Searching with Two-Speed Autonomous Mobile Robots by Fraser Lachlan MACQUARRIE A thesis submitted to the Faculty of Graduate and Postdoctoral Affairs in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Computer Science CARLETON UNIVERSITY OTTAWA,ONTARIO c 2015 ⃝ Fraser Lachlan MacQuarrie Abstract We introduce and study various search problems using mobile robots. Specifically we study the problems of exploration and patrolling using a new two-speed model of robots, as well as studying the problems of rendezvous and evacuation using a more traditional one-speed model of mobile robots. The problems are executed on a continuous finite domain by a swarm of n robots r1; r2; : : : ; rn. For the two-speed model, each robot ri has a walking speed wi and a searching speed si, where si < wi. At each moment a robot either moves through the domain with a speed not exceeding wi, or it performs a more elaborate task (such as searching) on the domain with speed not exceeding si. A search of an interval is completed when each of its points have been searched by at least one of the n two-speed robots. We want to develop efficient mobility schedules (algorithms) for the robots which complete the search of the interval as quickly as pos- sible. More exactly, we want to maximize the speed of the mobility schedule – defined as the ratio of the interval length compared to the time of completion of the schedule. For patrolling, a unit interval is to be patrolled collectively by n two-speed robots. A robot patrols a portion of the domain by searching it. Each robot is allowed to patrol only in one of the two directions (not necessarily the same for all robots). We want to schedule the perpetual movements of the robots to minimize the idle time, defined as the maximal time interval any point is not visited by some patrolling robot. We also study randomized rendezvous on a circle: Two one-speed robots begin at dif- ferent locations on a circle and want to minimize the time required to find each other. The robots have different speeds, are unaware of their own speed, are equipped with identical chronometers, and each have access to one or more random bits. Finally, we study the evacuation problem: Distributed on a unit circle are k exits; two identical one-speed robots are placed on the circle. The goal of the evacuation problem is to give an algorithm for the robots which minimizes the time required for both robots to reach an exit, in the worst case. ii Acknowledgements First and foremost, I would like to extend my deepest gratitude to my supervisor, Dr. Evangelos Kranakis, for his encouragement and advice during the course of my studies. His passion for computer science, extensive knowledge and positive outlook has been inspirational and helped me grow as researcher and person. I would also like to sincerely thank everyone who has collaborated on the research presented in this thesis. Finally, I am very thankful for the love and support of my friends and family who have given me the confidence and determination to achieve my academic and non-academic goals. iii Contents Abstract ii Acknowledgements iii Contents iv List of Tables vii List of Figures viii List of Algorithms ix 1 Introduction1 1.1 Searching with Two-Speed Robots....................2 1.2 Two-Speed Robot Model.........................4 1.3 Problem Domains.............................5 1.4 Problems Studied and Summary of Results................6 1.5 Related Work............................... 12 1.6 Publications................................ 22 2 Searching 25 2.1 Introduction................................ 25 2.1.1 Preliminaries and Notation.................... 26 2.1.2 Search Problems......................... 28 2.1.3 Robot Model for Searching.................... 30 2.1.4 Outline and Results........................ 30 2.2 The Beachcombers’ Problem....................... 32 2.2.1 Properties of Optimal Schedules................. 32 2.2.2 The Optimal Schedule...................... 40 2.2.3 Closed Formulas for the Optimal Schedule........... 42 iv Contents v 2.3 The Online Beachcombers’ Problem................... 45 2.3.1 Competitiveness of the Online Searching............ 48 2.3.2 Online Searching with W-uniform swarms............ 51 2.3.3 Online Searching for Totally Uniform Swarms......... 56 2.4 The Multi-source Beachcombers’ Problem................ 60 2.4.1 t-SBP as a Partitioning Problem and its Hardness........ 60 2.4.1.1 A Deterministic Approximation Guarantee for 2-SBP 65 2.4.1.2 An Exact Solution for t-SBP with S-uniform Swarms 68 2.4.2 A Randomized Algorithm for t-SBP............... 72 2.5 Conclusion and Open Problems...................... 78 3 Patrolling 80 3.1 Introduction................................ 80 3.1.1 Preliminaries and Notation.................... 82 3.1.2 Patrolling Problems........................ 82 3.1.3 Robot Model for Patrolling.................... 84 3.1.4 Outline and Results........................ 85 3.2 Optimal Fence Patrolling Centralized Algorithm............. 86 3.2.1 Schedule Obtained by a Centralized Algorithm......... 86 3.2.2 Lower Bound of the Centralized Schedule............ 87 3.2.3 Application to Transportation: Scheduling with Regular Delivery 89 3.3 Fence Patrolling in the Distributed Model................ 90 3.3.1 Distributed Optimal Schedule for Two Robots.......... 91 3.3.2 Distributed Schedule for n Primitive Robots........... 95 3.3.3 Dynamics for Regular Systems of Primitive Robots....... 97 3.3.4 Monotone Robot Collections, and Convergence......... 104 3.4 Conclusion................................ 111 4 Rendezvous 112 4.1 Introduction................................ 112 4.1.1 Preliminaries and Notation.................... 113 4.1.2 Rendezvous Problem....................... 113 4.1.3 Robot Model for Rendezvous.................. 114 4.1.4 Outline and Results........................ 114 4.2 Single Random Bit............................ 115 4.3 k + 1 Random Bits............................ 124 4.4 Rendezvous and Knowledge....................... 133 4.4.1 Unknown u and L ........................ 134 4.4.2 Known u and L .......................... 134 4.4.3 Unknown u and Known L .................... 138 4.4.4 Known u and Unknown L .................... 140 4.5 Conclusion................................ 141 Contents vi 5 Evacuation 143 5.1 Introduction................................ 143 5.1.1 Preliminaries........................... 144 5.1.2 Evacuation Problems....................... 145 5.1.3 Robot Model for Evacuation................... 145 5.1.4 Outline and Results........................ 147 5.2 Evacuation with a Single Exit....................... 148 5.3 Evacuation with k Exits.......................... 154 5.4 k-exit Evacuation with Arbitrary D and L ................ 157 5.5 Conclusion................................ 166 6 Conclusion and Future Work 167 6.1 Discussion................................. 167 6.2 Open Problems.............................. 172 6.3 Future Work................................ 172 Abbreviations 176 Nomenclature 177 Bibliography 178 Index 187 List of Tables 2.1 Competitive ratio of LeapFrog for collections of robots of size 2,3,4.. 57 4.1 Summary of performance of randomized rendezvous algorithms for two robots in a ring............................... 115 5.1 Costs of EvacuateL(D) for the Evacuation Problem with Fixed Initial Distance.................................. 146 5.2 Costs of Evacuate(D) for the Evacuation Problem with Chosen Ini- tial Distance................................ 146 vii List of Figures 2.1 Reducing the time of the first searcher as described in Lemma 2.4.... 33 2.2 Reducing the time of subsequent searchers as described in Lemma 2.4.. 34 2.3 Search configurations described in Lemma 2.5.............. 36 2.4 Search configurations described in Lemma 2.6.............. 37 2.5 Schedules produced by (a) Comb and (b) LeapFrog........... 49 2.6 Competitive ratio of LeapFrog as a function of the number of robots. 60 2.7 Effect of swapping a and b on outer intervals............... 70 3.1 Two robots r1; r2 patrolling in opposite directions............. 91 3.2 Two robots r1; r2 perpetually alternate between x(0) and x(1)...... 95 3.3 Two time consecutive bounces between robots ri; ri+1.......... 98 4.1 Initial configurations Aa and Ba for distance a L between robots... 117 ≤ 5.1 Example of an evacuation problem with 4 exits.............. 144 5.2 Single exit evacuation of colocated robots................. 149 1 L 5.3 Two robots starting at distance L explore the circle up to time t = 4 + 2 . 152 5.4 k-exit evacuation of colocated robots................... 154 5.5 Transforming a 4-exit evacuation with equally spaced exits to a single exit problem................................ 157 5.6 k-exit evacuation for EvacuateL(D).................. 159 5.7 Performance of Evacuate(D) for k = 2 exits............. 163 viii List of Algorithms 2.1 Comb .................................... 41 2.2 SwarmSpeed ............................... 47 2.3 LeapFrog ................................. 47 2.4 Swap-and-Expand ........................... 66 2.5 GreedyPartition ........................... 69 2.6 ObliviousRandomized ........................ 73 3.1 CentralizedPatrolling ...................... 86 3.2 OnlinePatrolling .......................... 92 3.3 DistributedPatrolling ...................... 96 3.4 DecideConverge ............................ 103 4.1 Rendezvous0 ............................... 116 4.2 Rendezvous0(t)............................. 118 4.3 Rendezvous0(~t = (t1; t2; :