi-quadratic time in the number of vertices of $G$.