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Formulae and Growth Rates of Animals on Cubical and Triangular Lattices Formulae and Growth Rates of Animals on Cubical and Triangular Lattices Mira Shalah Technion - Computer Science Department - Ph.D. Thesis PHD-2017-18 - 2017 Technion - Computer Science Department - Ph.D. Thesis PHD-2017-18 - 2017 Formulae and Growth Rates of Animals on Cubical and Triangular Lattices Research Thesis Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Mira Shalah Submitted to the Senate of the Technion | Israel Institute of Technology Elul 5777 Haifa September 2017 Technion - Computer Science Department - Ph.D. Thesis PHD-2017-18 - 2017 Technion - Computer Science Department - Ph.D. Thesis PHD-2017-18 - 2017 This research was carried out under the supervision of Prof. Gill Barequet, at the Faculty of Computer Science. Some results in this thesis have been published as articles by the author and research collaborators in conferences and journals during the course of the author's doctoral research period, the most up-to-date versions of which being: • Gill Barequet and Mira Shalah. Polyominoes on Twisted Cylinders. Video Review at the 29th Symposium on Computational Geometry, 339{340, 2013. https://youtu.be/MZcd1Uy4iv8. • Gill Barequet and Mira Shalah. Automatic Proofs for Formulae Enu- merating Proper Polycubes. In Proceedings of the 8th European Conference on Combinatorics, Graph Theory and Applications, 145{151, 2015. Also in Video Review at the 31st Symposium on Computational Geometry, 19{22, 2015. https://youtu.be/ojNDm8qKr9A. Full version: Gill Barequet and Mira Shalah. Counting n-cell Polycubes Proper in n − k Dimensions. European Journal of Combinatorics, 63 (2017), 146{163. • Gill Barequet, Gunter¨ Rote and Mira Shalah. λ > 4. In Proceedings of the 23rd European Symposium on Algorithms, 83{94, 2015. Full version: Gill Barequet, Gunter¨ Rote and Mira Shalah. λ > 4 : An improved lower bound on the growth constant of polyominoes, Communications of the ACM, 59 (2016), 88{95. • Gill Barequet, Mira Shalah and Yufei Zheng. An Improved Lower Bound on the Growth Constant of Polyiamonds. In Proceedings of The 22nd International Computing and Combinatorics Conference, 50{61, 2017. Technion - Computer Science Department - Ph.D. Thesis PHD-2017-18 - 2017 Acknowledgements First and foremost, I would like to extend my sincere gratitude to my advisor, Prof. Gill Barequet, for guiding and mentoring over the past five years, raising me to be the person and researcher I am today. Throughout my studies, you have always inspired me with your creativity, intuition, vision, and positivity, all which had a tremendous impact on me. Thank you for always being there to encourage me and offer me your kind advice on many scientific, academic, and personal matters, for pushing me to peruse directions which were challenging, yet promising, at the same time, and for giving me the freedom, at times, to try other directions. Finally, thank you for teaching me, by means of example, to be committed, respectful, and patient towards my work. Expressions of gratitude are also in order for Prof. Gunter Rote, for his support of this research and the opportunity to learn first-hand from one of the top researchers in the field. Many of the results presented in this thesis would not have been possible without the support of the facilities and staff of both the Hasso Plattner Institute Future SOC Lab and the Technion's Division of Computing and Information Systems. My time at the Technion gave me the opportunity to work with many amazing people who shared their wisdom, expertise and kind advice. I owe special thanks to Prof. Freddy Bruckstein, whose kindness and friendliness is only matched by his brilliance and enthusiasm. I also owe a lot of gratitude to Prof. Nader Bshouty, who was my temporary advisor when I started graduate school at the Technion, and whose advise and guidance were truly invaluable to me. I was also fortunate to have wonderful fellow peers around me. Special thanks to my friends Raeda, Majd and Areej, who were my office mates for five years combined, and to Nurit for all the enjoyable coffee breaks we shared. The journey towards becoming amongst the first Israeli-Arab women with a PhD in Computer Science was only possible thanks to the support and education I received from my school teachers and family, especially my parents, who always supported and believed in me, and who already at a very early age awoke in me the love of knowledge, the enjoyment of study, and the drive to success. Thank you for your love, patience, infinite dedication, and for the independence you trusted me with already at an early age. I am incredibly blessed to have you and my brothers Ramzy and Amir, my sisters Christina and Aml, my mother, father, brother, and sister in law Omaima, Eli, Rani and Lyan, and my dearest friends Suzan and Mariana. Thank you for your support, encouragement, and for standing by me and cheering me up through it all. I have been fortunate to begin my academic path as a student at the Etgar framework at the University of Haifa. I am forever indebted to the head of Etgar, Prof. Gadi Landau, for this educational experience, where I fell in love with Computer Science and my husband, soul mate, and best friend, Amir Abboud. Thank you Amir for your wisdom, patience, advice, scientific insight, love, and support, and for being a source of Technion - Computer Science Department - Ph.D. Thesis PHD-2017-18 - 2017 perspective, inspiration and happiness. This thesis is dedicated to you. The generous financial support of the Technion, the Israeli Science Foundation, the Israeli Council for Higher Education, the Google Anita Borg Memorial Scholarship program, and the Nevai, Zelig, and Levy Family Fellowships, is gratefully acknowledged. Technion - Computer Science Department - Ph.D. Thesis PHD-2017-18 - 2017 Dedicated to my husband Amir Abboud, and my mom and dad, Najat and Zuhair Shalah Technion - Computer Science Department - Ph.D. Thesis PHD-2017-18 - 2017 Contents List of Figures Abstract 1 1 Introduction 3 2 Preliminaries 15 2.1 The Triangular Lattice . 15 2.2 The Twisted Cylinder . 15 2.3 Motzkin Path . 15 2.4 Concatenation of Lattice Animals . 16 2.5 Composition of Lattice Animals . 16 2.6 Cayley Trees . 17 2.7 Integer Partition . 17 2.8 Graph Isomorphism . 17 3 Polyominoes 19 3.1 Introduction . 19 3.2 Lower Bound . 19 3.2.1 Counting Polyominoes on Twisted Cylinders . 19 3.2.2 Method . 22 3.2.3 Computing λ27 ............................ 25 3.2.4 Validity and Certification . 30 3.3 Upper Bound . 31 3.3.1 Polyominoes Viewed as Sequences of Twigs . 31 3.3.2 Mapping Lattice Animals to Binary Strings . 34 3.3.3 Klarner and Rivest's Enhancement . 35 3.3.4 Improved Upper Bound . 39 3.3.5 Comparison of the Results . 42 4 Polyiamonds 47 4.1 Introduction . 47 4.2 Lower Bound . 47 Technion - Computer Science Department - Ph.D. Thesis PHD-2017-18 - 2017 4.3 Upper Bound . 49 4.3.1 Number of Compositions . 49 4.3.2 Balanced Decompositions . 49 4.3.3 The Bound . 50 5 Polycubes 53 5.1 Introduction . 53 5.2 Overview of the Method . 56 5.2.1 Counting . 58 5.2.2 Distinguished Structures . 58 5.3 Inclusion-Exclusion Graph . 66 5.4 Counting Polycubes . 67 5.4.1 Trees . 68 5.4.2 Nontrees . 69 5.5 Proof of Main Theorem . 69 5.6 Results . 72 6 Conclusion and Open Problems 75 6.1 Polyominoes . 75 6.2 Polyiamonds . 79 6.3 Polycubes . 79 A Appendix 83 A.1 Certified Computations - Chapter 3 . 83 A.2 The Functions Wi(x; y) - Chapter 3 . 84 A.3 Disconnected Structures - Chapter 5 . 87 Hebrew Abstract i Technion - Computer Science Department - Ph.D. Thesis PHD-2017-18 - 2017 List of Figures 1.1 Polyiominoes of sizes 1 ≤ n ≤ 4.......................4 1.2 Polyiamonds of sizes 1 ≤ n ≤ 5 .......................5 2.1 Polyiamonds on the triangular lattice . 15 2.2 A twisted cylinder of perimeter W = 5. The wrap-around connections are indicated; for example, cells 1 and 2 are adjacent. 16 2.3 A Motzkin path of length 7. 16 2.4 The three Cayley trees of size 3. 17 3.1 A polyomino embedded on a twisted cylinder . 20 3.2 Bottom: A snapshot of the boundary line (solid line) during the transfer- matrix calculation. The dashed line indicates two adjacent cells which are connected \around the cylinder," where this is not immediately apparent. The boundary cells (top row) are shown darker. The numbers are the labels of the boundary cells. Middle: A symbolic encoding of the state. Top: the same state encoded as the Motzkin path (0,1,0,0,1,1,-1,1,0,-1,- 1,0,1,0,-1,0,-1). 21 3.3 The dependence between the different groups of ynew and yold...... 23 3.4 Extrapolating the sequence λW ........................ 24 3.5 Front view of the \supercomputer" which we used. It is only a box of about 45 × 35 × 70 cm3............................ 25 3.6 An unreachable state. 26 3.7 The process of computing succ0 and succ1 for an integer index. 27 3.8 Simplified pipeline of the successor computation. 28 3.9 text . 28 3.10 A Motzkin path of length 29, its symbolic representation as a sequence of steps `0', `+', and `-', and the associated bit pattern . 29 3.11 Results for W = 23;:::; 27.......................... 30 3.12 Eden's set of twigs E [KR73, Figure 3] . 32 3.13 A spanning tree generated by Eden's method [KR73, Figure 4] . 33 3.14 A snapshot of the hexagonal lattice. Cells a and b adjoin both cells i and j [Lun72a, Figure 11] .
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