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 \(\lambda |x|\), if the target is located at x, \(|x| \ge 1\), for \(\lambda \) as small as possible. We show that this cannot be achieved for

which is tight due to earlier work (see Czyzowitz et al. in Proc PODC’16, pp 405–414, 2016, 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 \(\rho := \nicefrac {m(f+1)}{k}\)). When specialized to the case \(f=0\), this resolves the question on parallel search on m rays, posed by three groups of scientists some 15–30 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.

假设我们从 0 开始派出k 个机器人以恒定速度（有转弯）搜索真实线路，以在未知位置找到目标；f的机器人有故障，这意味着它们在访问目标位置时未能报告目标（称为崩溃类型）。目标是最多及时找到目标\(\lambda |x|\)，如果目标位于x，\(|x| \ge 1\)，对于\(\lambda \)小到可能的。我们表明这无法实现

由于早期的工作，这很紧张（参见 Czyzowitz 等人在 Proc PODC'16, pp 405–414, 2016 中，在那里引入了这个问题）。这也为所谓的拜占庭型故障机器人提供了比以前已知的更好的下限，这些机器人实际上可能会错误地报告目标。在论文的第二部分，我们处理问题的m射线泛化，其中隐藏目标将在m射线上检测到，所有射线都在同一点发出。使用我们方法的推广，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\(\rho := \nicefrac {m(f+1)}{k }\) )。当专门用于案例\(f=0\) 时，这解决了关于并行搜索的问题m射线，由三组科学家在大约 15-30 年前提出：Baeza-Yates、Culberson 和 Rawlins；作者：Kao、Ma、Sipser 和 Yin；伯恩斯坦、芬克尔斯坦和齐伯斯坦。所述米射线泛化已知具有到其他，看似无关，问题，包括用于在线问题混合算法，和所谓的合同算法连接。