On the Construction of a Generalized Voronoi Inverse of a Rectangular Tessellation

Abstract

We introduce a new concept of constructing a generalized Voronoi inverse (GVI) of a given tessellation T of the plane. Our objective is to place a set Si of one or more sites in each convex region (cell) ti ∈ T , such that all the edges of T coincide with the edges of Voronoi diagram V (S), where S = i Si, and ∀i, j, i = j, Si Sj = ∅. In this paper, we study the properties of GVI for the special case when T is a rectangular tessellation and propose an algorithm that finds a minimal set of sites S. We also show that for a general tessellation, a solution of GVI always exists.

Fitting Voronoi Diagrams to Planar Tesselations

“…Our results show that the Generalized Inverse Voronoi Problem can be solved with a number of generators that is linear in the size of the input tesselation, provided that we enforce a lower bound on the size of the smallest angle. On the other hand, the algorithm described in [3] produces O(V 3 ) generators, where V is the number of vertices of an acute triangulation of G. As the performance of the two algorithms is given as a function of different parameters, a theoretical comparison between them is not straightforward. An experimental study could be helpful, but that would require an implementation of the algorithm in [3].…”

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Given a tesselation of the plane, defined by a planar straightline graph G, we want to find a minimal set S of points in the plane, such that the Voronoi diagram associated with S 'fits' G. This is the Generalized Inverse Voronoi Problem (GIVP), defined in [12] and rediscovered recently in [3]. Here we give an algorithm that solves this problem with a number of points that is linear in the size of G, assuming that the smallest angle in G is constant.

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“…However, the manuscript remained dormant for several years, and the algorithm was only published in Spanish in 2007 [12]. Recently, the problem was revisited in [3], where another algorithm for the GIVP in R 2 is given, and the special case of a rectangular tesselation is discussed in greater detail. The authors of [3] were unaware of [12], however the two algorithms turn out to have certain common aspects.…”

“…In comparison, the analysis given for the algorithm in [3] states that O(V 3 ) sites are generated, where V is the size of a refinement of G such that all faces are triangles with acute angles. Given an arbitrary PSLG, there does not appear to be any known polynomial upper bound on the size of its associated acute triangulation.…”

“…Given an arbitrary PSLG, there does not appear to be any known polynomial upper bound on the size of its associated acute triangulation. Even though it seems to us that the analysis in [3] should have given a tighter upper bound in terms of V , even a linear bound would not make much of a difference, given that V can be very large compared to the size of G. The analysis in [3] is purely theoretical, so it would be interesting to perform an experimental study to shed some light on the algorithm's performance in practice.…”