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DECOMPOSING AND PACKING POLYGONS
COMPUTER SCIENCE AND SOFTWARE ENGINEERING
PRESENTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY (COMPUTER SCIENCE)
MONTREAL, QUEBEC, CANADA
DANIA EL-KHECHEN, 2009
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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
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
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
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
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.
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