Graph Embeddings Motivated by Greedy Routing

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Graph Embeddings Motivated by Greedy Routing Graph Embeddings Motivated by Greedy Routing zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften von der KIT-Fakultät für Informatik des Karlsruher Instituts für Technologie (KIT) genehmigte Dissertation von Roman Prutkin aus Ekaterinburg Tag der mündlichen Prüfung: 29. November 2017 Erster Gutachter: Prof. Dr. Dorothea Wagner Zweiter Gutachter: Prof. Dr. Martin Nöllenburg Dritter Gutachter: Prof. Dr. Michael Kaufmann Roman Prutkin: Graph Embeddings Motivated by Greedy Routing, © 21. Februar 2018 To my family DANKSAGUNG Als Erstes möchte ich mich bei Dorothea Wagner bedanken für die Möglichkeit, an ihrem Lehrstuhl meiner Promotion nachzugehen. Ich weiß es sehr zu schätzen, dass ich ohne Zeitdruck und unter äußerst komfortablen Rahmenbedingungen Forschungsfragen untersuchen konnte, die mich interessieren. Bei der Wahl der Forschungsthemen war ich immer frei und bekam von Dorothea stets Rat und Unterstützung. Michael Kaufmann danke ich, dass er sich bereit erklärt hat, meine Disserta- tion zu begutachten, sowie für seine Flexibilität bei der Terminfindung für die Promotionsprüfung. Martin Nöllenburg und Ignaz Rutter haben mich sowohl bei meiner Diplom- arbeit als auch während meiner Promotion hervorragend betreut. Ich bin sehr froh, dass ich von eurem Fachwissen und Erfahrung profitieren und von euch viel lernen konnte. Die Zusammenarbeit mit euch hat mir Spaß gemacht. Ich bedanke mich bei meinen Koautoren Therese Biedl, Thomas Bläsius, Xiaoji Chen, Carsten Dachsbacher, Fabian Fuchs, Alexander E. Holroyd, Anton Kapla- nyan, Boris Klemz, Bongshin Lee, Moritz von Looz, Henning Meyerhenke, Lev Nachmanson, Benjamin Niedermann, Martin Nöllenburg, Nathalie Henry Riche und Ignaz Rutter. Bei meinen aktuellen und ehemaligen Kollegen Lukas Barth, Moritz Baum, Tho- mas Bläsius, Guido Brückner, Valentin Buchhold, Julian Dibbelt, Fabian Fuchs, Andreas Gemsa, Sascha Gritzbach, Michael Hamann, Tanja Hartmann, Andrea Kappes, Tamara Mchedlidze, Benjamin Niedermann, Martin Nöllenburg, Thomas Pajor, Marcel Radermacher, Ignaz Rutter, Ben Strasser, Markus Völker, Franziska Wegner, Matthias Wolf, Tim Zeitz und Tobias Zündorf bedanke ich mich für die angenehme und lockere Atmosphäre am Institut und die vielen Tischfußballspie- le, Kino-, Bastel- und Spieleabende, Wanderungen und Fußball-Sessions im Soccer Center. Besonderer Dank geht an Fabian Fuchs, der mit mir das Sensornetze-Büro geteilt hat. Ich danke meiner Frau Anna, meinen Eltern Olga und Ilya und dem Rest mei- ner Familie für eure Inspiration und grenzenlose Unterstützung während meiner Zeit als Doktorand, insbesondere in der Zeit des Schreibens dieser Arbeit. v . DEUTSCHEZUSAMMENFASSUNG In dieser Arbeit untersuche ich Probleme aus Algorithmischer Geometrie und Graphentheorie, die Greedy Routing betreffen. Ich konzentriere mich insbesonde- re auf Greedy Routing in geometrisch eingebetteten Graphen, welches wie folgt definiert ist. Gegeben sei ein Graph G = (V, E), dessen Knoten V Koordinaten zugeordnet wurden, beispielsweise Punkte in der euklidischen Ebene. Kanten E symbolisieren die Möglichkeit der direkten bidirektionalen Kommunikation zwi- schen Knoten. Jeder Knoten kennt seine eigenen Koordinaten und die seiner direk- ten Nachbarn in G. Beim Routing von Nachrichten in diesem Netzwerk nehmen wir zusätzlich an, dass jede Nachricht die Koordinaten des Ziels enthält. Unter den obigen Annahmen ist die folgende t2 einfache Routing-Strategie als Greedy Routing be- kannt. Für eine eingehende Nachricht berechnet je- der Knoten die euklidischen Distanzen zum Ziel ausgehend von sich selbst und von jedem seiner s1 t1 Nachbarn und gibt anschließend die Nachricht an s2 einen Nachbarn weiter, der näher am Ziel liegt als der Knoten selbst. Abbildung 1 zeigt einen geo- metrisch eingebetteten Graphen sowie einen mög- Abbildung 1: Greedy Routing lichen Pfad beim Greedy Routing zwischen dem ist erfolgreich zwischen dem Startknoten s1 und Ziel t1. Greedy Routing ist bei- Start s1 und Ziel t1 (roter Pfad). spielsweise einer der beiden Routing-Modi im Pro- Knoten s2 ist ein lokales Mini- tokoll GPSR (Greedy Perimeter Stateless Routing) für mum für das Ziel . t2 drahtlose Sensornetze. Das Grundproblem von Greedy Routing ist, t2 dass Nachrichten in lokalen Minima stecken blei- ben können, wo kein Nachbarknoten näher am Ziel liegt (siehe zum Beispiel Abbildung 1 für den Startknoten s2 und Zielknoten t2). Für einen gegebenen Graphen bestimmt die Wahl der Knotenkoordinaten die Erfolgsrate von s2 Greedy Routing. Für den Graphen in Abbildung 1 Abbildung 2: Eine Einbettung ist eine andere Koordinatenzuweisung in Abbil- des Graphen aus Abbildung 1 dung 2 dargestellt. In der letzteren Einbettung ist mit einer anderen Zuweisung Greedy Routing von jedem Start- zu jedem Ziel- der Knotenkoordinaten. Identi- sche Knoten haben die gleiche knoten erfolgreich. Grapheinbettungen mit die- Farbe in beiden Abbildungen. ser Eigenschaft werden Greedy-Einbettungen bzw. Greedy Routing ist nun immer Greedy-Zeichnungen genannt. Die Untersuchung erfolgreich. vii . der Greedy-Einbettungen wird in der Literatur durch Routing in drahtlosen Sen- sornetzwerken motiviert. Drahtlose Sensornetzwerke bzw. Sensornetze (engl. wireless sensor networks) sind Netzwerke von kleinen mit Sensoren ausgestatteten Rechenkno- ten. Die Knoten sind räumlich verteilt und können untereinander drahtlos kommunizieren. Obwohl Senke einzelne Knoten typischerweise nur über begrenz- te Rechenkapazitäten sowie begrenzte Batterien verfügen, können die Knoten ein Netzwerk bil- Abbildung 3: Sensornetze wer- den und eine Aufgabe in Zusammenarbeit erfül- den zur Erkennung von Wald- len. Sie können beispielsweise Temperatur, Feuch- bränden verwendet. tigkeit, Konzentration von Kohlenmonoxid in der Luft usw. überwachen und diese Daten an eine Basisstation weiterleiten als Teil eines Systems, das Waldbrände erkennt und überwacht (siehe Abbildung 3). Ein solches Netzwerk kann seine Aufgabe weiterführen, auch wenn einige Knoten ausfallen. Anwendungsgebiete von Sensornetzen sind Militär, Umwelt, Gesund- heitswesen und Sicherheit. Die Vision, wie man Greedy-Einbettungen für das Routing in Sensornetzen anwenden könnte, wird in der Literatur wie folgt geschildert. Das Sensornetz be- rechnet eine Greedy-Einbettung von seinem Kommunikationsgraphen und teilt jedem Knoten seine eigenen Koordinaten in dieser Einbettung mit (die sogenann- ten virtuellen Koordinaten) sowie die virtuellen Koordinaten der Nachbarknoten. Enthält nun jede Nachricht die virtuellen Koordinaten des Zielknotens, kann je- der Knoten die Kenntnis seiner virtuellen Koordinaten und der seiner Nachbarn nutzen, um die Nachricht mittels Greedy Routing weiterzuleiten. Da die virtuel- len Koordinaten aus einer Greedy-Einbettung stammen, ist Greedy Routing nun immer erfolgreich. In dieser Arbeit untersuche ich die Realisierbarkeit dieser Visi- on aus dem Blickwinkel der Graphentheorie und erhalte neue Erkenntnisse über die Frage, welche Graphen eine Greedy-Einbettung zulassen. Routing in Sensornetzen ist nicht die einzige Motivation für die Untersuchung von Greedy- Einbettungen von Graphen und anderen verwand- ten Einbettungsarten. Kriterien wie etwa mög- lichst wenige lokale Minima spielen eine Rolle, wenn eine Netzwerkzeichnung Nutzern helfen Abbildung 4: Eine Increasing- soll, Pfade im Netzwerk zu finden. Dazu wurde in Chord-Graphzeichnung. Für je- den letzten Jahren eine Reihe von verschiedenen den Start- und Zielknoten exis- Zeichnungskonventionen vorgeschlagen, nämlich tiert immer ein Pfad, entlang die bereits erwähnten Greedy-Zeichnungen sowie dessen Kanten die Distanz zum Ziel kontinuierlich abnimmt. (stark) monotone, Self-Approaching- und Increasing- Chord-Zeichnungen (siehe Abbildung 4). Ich fasse das Problem, eine für die Pfad- suche geeignete Netzwerkeinbettung zu konstruieren, wie folgt auf: Finde für einen gegebenen Graphen Knotenkoordinaten in R2, die verwendet werden kön- viii . nen, um auf dem Graphen mit lokalen Entscheidungen zu routen, und sodass man Pfade finden kann, die immer Fortschritte in Richtung ihrer Ziele machen. Dies ist das zentrale Problem, das in dieser Arbeit untersucht wird. überblick und beitrag Ich betrachte mehrere Arten von Graphzeichnungen, die durch Greedy Routing auf geometrisch eingebetteten Graphen motiviert sind. Das zentrale Problem, das ich untersucht habe, ist, zu verstehen, welche Graphen eine Greedy-, Self- Approaching- oder Increasing-Chord-Zeichnung zulassen. Meine Arbeit erweitert den aktuellen Kenntnisstand zu dieser Frage um neue Erkenntnisse. Auf dem Weg zu einer vollständigen Charakterisierung von Graphen, die solche Zeichnungen zulassen, konzentriere ich mich auf gängige und wichtige Graphklassen wie Bäu- me, Triangulierungen und dreifach-zusammenhängende planare Graphen, die in diesem Forschungsbereich häufig betrachtet werden. Außerdem untersuche ich die Komplexität des Problems, Polygone und Graph- zeichnungen in Teilbereiche zu zerlegen, die Greedy Routing unterstützen (siehe Abbildung 5). Dieses Zerlegungsproblem entstammt direkt aus einem für drahtlo- se Sensornetzwerke vorgeschlagenen Routing-Algorithmus (Tan und Kermarrec, IEEE/ACM Trans. Networking 20.3 (2012), 864–877) und ist stark verbunden mit Increasing-Chord-Zeichnungen. Euklidische Greedy-Zeichnungen von Bäumen Im Zusammenhang mit dem Einbetten von Graphen in R2, um Greedy Routing zu unterstützen, ist folgendes Problem der „Heilige Gral“: Charakterisiere die Graphen, die eine Greedy-Zeichnung in R2 besitzen. Dieses Problem zog großes Interesse der Graph-Drawing-Gemeinschaft auf
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