Sequences of spanning trees for L-infinity Delaunay triangulations

Abstract

We extend a known result about L2-Delaunay triangulations to L∞-Delaunay. Let TS be the set of all non-crossing spanning trees of a planar n-point set S. We prove that for each element T of TS, there exists a length-decreasing sequence of trees T0, . . . , Tk in the L∞-metric such that T0 = T, Tk = MST(S) and Ti does not cross Ti−1 for all i = 1, . . . , k, where MST(S) denotes the minimum spanning tree of S in the L∞ metric. We also give an Ω(log n) lower bound for the length of the sequence