Lower Bounds for Searching Robots, some Faulty

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 $\lambda |d|$, if the target is located at $d$, $|d| \ge 1$, for $\lambda$ as small as possible. We show that it cannot be achieved for $\lambda < 2\frac{(1+\rho)^{1+\rho}}{\rho^\rho}+1$, $\rho := 2\frac{f+1}{k} -1$, which is tight due to earlier work. This also gives some better than previously known lower bounds for so-called Byzantine-type faulty robots (that may actually wrongly report a target).