Title: Lower Bounds for Searching Robots, some Faulty,
Abstract: 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 this cannot be achieved for
$$\lambda < 2\frac{\rho^\rho}{(\rho-1)^{\rho-1}}+1,~~ \rho :=
\frac{2(f+1)}{k}~, $$ 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 $\rho := m(f+1)/k$). 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.