Parallel Linear Search with no Coordination for a Randomly Placed Treasure

TL;DR

This paper investigates the efficiency of non-coordinating parallel search algorithms for finding a randomly placed treasure, providing tight bounds on speed-up factors and demonstrating simple, memory-efficient algorithms that approach optimal performance.

Contribution

It introduces new non-coordinating algorithms with proven optimal speed-ups for multiple searchers in a random placement setting, improving understanding of coordination-free search efficiency.

Findings

Achieves a speed-up of $k(k+1)/(3k-1)$ for $k$ searchers.

Provides tight bounds showing no better speed-up is possible without coordination.

Develops simple algorithms using logarithmic memory that are asymptotically optimal.

Abstract

In STOC'16, Fraigniaud et al. consider the problem of finding a treasure hidden in one of many boxes that are ordered by importance. That is, if a treasure is in a more important box, then one would like to find it faster. Assuming there are many searchers, the authors suggest that using an algorithm that requires no coordination between searchers can be highly beneficial. Indeed, besides saving the need for a communication and coordination mechanism, such algorithms enjoy inherent robustness. The authors proceed to solve this linear search problem in the case of countably many boxes and an adversary placed treasure, and prove that the best speed-up possible by $k$ non-coordinating searchers is precisely $\frac{k}{4}(1+1/k)^2$. In particular, this means that asymptotically, the speed-up is four times worse compared to the case of full coordination. We suggest an important variant of the problem, where the treasure is placed uniformly at random in one of a finite, large, number of boxes. We devise non-coordinating algorithms that achieve a speed-up of $6/5$ for two searchers, a speed-up of $3/2$ for three searchers, and in general, a speed-up of $k(k+1)/(3k-1)$ for any $k \geq 1$ searchers. Thus, as $k$ grows to infinity, the speed-up approaches three times worse compared to the case of full coordination. Moreover, these bounds are tight in a strong sense as no non-coordinating search algorithm for $k$ searchers can achieve better speed-ups. We also devise non-coordinating algorithms that use only logarithmic memory in the size of the search domain, and yet, asymptotically, achieve the optimal speed-up. Finally, we note that all our algorithms are extremely simple and hence applicable.