# Efficient Algorithm For Identifying a Pair of Maximal Empty Rectangles of Maximum Total Area Amidst a Point Set.

December 2016

## Date of Award

Winter 12-12-2017

## 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

Nandy, Subhas Chandra (ACMU-Kolkata; ISI)

## Abstract (Summary of the Work)

Given an isothetic rectangular floor containing a set of n randomly placed points, an important problem that arises in computational geometry is to identify the largest maximal empty rectangle(MER). The best known algorithm for this problem has complexity of O(n log2 n). Here, our motivation is to develop an algorithm for a given set of n points on a rectangular floor, identify a pair of maximal empty rectangles such that the area of their union is maximum among all pairs of maximal empty rectangles that exist on the floor. We have observed that if we try to find two disjoint MER whose union is maximum, the by line sweep technique it can be solved in O(R + nogn) time, where R is the maximum number of MERâ€™s. We have also observed that the unconstrained version of this problem can be solved in less time if it is possible to devise a sub quadratic algorithm for the problem of finding the pair of rectangles whose union of area is maximum among m rectangles. In order to solve this problem we have observed that if we try to find two disjoint rectangles whose union is maximum among m rectangles, it can be solved in O(mlog m) time. We have also observed that if they are axis-parallel squares rather than rectangles, and they are of same area, then even for the non-disjoint case, the problem can be solved in O(m log m) time.We have approached to solve the same problem for rectangles with same area using matrix space, but have not found any way to approach the general case. This study is still in process.

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:28843075

## Control Number

ISI-DISS-2016-356