Abstract
Every graph $G$ can be represented by a collection of equi-radii spheres in a $d$-dimensional metric $\Delta$ such that there is an edge $uv$ in $G$ if and only if the spheres corresponding to $u$ and $v$ intersect. The smallest integer $d$ such that $G$ can be represented by a collection of spheres (all of the same radius) in $\Delta$ is called the sphericity of $G$, and if the collection of spheres are nonoverlapping, then the value $d$ is called the contact-dimension of $G$. In this paper, we study the sphericity and contact-dimension of the complete bipartite graph $K_{n,n}$ in various $L^p$-metrics and consequently connect the complexity of the monochromatic closest pair and bichromatic closest pair problems.
Highlights
This paper studies the geometric representation of a complete bipartite graph in Lp-metrics and connects the complexity of the closest pair and bichromatic closest pair problems beyond certain dimensions
Provided that the polar-pair of point-sets (A, B) in a d-dimensional metric can be constructed within a running time at least as fast as the time for computing Closest Pair in the same metric, this gives a reduction from bichromatic closest pair problem (BCP) to Closest Pair, implying that they have the same running time lower bound
We have studied the sphericity and contact dimension of the complete bipartite graph in various metrics
Summary
This paper studies the geometric representation of a complete bipartite graph in Lp-metrics and connects the complexity of the closest pair and bichromatic closest pair problems beyond certain dimensions. Given a point-set P in a d-dimensional Lp-metric, an α-distance graph is a graph G = (V, E) with a vertex set V = P and an edge set. The sphericity of a graph G in an Lp-metric, denoted by sphp(G), is the smallest dimension d such that G is isomorphic to some α-distance graph in a d-dimensional Lp-metric, for some constant α > 0. An α-contact graph G = (V, E) of a point-set P is an α-distance graph of P such that every edge uv of G has the same distance (i.e., u − v p = α). We call a pair of point-sets (A, B) polar if it is the partition of the vertex set of a contact graph isomorphic to Kn,n. For other Lp-metrics, contact dimension and sphericity are not well-studied
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
