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Disk Intersection Graphs: Models, Data Structures, and Algorithms Disk Intersection Graphs: Models, Data Structures, and Algorithms Dissertation zur Erlangung des Doktorgrades vorgelegt am Fachbereich Mathematik und Informatik der Freien Universität Berlin von Paul Seiferth 2016 Erstgutachter: Prof. Dr. Wolfgang Mulzer Zweitgutachter: Prof. Dr. Christian Knauer Tag der Disputation: 19.08.2016 iii Abstract 2 Let P R be a set of n point sites. The unit disk graph UD(P) on P has vertex ⊂ set P and an edge between two sites p, q P if and only if p and q have Euclidean 2 distance jpqj 6 1. If we interpret P as centers of disks with diameter 1, then UD(P) is the intersection graph of these disks, i.e., two sites p and q form an edge if and only if their corresponding unit disks intersect. Two natural generalizations of unit disk graphs appear when we assign to each point p P an associated radius r > 0. The first one is 2 p the disk graph D(P), where we put an edge between p and q if and only if jpqj 6 rp +rq, meaning that the disks with centers p and q and radii rp and rq intersect. The second one yields a directed graph on P, called the transmission graph of P. We obtain it by putting a directed edge from p to q if and only if jpqj 6 rp, meaning that q lies in the disk with center p and radius rp. For disk and transmission graphs we define the radius ratio Ψ to be the ratio of the largest and the smallest radius that is assigned to a site in P. It turns out that the radius ratio is an important measure of the complexity of the graphs and some of our results will depend on it. For these three classes of disk intersection graphs we present data structures and algorithms that solve four types of graph theoretic problems: dynamic connectivity, routing, spanner construction, and reachability oracles; see below for details. For disk and unit disk graphs, we improve upon the currently best known results, while the problems we consider for transmission graphs abstain non-trivial solutions so far. Dynamic Connectivity. First, we present a data structure that maintains the connected components of a unit disk graph UD(P) when P changes dynamically. It takes O(log2 n) amortized time to insert or delete a site in P and O(log n= log log n) worst-case time to determine if two sites are in the same connected component. Here, n is the maxi- mum size of P at any time. A simple variant improves the amortized update time to O(log n log log n) at the cost of a slightly increased worst-case query time of O(log n). Using more advanced data structures, we can extend our approach to disk graphs. While the worst-case query time remains O(log n= log log n), an update now requires O(Ψ22α(n) log10 n) amortized expected time, where Ψ is the radius ratio of the disk graph and α(n) is the inverse Ackermann function. Routing. As the second problem, we consider routing in unit disk graphs. A routing scheme R for UD(P) assigns to each site s P a label `(s) and a routing table ρ(s). For 2 any two sites s, t P, the scheme R must be able to route a packet from s to t in the 2 following way: given a current site r (initially, r = s), a header h (initially empty), and the target label `(t), the scheme R may consult the current routing table ρ(r) to compute v a new site r0 and a new header h0, where r0 is a neighbor of r in UD(P). The packet is then routed to r0, and the process is repeated until the packet reaches t. The resulting sequence of sites is called the routing path. The stretch of R is the maximum ratio of the (Euclidean) length of the routing path produced by R and the shortest path in UD(P), over all pairs of distinct sites in P. For any given " > 0, we show how to construct a routing scheme for UD(P) with stretch 1 + " using labels of O(log n) bits and routing tables of O("-5 log2 n log2 D) bits, where D is the (Euclidean) diameter of UD(P). The header size is O(log n log D) bits. Spanners. Next, we construct sparse approximations of transmission and disk graphs. Let G be a transmission graph. A t-spanner for G is a subgraph H G with vertex set ⊆ P so that for any two sites p, q P, we have d (p, q) td (p, q), where d and d 2 H 6 G H G denote the shortest path distance in H and G (with Euclidean edge lengths). We show how to compute a t-spanner for G with O(n) edges in O(n(log n + log Ψ)) time, where Ψ is the radius ratio of P. Utilizing advanced data structures, we obtain a construction that runs in O(n log5 n) time, independent of Ψ. This construction can be adapted to disk graphs and gives a t-spanner for D(P) in expected time O(n2α(n) log10 n), where α(n) is the inverse Ackermann function. As an application we show that our t-spanner can be used to find a BFS tree in a transmission or disk graph for any given start vertex in O(n log n) additional time. Reachability Oracles. Finally, we compute reachability oracles for transmission graphs. These are data structures that answer reachability queries: given two sites p and q, is there a directed path between them? The quality of an oracle is measured by the space S(n), the query time Q(n), and the preproccesing time. We present three reachability oracles whose quality depends on the radius ratio Ψ: the first one works only for Ψ < p3 and achieves Q(n) = O(1) with S(n) = O(n) and preprocessing time O(n log n); the second data structure gives Q(n) = O(Ψ3pn) and S(n) = O(Ψ3n3=2); the third data structure is randomized with Q(n) = O(n2=3(log n + log Ψ)) and S(n) = O(n5=3(log n + log Ψ)) and answers queries correctly with high probability. As a second application for our spanners, we employ them to achieve a fast prepro- cessing time for our reachability oracles. vi Selbständigkeitserklärung Ich erkläre hiermit, dass ich alle Hilfsmittel und Hilfen angegeben habe und versichere, auf dieser Grundlage die Arbeit selbständig verfasst zu haben. Die Arbeit habe ich nicht in einem frühren Promotionsverfahren eingereicht. Berlin, den 17. Mai 2016 vii Acknowledgements First of all I would like to thank my advisor Wolfgang Mulzer. He literally made my PhD a long journey and on the way he never got tired in teaching me geometry with endless passion, at any time and any place. Beyond this, he also gave me the opportunity to visit numerous exciting places around the world and to experience foreign cultures. Thanks for the wonderful and unique time we spend together, I really appreciate it and it won’t be forgotten. Furthermore, I am very grateful for my second referee Christian Knauer who, without any hesitation, agreed to review my thesis, even after learning about my tight time constraints. I would like to thank all my coauthors, it has been a pleasure to work with so many brilliant people. Especially, I need thank Liam Roditty and Haim Kaplan. They never run out of challenging and interesting problems to attack and they encouraged me to never stop hunting for more simple and more intuitive arguments. Also their hospitality at our regular visits in Israel was outstanding and it made me feel very welcome. Fur- thermore, I like to thank Yannik Stein, who was a worthy companion on all the trips we made together and with whom I had a lot of fun at and after work. He participated in all our common projects very diligently and I always enjoyed to work with him, no matter of time and date. I am much obliged to every single member of the AG Theoretische Informatik. All of them offered me assistance and guidance in any situation of my PhD-life and my life in general. They always managed to cheer me up, especially in the more cumbersome beginning of my studies. I also have to thank them for nurturing my common knowledge day by day with some of the, sometimes, uttermost strange discussions during the coffee round. Finally, and most of all, I like to thank my family and my friends for their support and for showing interest in what I am doing, even though it is tough to understand what all this is about. In particular, I thank my cohabitant Nadja Scharf. She was one of the major discoveries during my PhD and she managed keep me motivated and focused during the last weeks. ix Contents 1 Introduction1 1.1 Modeling Sensor Networks...........................2 1.2 The Problems Considered...........................5 1.3 Further Problems on Disk Intersection Graphs...............7 1.4 Results and Organization of the Thesis....................8 2 Preliminaries 11 2.1 Notation and Planar Grids.......................... 11 2.2 Well Separated Pair Decompositions..................... 12 2.2.1 The WSPD Spanner.......................... 13 2.2.2 WSPDs for General Metric Spaces.................. 14 2.3 Dynamic Lower Envelopes........................... 15 2.3.1 The Planar Case............................ 15 2.3.2 Dynamic Lower Envelopes in 3-space................
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