A note on relative neighborhood graphs

Abstract

f'u~o ne'tcl algorithms jinding relative neighborhood graph RNGc(V) /or a set V oj n points are presented. The first algorithm solves this problem /or input points in (R',L,) metric space in time O(na(n,n)) ij the Delaunay triangulation DT(V) is given. This time perjormance is achieved due to attractive and natural application oj FIND-UNION data structure to represent so-called elimination forest oj edges in DT(V). The second algorithm solves the relative neighborhood graph problem in (Rd,Lp), l<p< 00, metric space in time O(n2) when no three points in V form an isosceles triangle. The complexity analysis oj this algorithm is based on some general /acts pertaining to properties oJ equilateral triangles ira the metric space (Rd,L,).Relative neighborhood graph for a set of points V in a metric space M with a metric dist is a graph G(V,E) such that there is an edge between points p and q if and only if dist(p,q)<max(dist(p,r),dist(q,z):zfV-{p,q}) Relative neighborhood graph for V will be denoted by RNG(V) and the problem of finding relative neighborhood graph will be referred as RNG problem.In this paper two following new results will be presented. Theorem 1: The RNG(V) for a set V of n points in R2 with the norm tP can be found in O(ncY(n,n)) time and O(n) space provided that the Delaunay triangulation DT(V) is given.0 The best known algorithm for R* with the norm L, was presented by Supowit, see [S]. His algorithm takes time O(nlogn) to construct RNG(V) starting from Delaunay Triangulation of V. 233

Essential Constraints of Edge-Constrained Proximity Graphs

“…The elimination path for a vertex p (starting from an adjacent triangle (p, u, v) ∈ CDT (I)) is an ordered list of edges, such that p ∈ U e (β) for each edge e of this list. In the work [5] an edge e belongs to the elimination path induced by some point p only if e is eliminated by p. In our problem this is not the case. The point p eliminates e if and only if p ∈ U e (β) and p is visible to both endpoints of e. We show how to adapt the original elimination forest to our problem later in this section.…”

“…To solve our main problem we will use two geometric structures: elimination path and elimination forest, introduced by Jaromczyk and Kowaluk in [5]. The elimination path for a vertex p (starting from an adjacent triangle (p, u, v) ∈ CDT (I)) is an ordered list of edges, such that p ∈ U e (β) for each edge e of this list.…”

“…Notice, that the construction of the elimination path introduced by Jaromczyk and Kowaluk in [5] does not contain step 4. if e is not locally Delaunay then terminate while loop Take triangle t ∈ CDT (I) adjacent to e, such that t and p are separated by the line passing through e; 5 if p ∈ U e (β) such that e ∈ t then set e := e ; Jaromczyk and Kowaluk show that a vertex cannot belong to the β-neighbourhood U (β) of more than two edges for a particular triangle. Thus elimination paths do not split.…”

“…Jaromczyk and Kowaluk showed that RN G of a set V of points can be constructed from the Delaunay triangulation of V in time O(nα(n, n)), where α(·) is the inverse of the Ackerman function [5]. These two authors, together with Yao, improved the running time of their algorithm to linear [6].…”

“…For now, note that the 1-skeleton corresponds to the Gabriel graph and the 2-skeleton corresponds to the relative neighbourhood graph. In this paper, we use two geometric structures: elimination path and elimination forest, introduced by Jaromczyk and Kowaluk [5].…”