# Approximation Algorithm for Base Station Placement on Vertices of a Convex Region.

## Date of Submission

December 2014

## Date of Award

Winter 12-12-2015

## Institute Name (Publisher)

Indian Statistical Institute

## Document Type

Master's Dissertation

## Degree Name

Master of Technology

## Subject Name

Computer Science

## Department

Advance Computing and Microelectronics Unit (ACMU-Kolkata)

## Supervisor

Das, Sandip (ACMU-Kolkata; ISI)

## Abstract (Summary of the Work)

Facility Location problem has been an interesting research topic in the area of Wireless (Sensor) Networks, Operation Research, Computational Geometry and other related area of Computer Science. A lot of work have been done in this area where the objective is to place k base station of equal(minimum) range to cover entire interior region of a given polygon. Many variations have been studied in the literature by restricting the base station location on the boundary of the polygon, on a given edge etc.We focus on the problem in other way round i.e. given a Convex Region P and a real number R the objective is to find minimum number (say k) of base station position(s), if possible, where we can place our base station(s) such that each and every interior point of P is covered by atleast one of the k base station(s). Here the constraint is that the base stations can only be placed on vertices of the Convex Region. This problem of finding k base station position(s)to cover polygonal region is named as MinRegionCover(P, R). The minimum value of given R for which MinRegionCover can be found out, we name that value as L(P). In this work we find L(P) in O(n) time and decide whether given R is sufficient to find MinRegionCover of given P along with an approach for MinRegionCover(P,L(P)). We also have proved that at most 5 base stations are sufficient to cover a regular region P when R â‰¥ L(P) and that can be found in O(n) time. In general none of the variation of Base Station Placement problems is known to be NP-hard or Polynomial time solvable though polynomial time algorithms for some special cases are known. Our main work is an optimal O(n2 ) algorithm to find MinRegionCover of a Convex Polygon P such that line joining farthest pair is an edge of the P, which leads to an approach to find constant factor approximation algorithm forMinRegionCoverfor general Convex Polygon in polynomial time.

## Control Number

ISI-DISS-2014-273

## Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

## DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/6429

## Recommended Citation

Mishra, Gopinath, "Approximation Algorithm for Base Station Placement on Vertices of a Convex Region." (2015). *Master’s Dissertations*. 341.

https://digitalcommons.isical.ac.in/masters-dissertations/341

## Comments

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:28843400