Solving ANTS with Loneliness Detection and Constant Memory Casey O'Brien

105 Pages·2015·0.6 MB·English

Solving ANTS with Loneliness Detection and
Constant Memory
by
Casey O’Brien
B.S., Computer Science, B.S., Mathematics
Massachusetts Institute of Technology (2014)
Submitted to the Department of Electrical Engineering and Computer
Science
in partial fulfillment of the requirements for the degree of
Master of Engineering in Electrical Engineering and Computer Science
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
September 2015
○c Massachusetts Institute of Technology 2015. All rights reserved.
Author ................................................................
Department of Electrical Engineering and Computer Science
August 17, 2015
Certified by............................................................
Nancy Lynch
Professor of Electrical Engineering and Computer Science
Thesis Supervisor
Accepted by ...........................................................
Christopher J. Terman
Chairman, Department Committee on Graduate Theses
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Solving ANTS with Loneliness Detection and Constant
Memory
by
Casey O’Brien
Submitted to the Department of Electrical Engineering and Computer Science
on August 17, 2015, in partial fulfillment of the
requirements for the degree of
Master of Engineering in Electrical Engineering and Computer Science
Abstract
In 2012, Feinerman et al. introduced the Ants Nearby Treasure Search (ANTS)
problem [1]. In this problem, 𝑘 non-communicating agents with unlimited memory,
initially located at the origin, try to locate a treasure distance 𝐷 from the origin.
They show that if the agents know 𝑘, then the treasure can be located in the optimal
𝑂(𝐷+𝐷2/𝑘) steps. Furthermore, they show that without knowledge of 𝑘, the agents
need Ω((𝐷 +𝐷2/𝑘)·log1+𝜖𝑘) steps for some 𝜖 > 0 to locate the treasure. In 2014,
Emek et al. studied a variant of the problem in which the agents use only constant
memorybutareallowedasmallamountofcommunication[2]. Specifically, theyallow
an agent to read the state of any agent sharing its cell. In this paper, we study a
variant of the problem similar to that in [2], but where the agents have even more
limited communication. Specifically, the only communication is loneliness detection,
in which an agent in able to sense whether it is the only agent located in its current
cell. To solve this problem we present an algorithm Hybrid-Search, which locates
the treasure in 𝑂(𝐷·log𝑘+𝐷2/𝑘) steps in expectation. While this is slightly slower
than the straightforward lower bound of Ω(𝐷 + 𝐷2/𝑘), it is faster than the lower
bound for agents locating the treasure without communication.
Thesis Supervisor: Nancy Lynch
Title: Professor of Electrical Engineering and Computer Science
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Acknowledgments
I would like to thank Mira Radeva for her help throughout the process of completing
this thesis. Our discussions led to many of the results presented in this paper, and
she helped me sort through many difficult proofs.
I would also like to thank my advisor, Professor Nancy Lynch, for the effort she
spent guiding me through this thesis. Her help was instrumental in developing the
workpresentedinthispaper, especiallyinformalizingalltheprobabilisticproofs. Her
comments on my many drafts helped me learn so much and were key in developing
my arguments.
Finally, I would like to thank all the others who took the time to discuss my
thesiswithmeandprovidevalueableinsight, specificallyCameronMuscoandMohsen
Ghaffari. I really appreciate the opportunity to have worked with this group.
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Contents
1 Introduction 11
1.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 Model 17
2.1 Agents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Executions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 The Rectangle-Search Algorithm 21
3.1 Separation Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.1.1 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.1.2 Correctness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2 Allocation Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2.1 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2.2 Correctness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.3 Search Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.3.1 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 64
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3.3.2 Correctness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.3.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.4 Putting it All Together . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4 Improving the Runtime 93
4.1 The Geometric-Search Algorithm . . . . . . . . . . . . . . . . . . . . 93
4.1.1 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.1.2 Correctness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.1.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.2 The Hybrid-Search Algorithm . . . . . . . . . . . . . . . . . . . . . . 97
4.2.1 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.2.2 Correctness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.2.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5 Conclusion 103
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List of Figures
2-1 Center and Axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3-1 Region for Separation Phase . . . . . . . . . . . . . . . . . . . . . . . 30
3-2 Path for Part A of the Allocation Phase . . . . . . . . . . . . . . . . 41
3-3 Path for Part B of the Allocation Phase . . . . . . . . . . . . . . . . 42
3-4 Sample Execution of Part B of the Allocation Phase . . . . . . . . . 47
3-5 Allocation Phase North Guide Protocol . . . . . . . . . . . . . . . . . 53
3-6 Allocation Phase West Guide and Explorer Protocol . . . . . . . . . . 54
3-7 Region for the Allocation Phase . . . . . . . . . . . . . . . . . . . . . 61
3-8 Exploring Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3-9 Region for the Search Phase . . . . . . . . . . . . . . . . . . . . . . . 79
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