arXiv 1707.050772017-07cs.ROcs.DCcs.DMmath.CO

搜索机器人的下界，一些故障

Lower Bounds for Searching Robots, some Faulty

University of Birmingham（伯明翰大学）；Moscow Inst. of Physics and Technology（莫斯科物理技术学院）；ETH Zurich（苏黎世联邦理工学院）

Andrey Kupavskii, Emo Welzl

备注：: Appears in the proceedings of PODC'18. Compared to the previous version, a generalization to m rays is added

摘要：

假设我们从$0$派出$k$机器人以恒定速度（带转弯）搜索真实线路，以找到未知位置的目标; $f$的机器人出现故障，这意味着它们在访问目标位置时未能报告目标（称为碰撞类型）。目标是在最多$λ|d|$时找到目标，如果目标位于$d$，则为$|d| \ge 1$，对于$λ$尽可能小。我们表明，这是不能实现的$$λ< 2\frac{ρ^ρ}{(ρ-1)^{ρ-1}}+1,~~ ρ:= \frac{2(f+1)}{k}~, $$，这是紧密的，由于早期的工作（见J. Czyzowitz，E. Kranakis，D.克里赞克湖Narayanan，J. Opatrny，PODC'16，其中介绍了这个问题）。这也为所谓的拜占庭式故障机器人提供了一些比以前更好的下限，这些机器人实际上可能会错误地报告目标。在本文的第二部分中，我们处理的$m$ -射线泛化的问题，其中隐藏的目标是检测$m$射线都在同一点发出。使用我们的方法的推广，以及对原始问题的有用放松，我们也为这个设置建立了一个紧下限（如上所述，使用$ρ:= m(f+1)/k$）。当专门用于$f=0$的情况下，这解决了关于并行搜索$m$射线的问题，由三组科学家在大约15到30年前提出：由Baeza-Yates，Culberson和Rawlins;由Kao，Ma，Sipser和Yin;由Bernstein，Finkelstein和Zilberstein。已知$m$射线的推广与其他看似无关的问题有联系，包括在线问题的混合算法和所谓的合同算法。

英文摘要：

Suppose we are sending out $k$ robots from $0$ to search the real line at constant speed (with turns) to find a target at an unknown location; $f$ of the robots are faulty, meaning that they fail to report the target although visiting its location (called crash type). The goal is to find the target in time at most $λ|d|$, if the target is located at $d$, $|d| \ge 1$, for $λ$ as small as possible. We show that this cannot be achieved for $$λ< 2\frac{ρ^ρ}{(ρ-1)^{ρ-1}}+1,~~ ρ:= \frac{2(f+1)}{k}~, $$ which is tight due to earlier work (see J. Czyzowitz, E. Kranakis, D. Krizanc, L. Narayanan, J. Opatrny, PODC'16, where this problem was introduced). This also gives some better than previously known lower bounds for so-called Byzantine-type faulty robots that may actually wrongly report a target. In the second part of the paper, we deal with the $m$-rays generalization of the problem, where the hidden target is to be detected on $m$ rays all emanating at the same point. Using a generalization of our methods, along with a useful relaxation of the original problem, we establish a tight lower for this setting as well (as above, with $ρ:= m(f+1)/k$). When specialized to the case $f=0$, this resolves the question on parallel search on $m$ rays, posed by three groups of scientists some 15 to 30 years ago: by Baeza-Yates, Culberson, and Rawlins; by Kao, Ma, Sipser, and Yin; and by Bernstein, Finkelstein, and Zilberstein. The $m$-rays generalization is known to have connections to other, seemingly unrelated, problems, including hybrid algorithms for on-line problems, and so-called contract algorithms.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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