{"title":"Overcoming probabilistic faults in disoriented linear search","authors":"","doi":"10.1016/j.tcs.2024.114761","DOIUrl":null,"url":null,"abstract":"<div><p>We consider search by mobile agents for a hidden, idle target, placed on the infinite line. Feasible solutions are agent trajectories in which all agents reach the target sooner or later. A special feature of our problem is that the agents are <em>p</em>-faulty, meaning that every attempt to change direction is an independent Bernoulli trial with known probability <em>p</em>, where <em>p</em> is the probability that a turn fails. We are looking for agent trajectories that minimize the worst-case expected termination time, relative to the distance of the hidden target to the origin (competitive analysis). Hence, searching with one 0-faulty agent is the celebrated linear search (cow-path) problem that admits optimal 9 and 4.59112 competitive ratios, with deterministic and randomized algorithms, respectively.</p><p>First, we study linear search with one deterministic <em>p</em>-faulty agent, i.e., with no access to random oracles, <span><math><mi>p</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>)</mo></math></span>. For this problem, we provide trajectories that leverage the probabilistic faults into an algorithmic advantage. Our strongest result pertains to a search algorithm (deterministic, aside from the adversarial probabilistic faults) which, as <span><math><mi>p</mi><mo>→</mo><mn>0</mn></math></span>, has optimal performance <span><math><mn>4.59112</mn><mo>+</mo><mi>ϵ</mi></math></span>, up to the additive term <em>ϵ</em> that can be arbitrarily small. Additionally, it has performance less than 9 for <span><math><mi>p</mi><mo>≤</mo><mn>0.390388</mn></math></span>. When <span><math><mi>p</mi><mo>→</mo><mn>1</mn><mo>/</mo><mn>2</mn></math></span>, our algorithm has performance <span><math><mi>Θ</mi><mo>(</mo><mn>1</mn><mo>/</mo><mo>(</mo><mn>1</mn><mo>−</mo><mn>2</mn><mi>p</mi><mo>)</mo><mo>)</mo></math></span>, which we also show is optimal up to a constant factor.</p><p>Second, we consider linear search with two <em>p</em>-faulty agents, <span><math><mi>p</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>)</mo></math></span>, for which we provide three algorithms of different advantages, all with a bounded competitive ratio even as <span><math><mi>p</mi><mo>→</mo><mn>1</mn><mo>/</mo><mn>2</mn></math></span>. Indeed, for this problem, we show how the agents can simulate the trajectory of any 0-faulty agent (deterministic or randomized), independently of the underlying communication model. As a result, searching with two agents allows for a solution with a competitive ratio of <span><math><mn>9</mn><mo>+</mo><mi>ϵ</mi></math></span> (which we show can be achieved with arbitrarily high concentration) or a competitive ratio of <span><math><mn>4.59112</mn><mo>+</mo><mi>ϵ</mi></math></span>. Our final contribution is a novel algorithm for searching with two <em>p</em>-faulty agents that achieves a competitive ratio <span><math><mn>3</mn><mo>+</mo><mn>4</mn><msqrt><mrow><mi>p</mi><mo>(</mo><mn>1</mn><mo>−</mo><mi>p</mi><mo>)</mo></mrow></msqrt></math></span>, in expectation and with arbitrarily high concentration.</p></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Theoretical Computer Science","FirstCategoryId":"94","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0304397524003785","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0
Abstract
We consider search by mobile agents for a hidden, idle target, placed on the infinite line. Feasible solutions are agent trajectories in which all agents reach the target sooner or later. A special feature of our problem is that the agents are p-faulty, meaning that every attempt to change direction is an independent Bernoulli trial with known probability p, where p is the probability that a turn fails. We are looking for agent trajectories that minimize the worst-case expected termination time, relative to the distance of the hidden target to the origin (competitive analysis). Hence, searching with one 0-faulty agent is the celebrated linear search (cow-path) problem that admits optimal 9 and 4.59112 competitive ratios, with deterministic and randomized algorithms, respectively.
First, we study linear search with one deterministic p-faulty agent, i.e., with no access to random oracles, . For this problem, we provide trajectories that leverage the probabilistic faults into an algorithmic advantage. Our strongest result pertains to a search algorithm (deterministic, aside from the adversarial probabilistic faults) which, as , has optimal performance , up to the additive term ϵ that can be arbitrarily small. Additionally, it has performance less than 9 for . When , our algorithm has performance , which we also show is optimal up to a constant factor.
Second, we consider linear search with two p-faulty agents, , for which we provide three algorithms of different advantages, all with a bounded competitive ratio even as . Indeed, for this problem, we show how the agents can simulate the trajectory of any 0-faulty agent (deterministic or randomized), independently of the underlying communication model. As a result, searching with two agents allows for a solution with a competitive ratio of (which we show can be achieved with arbitrarily high concentration) or a competitive ratio of . Our final contribution is a novel algorithm for searching with two p-faulty agents that achieves a competitive ratio , in expectation and with arbitrarily high concentration.
期刊介绍:
Theoretical Computer Science is mathematical and abstract in spirit, but it derives its motivation from practical and everyday computation. Its aim is to understand the nature of computation and, as a consequence of this understanding, provide more efficient methodologies. All papers introducing or studying mathematical, logic and formal concepts and methods are welcome, provided that their motivation is clearly drawn from the field of computing.