Dynamic Schnyder woods

A Schnyder wood of a triangulation is a partition of its interior edges into three oriented rooted trees (i.e., a three-colored and oriented triangulation). A flip in a Schnyder wood is a local operation that transforms one Schnyder wood into another, possibly of another triangulation. Two types of flips in a Schnyder wood have been introduced: colored flips, that change the underlying triangulation, and cycle flips, that transform a Schnyder wood into another Schnyder wood of the same triangulation. A flip graph is defined for each of these two types of flips. In this paper, we study the relationship 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 give an explicit upper bound of O(n²) on the diameter of the colored flip graph. Moreover, a data structure is given to dynamically maintain a Schnyder wood over a sequence of colored flips which supports queries in O(log n) time per flip or query.