Transforming spanning trees: A lower bound

Abstract

For a planar point set we consider the graph whose vertices are the crossing-free straightline spanning trees of the point set, and two such spanning trees are adjacent if their union is crossing-free. An upper bound on the diameter of this graph implies an upper bound on the diameter of the flip graph of pseudo-triangulations of the underlying point set.We prove a lower bound of Ω(log n/ log log n) for the diameter of the transformation graph of spanning trees on a set of n points in the plane. This nearly matches the known upper bound of O (log n). If we measure the diameter in terms of the number of convex layers k of the point set, our lower bound construction is tight, i.e., the diameter is in Ω(log k) which matches the known upper bound of O (log k). So far only constant lower bounds were known.

Bichromatic compatible matchings

“…, G k = G where each successive pair of PSLGs Gi−1, Gi jointly satisfy some geometric constraints. In some situations, a bound on the value of k is desired as well [2,3,4,6,8,10,14].…”

LATIN 2018: Theoretical Informatics

“…A simultaneous compatible exchange graph is typically not a complete graph. Buchin et al [12] constructed a set S of n points and a pair of trees T 1 , T 2 ∈ T (S) such that Ω(log n/ log log n) simultaneous compatible exchanges are required to transform T 1 into T 2 . Aichholzer et al [3] proved that, for every set S of n points, every T ∈ T (S) can be transformed into the minimum spanning tree of S using O(log n) simultaneous compatible exchanges; moreover, each operation decreases the Euclidean weight of the tree.…”

Compatible geometric matchings

“…Partially motivated by connections with pseudo-triangulations, Aichholzer et al [2] conjectured that there is a transformation of length o(log n) between any two spanning trees. Recently Buchin et al [9] proved an (log n/ log log n) lower bound for this question.…”