A GEOMETRIC ROUTING SCHEME IN WORD- METRIC SPACES FOR DATA NETWORKS Miguel Hernando Camelo Botero Dipòsit legal: Gi. 1844-2014 http://hdl.handle.net/10803/283749 ADVERTIMENT. L'accés als continguts d'aquesta tesi doctoral i la seva utilització ha de respectar els drets de la persona autora. Pot ser utilitzada per a consulta o estudi personal, així com en activitats o materials d'investigació i docència en els termes establerts a l'art. 32 del Text Refós de la Llei de Propietat Intel·lectual (RDL 1/1996). Per altres utilitzacions es requereix l'autorització prèvia i expressa de la persona autora. En qualsevol cas, en la utilització dels seus continguts caldrà indicar de forma clara el nom i cognoms de la persona autora i el títol de la tesi doctoral. No s'autoritza la seva reproducció o altres formes d'explotació efectuades amb finalitats de lucre ni la seva comunicació pública des d'un lloc aliè al servei TDX. 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DOCTORAL THESIS A GEOMETRIC ROUTING SCHEME IN WORD-METRIC SPACES FOR DATA NETWORKS Miguel Hernando Camelo Botero 2014 DOCTORAL THESIS A GEOMETRIC ROUTING SCHEME IN WORD-METRIC SPACES FOR DATA NETWORKS Miguel Hernando Camelo Botero 2014 DOCTORAL PROGRAM IN TECHNOLOGY Supervisors: Ph.D. Llu´ıs Fabrega` and Ph.D. Pere Vila` This thesis is presented in fulfillment of the requirement for the conferral of the degree of Doctor of Philosophy by the University of Girona Acknowledgments First at all, all the credit of this thesis is to God, who made all this possible. In the last years, many persons have influenced, encouraged, and helped me to the completion of this thesis. I would like to start my list of thanks with my supervisors Llu´ıs Fabrega` and Pere Vila` for the guidance, encouragements, support, patience and also for having faith in me. I also would like to thank Dimitri Papadimitriou, who hosted me at Alcatel Lucent Bell labs (Antwerp, Belgium) and provided rigorous and valuable feedback during all the stages of my research. I thank to Ramon´ Fabregat and Jose´ Lu´ıs Marzo, who gave me the opportunity to work in the Broadband Communication and Distributed Systems (BCDS) research group, and to Yezid Donoso for being always a guidance in all my academical processes. In this list, I can not forget my co-workers (and ex co-workers) at BCDS research group (Silvi, Carlitos, Fer, Nico, Ludy, Juan, Marc), who I had the opportunity of sharing good moments of friendship with them and interesting discussions and support about the work in this dissertation. I would like to include in this list to Oscar, Alejo, Fabis, Jito, Juanda and Isa who have been as my family during all this time in Girona and Alejo (kvi), Luis and Julian, my best friends who always have been there to support me. I am grateful to Montse, Cristina y Anna for sorting out conference trips, research stays and all the paperwork needed during this research work. I appreciate the valuable feedback that the anonymous reviewers of my PhD thesis document and of our papers have provided throughout these years. Finally, I would like to express the most special thank you to my family for all its unconditional love, which brings me the strength to achieve any personal goal, and to my lovely Christine, who now is an important part of my life and who give me the love, support and motivation to complete this process. This thesis is especially dedicated to them. This research work is partially funded by the European Commission (EULER project, FP7 258307), the Spanish Government (RoGER project, TEC 2012-32336), the Generalitat of Catalonia (CSI project, SGR-1202), and the SUR of the Generalitat of Catalonia and the European Social Fund (PhD grant FI-DGR 2011). iii iv List of Acronyms AuS Automatic Structure AS Autonomous System FSA Finite State Automata MWP minimum-length word problem AG Automatic Group SAS Shortlex Automatic Structure CG Cayley Graph ERT Equivalent Routing Table IA Index Automaton CRS Confluent Rewriting System WA Word-Acceptor Automaton GM General-Multiplier Automaton WD Word-Differences Automaton DC Data Center CRP Compact Routing Problem GGR Greedy Geometric Routing CR Compact Routing GR Geometric Routing BFS Breadth-First Search SPRT Shortest Path Routing in Trees GPS Global Positioning System v vi GPSR Greedy Perimeter Stateless Routing GOAFR Greedy Other Adaptive Face Routing GPGF Gravity-Pressure Greedy Forwarding GNP Global Network Positioning SWDC Small-World Datacenter APSP All Pairs Shortest Path SP Shortest Path MDST Minimum Diameter Spanning Tree MdST Minimum Degree Spanning Tree AMST Approximated Minimum Spanning Tree DFS Depth-First Search RT Routing Table BGP Border Gateway Protocol IR Internet Router CIDR Classless Inter-Domain Routing IoT Internet of Things SPRT Shortest Path Routing on Trees CDF Cumulative Distribution Function BA Barabasi-Albert preferential attachment WM Word-Metric GRH2 Greedy Geometric Routing in H2 HK Holme-Kim preferential attachment RTR Reducing Table Ratio C-GGR Compact Greedy Geometric Routing GF Greedy Forwarding List of Tables 3.1 State-of-art of Greedy Geometric Routing (GGR) schemes for both general and scale-free graphs............................ 36 3.2 State-of-art of Cayley Graph (CG) Data Center (DC) architectures and their routing scheme properties.......................... 39 4.1 Strategies for distributed computing of a rooted spanning tree from an undirected, unweigthed and connected graph................ 48 4.2 Main properties of some well-known DC topologies based on Cayley Graphs.................................... 52 5.1 Complexity upper bounds of the proposed Compact Greedy Geometric Routing (C-GGR) for scale-free graphs.................... 55 5.2 Properties of the Barabasi-Albert preferential attachment (BA) graphs and the diameter of the computed Breadth-First Search (BFS) tree........ 56 5.3 Mean and 95% CI of the stochastic topological parameters of the BA topologies................................... 57 5.4 Properties of the Holme-Kim preferential attachment (HK) graphs and the diameter of the computed BFS tree...................... 58 5.5 Mean and 95% CI of the stochastic topological parameters of the HK topologies................................... 58 5.6 Graph properties of the evaluated AS topologies.............. 59 5.7 Mean and 95% CI of stretch in the BA topologies.............. 60 5.8 Mean and 95% CI of stretch in the HK topologies.............. 64 5.9 Theoretical vs experimental stretch in the evaluated Internet Router (IR) level topologies................................ 68 5.10 Experimental vs theoretical maximum vertex label size in the evaluated IR level topologies.............................. 68 5.11 Theoretical vs experimental vs Shortest Path Routing Table (RT) size in the evaluated IR level topologies....................... 68 vii LIST OF TABLES viii 5.12 The value of n and the resulting number of vertices for H(k), BF(k), TP(k), BS(k), ST(k) and P(k)........................ 70 List of Figures 1.1 The Internet topology represented at different level: a) Internet Router and b) Autonomous Systems.........................2 1.2 The growth of the Routing Table (RT) between 1994 and 2014.......3 1.3 A Cloud Computing Infrastructure......................4 3.1 The embedding of the 3-cube graph in several metric spaces........ 31 4.1 An example of graph embedding in the Word-Metric (WM) metric space for Greedy Geometric Routing (GGR).................... 45 5.1 Stretch Cumulative Distribution Function (CDF) for WM, Greedy Geometric Routing in H2 (GRH2) and Shortest Path Routing on Trees (SPRT) on a Barabasi-Albert preferential attachment (BA) graph with 1000 nodes..................................... 59 5.2 Stretch as a function of the number of nodes on BA graphs......... 60 5.3 Average path stretch as a function of the number of nodes on BA graphs.. 61 5.4 Stretch CDF for WM GGR scheme in several BA graphs........... 61 5.5 Theoretical upper bound vs experimental vertex label size on BA graphs.