A Low Arithmetic-Degree Algorithm for Computing Proximity Graphs

In this paper, we study the problem of designing robust algorithms for computing the minimum spanning tree, the nearest neighbor graph, and the relative neighborhood graph of a set of points in the plane, under the Euclidean metric. We use the term “robust” to denote an algorithm that can properly handle degenerate configurations of the input (such as co-circularities and collinearities) and that is not affected by errors in the flow of control due to round-off approximations. Existing asymptotically optimal algorithms that compute such graphs are either suboptimal in terms of the arithmetic precision required for the implementation, or cannot handle degeneracies, or are based on complex data structures. We present a unified approach to the robust computation of the above graphs. The approach is a variant of the general region approach for the computation of proximity graphs based on Yao graphs, first introduced in Ref. 43 (A. C.-C. Yao, On constructing minimum spanning trees in [Formula: see text]-dimensional spaces and related problems, SIAM J. Comput. 11(4) (1982) 721–736). We show that a sparse supergraph of these geometric graphs can be computed in asymptotically optimal time and space, and requiring only double precision arithmetic, which is proved to be optimal. The arithmetic complexity of the approach is measured by using the notion of degree, introduced in Ref. 31 (G. Liotta, F. P. Preparata and R. Tamassia, Robust proximity queries: An illustration of degree-driven algorithm design, SIAM J. Comput. 28(3) (1998) 864–889) and Ref. 3 (J. D. Boissonnat and F. P. Preparata, Robust plane sweep for intersecting segments, SIAM J. Comput. 29(5) (2000) 1401–1421). As a side effect of our results, we solve a question left open by Katajainen27 (J. Katajainen, The region approach for computing relative neighborhood graphs in the [Formula: see text] metric, Computing 40 (1987) 147–161) about the existence of a subquadratic algorithm, based on the region approach, that computes the relative neighborhood graph of a set of points [Formula: see text] in the plane under the [Formula: see text] metric.

Connectivity of Random Geometric Graphs Related to Minimal Spanning Forests

The almost-sure connectivity of the Euclidean minimal spanning forest MSF(X) on a homogeneous Poisson point process X ⊂ ℝd is an open problem for dimension d>2. We introduce a descending family of graphs (Gn)n≥2 that can be seen as approximations to the MSF in the sense that MSF(X)=∩n=2∞Gn(X). For n=2, one recovers the relative neighborhood graph or, in other words, the β-skeleton with β=2. We show that almost-sure connectivity of Gn(X) holds for all n≥2, all dimensions d≥2, and also point processes X more general than the homogeneous Poisson point process. In particular, we show that almost-sure connectivity holds if certain continuum percolation thresholds are strictly positive or, more generally, if almost surely X does not admit generalized descending chains.

Connectivity of Random Geometric Graphs Related to Minimal Spanning Forests

The almost-sure connectivity of the Euclidean minimal spanning forest MSF(X) on a homogeneous Poisson point processX⊂ ℝdis an open problem for dimensiond>2. We introduce a descending family of graphs (Gn)n≥2 that can be seen as approximations to the MSF in the sense that MSF(X)=∩n=2∞Gn(X). Forn=2, one recovers the relative neighborhood graph or, in other words, the β-skeleton with β=2. We show that almost-sure connectivity ofGn(X) holds for alln≥2, all dimensionsd≥2, and also point processesXmore general than the homogeneous Poisson point process. In particular, we show that almost-sure connectivity holds if certain continuum percolation thresholds are strictly positive or, more generally, if almost surelyXdoes not admit generalized descending chains.