PEdom.num.nondeg {pcds}

The domination number of Proportional Edge Proximity Catch Digraph (PE-PCD) with non-degeneracy centers - multiple triangle case

Description

Returns the domination number, indices of a minimum dominating set of PE-PCD whose vertices are the data points in Xp in the multiple triangle case and domination numbers for the Delaunay triangles based on Yp points when PE-PCD is constructed with vertex regions based on non-degeneracy centers.

PE proximity regions are defined with respect to the Delaunay triangles based on Yp points with expansion parameter r \ge 1 and vertex regions in each triangle are based on the center M which is one of the 3 centers that renders the asymptotic distribution of domination number to be non-degenerate for a given value of r in (1,1.5) and M is center of mass for r=1.5.

Convex hull of Yp is partitioned by the Delaunay triangles based on Yp points (i.e., multiple triangles are the set of these Delaunay triangles whose union constitutes the convex hull of Yp points). Loops are allowed for the domination number.

See (Ceyhan (2005); Ceyhan and Priebe (2007); Ceyhan (2011, 2012)) more on the domination number of PE-PCDs. Also, see (Okabe et al. (2000); Ceyhan (2010); Sinclair (2016)) for more on Delaunay triangulation and the corresponding algorithm.

Usage

PEdom.num.nondeg(Xp, Yp, r)

Arguments

Xp

A set of 2D points which constitute the vertices of the PE-PCD.

Yp

A set of 2D points which constitute the vertices of the Delaunay triangles.

r

A positive real number which serves as the expansion parameter in PE proximity region; must be in (1,1.5] here.

Value

A list with three elements

dom.num

Domination number of the PE-PCD whose vertices are Xp points. PE proximity regions are constructed with respect to the Delaunay triangles based on the Yp points with expansion parameter r in (1,1.5].

#

mds

A minimum dominating set of the PE-PCD whose vertices are Xp points.

ind.mds

The data indices of the minimum dominating set of the PE-PCD whose vertices are Xp points.

tri.dom.nums

Domination numbers of the PE-PCD components for the Delaunay triangles.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2005). An Investigation of Proximity Catch Digraphs in Delaunay Tessellations, also available as technical monograph titled Proximity Catch Digraphs: Auxiliary Tools, Properties, and Applications. Ph.D. thesis, The Johns Hopkins University, Baltimore, MD, 21218.

Ceyhan E (2010). “Extension of One-Dimensional Proximity Regions to Higher Dimensions.” Computational Geometry: Theory and Applications, 43(9), 721-748.

Ceyhan E (2011). “Spatial Clustering Tests Based on Domination Number of a New Random Digraph Family.” Communications in Statistics - Theory and Methods, 40(8), 1363-1395.

Ceyhan E (2012). “An investigation of new graph invariants related to the domination number of random proximity catch digraphs.” Methodology and Computing in Applied Probability, 14(2), 299-334.

Ceyhan E, Priebe CE (2007). “On the Distribution of the Domination Number of a New Family of Parametrized Random Digraphs.” Model Assisted Statistics and Applications, 1(4), 231-255.

Okabe A, Boots B, Sugihara K, Chiu SN (2000). Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. Wiley, New York.

Sinclair D (2016). “S-hull: a fast radial sweep-hull routine for Delaunay triangulation.” 1604.01428.

See Also

PEdom.num.tri, PEdom.num.tetra, dom.num.exact, and dom.num.greedy

Examples

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-5;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

r<-1.5  #try also r<-2

set.seed(1)
Xp<-cbind(runif(nx,0,1),runif(nx,0,1))
Yp<-cbind(runif(ny,0,.25),
runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1))
#try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

PEdom.num.nondeg(Xp,Yp,r)