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DECOMPOSING AND PACKING POLYGONS

DANIA EL-KHECHEN

A THESIS

IN

THE DEPARTMENT

OF

COMPUTER SCIENCE AND SOFTWARE ENGINEERING

PRESENTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF DOCTOR OF PHILOSOPHY (COMPUTER SCIENCE)

CONCORDIA UNIVERSITY

MONTREAL, QUEBEC, CANADA

APRIL 2009

DANIA EL-KHECHEN, 2009

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Abstract

Decomposing and packing polygons

Dania El-Khechen, Ph.D.

Concordia University, 2009

In this thesis, we study three different problems in the field of computational geometry: the

partitioning of a simple polygon into two congruent components, the partitioning of squares

and rectangles into equal area components while minimizing the perimeter of the cuts, and

the packing of the maximum number of squares in an orthogonal polygon.

To solve the first problem, we present three polynomial time algorithms which given

a simple polygon P partitions it, if possible, into two congruent and possibly nonsimple

components Pi and Pi: an 0(n2 logn) time algorithm for properly congruent components

and an 0(n3) time algorithm for mirror congruent components.

In our analysis of the second problem, we experimentally find new bounds on the optimal

partitions of squares and rectangles into equal area components. The visualization of the

best determined solutions allows us to conjecture some characteristics of a class of optimal

solutions.

Finally, for the third problem, we present three linear time algorithms for packing the

maximum number of unit squares in three subclasses of orthogonal polygons: the staircase

polygons, the pyramids and Manhattan skyline polygons. We also study a special case of

the problem where the given orthogonal polygon has vertices with integer coordinates and

the squares to pack are (2 x 2) squares. We model the latter problem with a binary integer

program and we develop a system that produces and visualizes optimal solutions. The

observation of such solutions aided us in proving some characteristics of a class of optimal

solutions.

iii

Acknowledgements

This work would not have been possible without my supervisors Thomas Fevens and John

Iacono. I thank them for their constant encouragement, motivation and advices. I also

thank them for giving me the great opportunity to travel and attend many conferences and

workshops

I thank all my co-authors. It has been a pleasure to work with every one of them.

In particular, I thank Godfried Toussaint for giving me the opportunity to at tend his

wonderful annual workshop on Computational Geometry in Barbados: an occasion to work

on challenging problems, meet great researchers and enjoy the beach during Montreal's

cold winter. I thank all the Mcgill lunch group with whom I enjoyed lunch from time to

time. I thank all the researchers with whom I worked on the interdisciplinary project with

the faculty of fine arts, professors: Cheryl Dudek, Thomas Fevens, Sudhir Mudur, Lydia

Sharman and Fred Szabo. I also thank Ramgopal Rajagopalan and Eric Hortop with whom

it was a lot of fun to drink coffee and discuss symmetry groups.

I thank Giinter Rote for his unpublished manuscript which inspired the material in

Chapter 4. I thank Ken Brakke for his software Surface Evolver that we used in Chapter 5.

I also thank Tobias Achterberg for his solver SCIP that aided us for Chapter 6 results.

I thank the graduate program advisor Halina Monkiewicz and the office assistant Hirut

Adugna for always answering my numerous questions with a smile. I thank the teaching

assistants coordinator Pauline Dubois who gave me the priceless opportunity to teach.

I thank Vasek Chvatal for lending me so many (excellent) books and movies, making me

discover many (good) restaurants in Montreal, introducing me to so many amazing people

and transmitting a great enthusiasm for Mathematics and a great joy of life. I also thank

him for his entertaining classes and his (crystal clear) way of transmitting information.

I thank all my friends for their constant support. Fatme el-Moukaddem for discussing

our research problems, reading my nagging over MSN and for or never-ending-after-defence

plans, Simon Kouyoumdjian for helping me recover most of the material in Chapter 5 after

my hard disk crashed and for sharing many precious moments, and Alessandro Zanarini, with

iv

his incredible sweetness, for many useful Mathematical discussions. I thank my long distance

friends: Nisrine Jaafar for not only helping me get through the first year in Montreal but for

turning it into a wonderful one, Malak Jalloul for her unlimited phone calls plan to Canada,

Abir Baz and Rouba Choueiry for continuously yeling "yalla, khalssina!", Alaa Abi-Haidar

for our great exchanges, Narjess Fathalla, Mayssan Maarouf, Ali Mourad, Houssam Nassif

and Rima Sleiman for many many reasons. I thank my Montreal friends: Khaled AbdelHay

for our long studying sessions, Ruddy Avalos for his super parties, Tamara Diaz (with her

unique laugh) and Duhamel Xolot for the great overnight discussions we had, Francois

Grandchamp for his Quebecois lessons, John Alexander Lopez for the many things he taught

me, Mahitab Seddik and Rania Khattab for always listening. I also thank Chloe Guillaume,

Layla Hussain, Bassem Hussami, Marie-Andre L'esperance, Daria Madjidian and George

Peristerakis for their continuous attention. I thank my two dear and constantly-traveling

friends Lama Kabbanji and Hicham Safieddine for being there even when they are not.

Finally, without my friend JJ, the thesis journey would have been less fun and much harder.

I thank all my dance and literature teachers who made my life richer. In particular, I

thank my first and current bellydancing teachers Sheila Ribeiro and Any Massicotte for

being an inspiration. I also thank my sweet "bellysisters" Wendy Corner and Ruth Gover.

I thank my family. The Atwi family: my uncles Wajih, Said, Bassam, Bassel and

Houssam (who accompanied me here the first month) and my aunt Samia (who supported

me in my first years in Montreal). I thank all my cousins! In particular, I thank Douaa,

Mayssa, Mohamad, Mostafa and Ahmad for being the siblings I never had. Their mother

Safaa Serhan is a precious gift to all of us. I thank also my caring uncle Ali El-Khechen.

Je remercie Nikolaj van Omme pour tout ce qu'il m'a appris sur la programation

mathemathiques et pour les discussions enrichissantes. Je le remercie d'etre si patient et

attentione. Je le remercie d'avoir une passion contagieuse pour toutes les choses de la vie.

Je le remercie aussi de m'avoir presente son pere Albert Carton, un homme extraordinaire.

Merci mon chanteur prefere.

I dedicate this thesis to Hind Atwi. A brilliant woman. A great militant. A silent

inspiration. C'est grace a elle si je suis devenue qui je suis. Merci Mama.

v

Contents

List of Figures ix

List of Tables xiii

1 Introduction 1

2 Background information and notat ion 7

2.1 Polygon definitions 7

2.2 Graph definitions 14

2.3 Complexity classes and algorithmic techniques 15

3 State of the art 19

3.1 Partitioning 20

3.1.1 General polygons 21

Triangles 21