# Some Problems on Guarding Monotone Polygons.

## 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)

The art gallery problem is a well-studied visibility problem in computational geometry. It originates from a real-world problem of guarding an art gallery with the minimum number of guards who together can observe the whole gallery. In the computational geometry version of the problem the layout of the art gallery is represented by a simple polygon and each guard is represented by a point in the polygon. A set S of points is said to guard a polygon if, for every point p in the polygon, there is some q âˆˆ S such that the line segment between p and q does not leave the polygon. Finding the smallest cardinality of guarding set of simple polygon is known to be NP-hard. Many researcher approached for an approximation algorithm. Subhir K. Ghosh [reference 1] proposed log(n)-factor approximation algorithm for simple polygon in O(n ) in 2010. It is also known that There exist a constant Îµ â‰¥ 0 such that an approximation ratio of 1 + Îµ can not guaranteed by any polynomial time approximation algorithm unless P = NP. In a recent paper, B. J. Nilsson [2013] proposed a 30-factor approximation algorithm for monotone polygon. L. Gewali [1992] proposed an O(n) time algorithm for covering a horizontally convex orthogonal polygon with minimum number of orthogonal star-shaped polygons. In this thesis, we are dealing with the art gallery problem for uni-monotone and special case of monotone polygon. For simple uni-monotone, we are assuming that there is some fixed guards G already placed inside the polygon P. If G covers the whole polygon then can we partition it to k-sets such that each set individually covers P. If we get such a partition, G is said to be fault tolerant at level k. In case the preplaced guard can not partition in to k-sets, find the smallest number of extra guards that is to be added to G such that it can be partitioned into k-sets which individually covers P. For orthogonal uni-monotone polygon, we are showing O(nlogn) time algorithm for optimal guarding. For monotone polygon, If we are considering only upper chain(lower chain) to form a graph based on visibility region of convex pieces inside the polygon then this graph is not chordal. We are also showing sub case of monotone polygon for optimal guarding. Our result can be used in the geometrical application where there is requirement to maintaining fault tolerant. For example, In ad-hoc network there is need to maintain fault tolerant at some level so by modifying the definition of coverage of point, our result may be used.

## Control Number

ISI-DISS-2014-280

## 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/10263/6436

## Recommended Citation

Lal, Ratan, "Some Problems on Guarding Monotone Polygons." (2015). *Master’s Dissertations*. 235.

https://digitalcommons.isical.ac.in/masters-dissertations/235

## 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:28843258