# Facility Location on Spherical Surfaces.

December 2015

## Date of Award

Winter 12-12-2016

## Institute Name (Publisher)

Indian Statistical Institute

## Document Type

Master's Dissertation

## Degree Name

Master of Technology

Computer Science

## Department

Advance Computing and Microelectronics Unit (ACMU-Kolkata)

## Supervisor

Das, Sandip (ACMU-Kolkata; ISI)

## Abstract (Summary of the Work)

The facility location problem in Computational Geometry deals with the placement of facilities with respect to some optimizing criteria. One of the most common facility location problems is the k-center problem or the minimax problem. The problem is defined as, given n sites, we need to find k facilities such that the maximum distance of any site to its nearest facility is the minimum over all placements of the k facilities. More formally, let C be the set of sites, then the problem is to find a set of facilities F such that the value p = maxfâˆˆF (mincâˆˆC(d(f, c))) is minimized; d(f, c), f âˆˆ F and c âˆˆ C is the distance metric. This problem is NP-hard if the value of k is a part of the input.The 1-center problem, also know as the minimum enclosing circle problem involves with placing only 1 facility. In Eucledian plane this problem can be solved in O(n) time, using the famous Megiddo prune and search technique. Here we consider a different version of the 1-center problem. The sites are located on the surface of a sphere, and the problem is to find a facility (on the surface of the sphere) with the same criteria and the distance metric set as d(pÌ‚q) = arccos (pÌ‚q) where and q are points on the surface of the sphere. The solution uses the Voronoi Diagram construction of points on spherical surfaces, which is an extension of the fortuneâ€™s algorithm.[Sweeping the Sphere, Denis and Memede] The time complexity of our algorithm is O(n log n) and the space complexity is O(n).We also look into the problem of 2-disk cover, which is, given a set of sites on the spherical surface, to find two circles whose union can cover all the sites whose radius are not more than a given value r. This may then be extended to solve the 2-center problem for spherical surfaces.

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28843102

## Control Number

ISI-DISS-2015-321