@article{e3a6985c8bc54cd781023780286efe2c,

title = "On the Complexity of Closest Pair via Polar-Pair of Point-Sets",

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,K- n in various L-p-metrics and consequently connect the complexity of the monochromatic closest pair and bichromatic closest pair problems.",

author = "Roee David and Karthik, {C. S.} and Bundit Laekhanukit",

note = "We are grateful to the anonymous reviewers for their detailed comments and for identifying a gap in the proof of Theorem 2 and helping us fix it (and even strengthen it). We would like to thank Aviad Rubinstein for sharing with us [26] and also for pointing out a mistake in an earlier version of the paper. We would like to thank Petteri Kaski and Rasmus Pagh for useful discussions and also for pointing out the reference [3]. We would like to thank Eylon Yogev and Amey Bhangale for some preliminary discussions. We would like to thank Uriel Feige for a lot of useful comments and discussions. Finally, we would like to thank Inbal Livni Navon, Orr Paradise, and Roei Tell for helping us improve the presentation of the paper.",

year = "2019",

doi = "https://doi.org/10.1137/17M1128733",

language = "الإنجليزيّة",

volume = "33",

pages = "509--527",

journal = "SIAM Journal on Discrete Mathematics",

issn = "0895-4801",

publisher = "Society for Industrial and Applied Mathematics Publications",

number = "1",

}