Dynamic Schnyder Woods

TL;DR

This paper explores the relationship between two flip operations in Schnyder woods, establishing bounds on flip graph diameters and providing a dynamic data structure for realizer maintenance.

Contribution

It demonstrates how cycle flips relate to colored flips, bounds the flip graph diameter, and introduces a data structure for dynamic realizer updates and queries.

Findings

Cycle flips can be simulated with linearly many colored flips.

The flip graph of realizers has an O(n^2) diameter bound.

A data structure supports dynamic realizer updates and efficient queries.

Abstract

A realizer, commonly known as Schnyder woods, of a triangulation is a partition of its interior edges into three oriented rooted trees. A flip in a realizer is a local operation that transforms one realizer into another. Two types of flips in a realizer have been introduced: colored flips and cycle flips. A corresponding flip graph is defined for each of these two types of flips. The vertex sets are the realizers, and two realizers are adjacent if they can be transformed into each other by one flip. In this paper we study the relation between these two types of flips and their corresponding flip graphs. We show that a cycle flip can be obtained from linearly many colored flips. We also prove an upper bound of $O(n^2)$ on the diameter of the flip graph of realizers defined by colored flips. In addition, a data structure is given to dynamically maintain a realizer over a sequence of colored flips which supports queries, including getting a node's barycentric coordinates, in $O(\log n)$ time per flip or query.