Black Hole Search in the Network and Subway Models Matthew Kellett Thesis submitted to the Faculty of Graduate and Postdoctoral Studies in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Computer Science† School of Information Technology and Engineering Faculty of Engineering University of Ottawa © Matthew Kellett, Ottawa, Canada, 2012 † The Ph.D. program is a joint program with Carleton University, administered by Ottawa-Carleton Institute for Computer Science. Abstract In this thesis we look at mobile agent solutions to black hole search and related problems. Mobile agents are computational entities that are autonomous, mobile, and can interact with their environment and each other. The black hole search problem is for a team of these agents to work together to map or explore a graph- like network environment where some elements of the network are dangerous to the agents. Most research into black hole search has focussed on finding a single dangerous node: a black hole. We look at the problem of finding multiple black holes and, in the case of dangerous graph exploration, multiple black links as well. We look at the dangerous graph exploration problem in the network model. The network model is based on a normal static computer network modelled as a simple graph. We give an optimal solution to the dangerous graph exploration problem using agents that start scattered on nodes throughout the network. We then make the problem more difficult by allowing an adversary to delete links during the execution of the algorithm and provide a solution using scattered agents. In the last decade or two, types of networks have emerged, such as ad hoc wireless networks, that are by their nature dynamic. These networks change quickly over time and can make distributed computations difficult. We look at black hole search in one type of dynamic network described by the subway model, ii iii which we base on urban subway systems. The model allows us to look at the cost of opportunistic movement by requiring the agents to move using carriers that follow routes among the network’s sites, some of which are black holes. We show that there are basic limitations on any solution to black hole search in the subway model and prove lower bounds on any solution’s complexity. We then provide two optimal solutions that differ in the agents’ starting locations and how they communicate with one another. Our results provide a small window into the cost of deterministic distributed computing in networks that have dynamic elements, but which are not fully ran- dom. For Jennifer iv Acknowledgements My first thanks go to my supervisors, Paola Flocchini, Nicola Santoro, and Peter Mason. Collectively, they were incredibly supportive and helpful in guiding my studies. I could not have asked for a more agreeable and professional group of people. I would like to thank Paola for taking a chance on me five years ago when she agreed to supervise my master’s and for her unwavering support and encouragement since then. I would like to thank Nicola for agreeing to co-supervise me. It was honour both to be taught by and collaborate with one of the most senior people in the field. I would like to thank Peter for agreeing to represent Defence R&D Canada – Ottawa. As my thesis co-supervisor, he gave me invaluable insight into the scientific process. As my work supervisor, he gave me the leeway I needed to succeed at both work and school. I am proud to count all three of my supervisors not only as mentors but as friends. My studies would not have been possible without the support of Julie Lefebvre, my manager at DRDC Ottawa. I would like to thank her for taking a chance on me when it was not at all clear that I would succeed. Finally, I would like to thank my parents, George and Cherry Kellett, for their love and support. This thesis is dedicated to my partner (my wife), Jennifer Panek, with all my love. v Table of Contents Abstract ii Dedication iv Acknowledgementsv Table of Contents vi List of Figures ix List of Tablesx List of Algorithms xi 1 Introduction1 1.1 Our contributions...........................6 1.2 Thesis organization.......................... 10 2 Definitions, Models, and Problems 12 2.1 Definitions................................ 13 2.1.1 Mobile agents............................ 14 2.1.2 Black holes............................. 15 2.2 Network model and problems.................... 17 2.3 Subway model and problems..................... 21 3 Related Work 28 3.1 Distinguishing assumptions..................... 29 3.2 Exploration............................... 35 3.2.1 Maze solving and regular planar graphs............ 37 3.2.2 Directed graphs.......................... 38 3.2.3 Trees and rings.......................... 40 3.2.4 Undirected graphs......................... 43 3.3 Black hole search........................... 46 vi TABLE OF CONTENTS vii 3.3.1 Asynchronous arbitrary networks............... 47 3.3.2 Synchronous arbitrary networks................ 49 3.3.3 Specific topologies......................... 51 3.4 Dynamic networks........................... 54 3.5 Other mobile agent problems..................... 57 4 Scattered Agent DGE 60 4.1 Model.................................. 61 4.2 Algorithm................................ 62 4.2.1 Overview.............................. 62 4.2.2 Initialization............................ 63 4.2.3 Looking for work.......................... 63 4.2.4 Exploration............................. 65 4.2.5 Verification............................. 67 4.2.6 Merging............................... 71 4.2.7 Internal link verification problem................ 71 4.3 Correctness and complexity..................... 74 4.4 Conclusion............................... 81 5 Scattered Agent DGE with Link Deletions 82 5.1 Model.................................. 83 5.2 Algorithm................................ 84 5.2.1 Overview.............................. 85 5.2.2 Operations without deletions.................. 85 5.2.3 Operations with deletions.................... 93 5.3 Correctness and complexity..................... 96 5.4 Conclusion............................... 107 6 Subway BHS Limitations and Lower Bound 109 6.1 Model.................................. 111 6.2 Basic Limitations............................ 113 6.3 Lower bounds.............................. 118 6.4 Conclusion............................... 122 7 Subway BHS with Site Whiteboards 123 7.1 Model.................................. 124 7.2 Algorithm................................ 129 7.2.1 Overview.............................. 129 7.2.2 Example subway graph...................... 130 7.2.3 Initialization............................ 130 7.2.4 Do work............................... 133 7.2.5 Compete to add work....................... 137 TABLE OF CONTENTS viii 7.2.6 Find work.............................. 140 7.3 Correctness and complexity..................... 143 7.4 Conclusion............................... 146 8 Subway BHS with Carrier Whiteboards 148 8.1 Model.................................. 149 8.2 Algorithm................................ 151 8.2.1 Overview.............................. 151 8.2.2 Initialization............................ 153 8.2.3 Find work.............................. 154 8.2.4 Do work............................... 155 8.3 Correctness and complexity..................... 157 8.4 Conclusion............................... 167 9 Conclusion 169 9.1 Future work............................... 174 Bibliography 177 A Harry Potter without the Marauder’s Map 193 List of Figures 1.1 Example computer network.......................2 1.2 Thesis roadmap.............................. 11 2.1 Example network model graph..................... 20 4.1 Exploration of a link............................ 69 4.2 Verification of a link............................ 69 4.3 Chain of l 1 trees............................. 75 Å 5.1 Exploration of a link with no deletions................. 87 5.2 Internal link verification with no deletions............... 87 5.3 External link verification and merging with no deletions...... 90 5.4 Exploration of a link with one deletion................. 94 5.5 Internal link verification with one deletion............... 94 5.6 External link verification and merging with one deletion...... 95 5.7 Exploration, verification, and merging with a second deletion... 96 6.1 Transfer graph for Theorem 6.5..................... 119 7.1 Subway graph of example........................ 131 7.2 Transfer graph of example........................ 132 7.3 Carrier graphs for example........................ 132 7.4 Doing work in example.......................... 135 7.5 Competing to add work in example................... 138 7.6 Finding work in example......................... 142 ix List of Tables 1.1 Thesis and related results........................9 2.1 Network model notation......................... 19 2.2 Subway model notation.......................... 25 3.1 Related work notation........................... 29 4.1 Results related to Dge solution..................... 61 5.1 Results related to Dge-ld solution................... 83 6.1 Results related to lower bounds on subway Exp ........... 110 7.1 Results related to subway Bhs with site whiteboards........ 124 7.2 Example graph in subway model.................... 131 8.1 Results related to subway Bhs with carrier whiteboards...... 149 x List of Algorithms 4.1 ExploreDG ................................. 64 4.2 Look for work................................ 66 4.3 Grab root marker............................. 67 4.4