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, \(p\in (0,1/2)\). 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 \(p\rightarrow 0\), has optimal performance \(4.59112+\epsilon \), up to the additive term \(\epsilon \) that can be arbitrarily small. Additionally, it has performance less than 9 for \(p\le 0.390388\). When \(p\rightarrow 1/2\), our algorithm has performance \(\Theta (1/(1-2p))\), which we also show is optimal up to a constant factor.
Second, we consider linear search with two p-faulty agents, \(p\in (0,1/2)\), for which we provide three algorithms of different advantages, all with a bounded competitive ratio even as \(p\rightarrow 1/2\). 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 \(9+\epsilon \) (which we show can be achieved with arbitrarily high concentration) or a competitive ratio of \(4.59112+\epsilon \). Our final contribution is a novel algorithm for searching with two p-faulty agents that achieves a competitive ratio \(3+4\sqrt{p(1-p)}\), with arbitrarily high concentration.
Research supported in part by NSERC, and by the Fields Institute for Research in Mathematical Sciences
Extended Abstract—The full version of this paper appears on arXiv [17].
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Notes
- 1.
For simplicity, we only give numerical bounds on p. All mentioned bounds of p in this section have explicit algebraic representations that will be discussed later.
- 2.
If one wants to use the original definition of the competitive ratio, then by properly re-scaling the searched space (just by scaling the intended turning points), one can achieve competitive ratio which is additively off by at most \(\epsilon \) from the achieved value, for any \(\epsilon >0\).
- 3.
The Lambert W-Function is the inverse function of \(L(x)=x e^x\).
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Georgiou, K., Giachoudis, N., Kranakis, E. (2023). Overcoming Probabilistic Faults in Disoriented Linear Search. In: Rajsbaum, S., Balliu, A., Daymude, J.J., Olivetti, D. (eds) Structural Information and Communication Complexity. SIROCCO 2023. Lecture Notes in Computer Science, vol 13892. Springer, Cham. https://doi.org/10.1007/978-3-031-32733-9_23
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