FLIPSANDSPANNERS alexander jozef hubertus verdonschot A thesis submitted to the Faculty of Graduate and Post Doctoral Affairs in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Computer Science Carleton University Ottawa, Ontario, Canada arXiv:1509.02563v1 [cs.CG] 8 Sep 2015 © 2015 Alexander Jozef Hubertus Verdonschot ii ABSTRACT In this thesis, we study two different graph problems. The first problem revolves around geometric spanners. Here, we have a set of points in the plane and we want to connect them with straight line segments, such that there is a path between each pair of points and these paths do not require large detours. If we achieve this, the resulting graph is called a spanner. We focus our attention on two graphs (the Θ-graph and Yao-graph) that are constructed by connect- ing each point with its nearest neighbour in a number of cones. Al- though this construction is very straight-forward, it has proven chal- lenging to fully determine the properties of the resulting graphs. We show that if the construction uses 5 cones, the resulting graphs are still spanners. This was the only number of cones for which this ques- tion remained unanswered. We also present a routing strategy (a way to decide where to go next, based only on our current location, its direct neighbourhood, and our destination) on the half-Θ6-graph, a variant of the graph with 6 cones. We show that our routing strat- egy avoids large detours: it finds a path whose length is at most a constant factor from the straight-line distance between the endpoints. Moreover, we show that this routing strategy is optimal. In the second part, we turn our attention to flips in triangulations. A flip is a simple operation that transforms one triangulation into an- other. It turns out that with enough flips, we can transform any trian- gulation into any other. But how many flips is enough? We present an improved upper bound of 5.2n - 33.6 on the maximum flip distance between any pair of triangulations with n vertices. Along the way, we prove matching lower bounds on each step in the current algorithm, including a tight bound of (3n - 9)=5 flips needed to make a trian- b c gulation 4-connected. In addition, we prove tight Θ(n log n) bounds on the number of flips required in several settings where the edges have unique labels. iii iv ACKNOWLEDGMENTS I would not have been able to write this thesis without the help and support of many people. First of all, I would like to thank my supervisors – Prosenjit Bose, Pat Morin, and Vida Dujmovi´c.They are all I could have wished for in my supervisors and more. Combining a wealth of knowledge with a burning curiosity and a penchant for finding fascinating, yet ap- proachable, open problems, they made these past five years into a journey of exploration and excitement. I also want to thank the other members and students of the Com- putational Geometry lab for making it such a nice place to work (and occasionally not work). I am especially grateful to fellow PhD stu- dents André, Carsten, Dana, and Luis, for being great friends and collaborators. In fact, I am very grateful to all the researchers and students who I got to work with during my PhD studies. Working to- gether was always a pleasure, and they taught me more than classes ever could. Finally, I would like to thank my family and friends for their sup- port during these long, and at times stressful, years. I am especially grateful to my mother for encouraging me to take the leap of faith that is an international PhD, and to my partner, Gehana, for her un- failing love and support. Thank you! v vi CONTENTS abstract iii acknowledgments v contents vii 1 summary of the thesis1 1.1 Geometric spanners 1 1.2 Flips in triangulations 2 i geometric spanners5 2 an introduction to yao- and Θ-graphs7 2.1 Geometric spanners 7 2.2 Preliminaries 9 2.3 Yao-graphs 10 2.4 Θ-graphs 14 2.5 History 16 3 the Θ5 -graph is a spanner 21 3.1 Introduction 21 3.2 Connectivity 23 3.3 Spanning ratio 25 3.4 Lower bound 35 3.5 Conclusions 38 4 competitive routing in the half-Θ6 -graph 41 4.1 Introduction 41 4.2 Preliminaries 43 4.3 Spanning ratio of the half-Θ6-graph 45 4.4 Remarks on the spanning ratio 48 4.5 Routing in the half-Θ6-graph 49 4.5.1 Positive routing 51 4.5.2 Negative routing 57 4.6 A stateful algorithm 60 4.7 Bounding the maximum degree 61 4.7.1 Routing in G12 63 4.7.2 Routing in G9 67 4.8 Conclusions 68 ii flips in triangulations 73 5 a history of flips in combinatorial triangula- tions 75 5.1 Introduction 75 5.2 Wagner’s bound 77 5.3 Komuro’s bound 79 5.4 Mori et al.’s bound 81 5.5 Lower bounds 84 vii viii contents 6 making triangulations 4-connected using flips 87 6.1 Introduction 87 6.2 Upper bound 88 6.3 Lower bound 102 6.4 Conclusions and open problems 104 6.5 Lemmas and proofs 105 7 edge-labelled flips 109 7.1 Introduction 109 7.2 Convex polygons 111 7.2.1 Upper bound 111 7.2.2 Lower bound 114 7.2.3 Simultaneous flips 117 7.3 Combinatorial triangulations 120 7.3.1 Upper bound 120 7.3.2 Lower bound 123 7.3.3 Simultaneous flips 124 7.4 Pseudo-triangulations 126 7.4.1 Pointed pseudo-triangulations 128 7.4.2 General pseudo-triangulations 134 7.5 Conclusions and open problems 139 SUMMARYOFTHETHESIS 1 This thesis is comprised of two main parts. The first part, found in Chapters 2 through 4, deals with geometric spanners. Chapters 5 through 7 contain the second part, which focuses on flips in triangula- tions. A brief introduction and summary of each part is given below. The first chapter of each part provides a more detailed introduction. The common theme in the two parts is that both deal with graphs. A graph consists of a set of vertices, some of which are connected by edges. In this thesis, all graphs will be simple, which means that there is at most one edge connecting each pair of vertices, and edges cannot connect a vertex to itself. 1.1 geometric spanners Spanners can be informally described as graphs in which one never needs to make a large detour. That is, the shortest path between two vertices is proportional to their actual distance. Road networks are a good example; nearby cities are typically connected by a direct road, so that the total distance travelled is not much more than the distance ‘as the crow flies’. Spanners have been studied in many different con- texts, but we will focus on geometric spanners, where the vertices are points in the plane, and the length of an edge is the Euclidean dis- tance between its endpoints. The spanning ratio is the maximum ratio between the shortest path in the graph and the straight-line distance between any pair of vertices. Chapter 2 gives an in-depth introduction to geometric spanners in general, and simple cone-based spanners in particular. The Θ-graph is one such cone-based spanner. To construct it, we partition the plane around each vertex into a number of equiangular cones and add an edge to the ‘closest’ vertex in each cone, where the closest vertex is defined as the vertex whose projection on the bisector of the cone is closest. It has been shown that for any desired spanning ratio t, there is a number of cones k such that the Θ-graph with k cones (typically written as Θk) is guaranteed to have spanning ratio t. However, it was not known exactly for which values of k the span- ning ratio of Θk is bounded by a constant. It was known that Θ3 and below are not constant spanners, while Θ6 and up are. Recently, Θ4 was shown to be a constant spanner as well, leaving the question unanswered only for Θ5. In Chapter 3, we prove that Θ5 is, indeed, a constant spanner. With the earlier results, this implies that Θk is a spanner for all k > 4. This result was first published in the proceed- 1 2 summaryofthethesis ings of the 39th International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2013)[8] and later appeared in Computa- tional Geometry: Theory and Applications [9]. Of course, knowing that there exists a short path to where you want to go is not the end of the story: you also have to know how to find it. This is called routing, or competitive routing if the spanning ratio of the resulting path is bounded by a constant. If you know the entire graph, routing is nothing more than computing a path, but most settings consider the more restricted scenario where you know your destination, but you can only see your current location and its neighbours. This is referred to as local routing. In Chapter 4, we present a local, competitive routing strategy for the half-Θ6-graph, which is closely related to Θ6. Our strategy achieves a routing ratio of 5=p3 = 2.886 : : : , which seems slightly disappointing compared to the spanning ratio of 2. This makes it all the more surprising that we managed to show that our algorithm is, in fact, optimal: no other routing strategy can achieve a better routing ratio, under the same restrictions.