Parallel Bayesian Search with No Coordination

• Published in: Journal of the ACM Vol. 66; no. 3; pp. 1 - 28
• Main Authors: Fraigniaud, Pierre, Korman, Amos, Rodeh, Yoav
• Format: Journal Article
• Language: English
• Published: New York, NY, USA ACM 01.06.2019; Association for Computing Machinery
• Subjects: Algorithms; Bayesian analysis; Boxes; Computer Science; Coordination compounds; Crashes; Design and analysis of algorithms; Distributed, Parallel, and Cluster Computing; Integers; Parallel algorithms; Radio signals; Random search heuristics; Randomness, geometry and discrete structures; Run time (computers); Search algorithms; Self-organization; Theory of computation; treasure hunt; Bayesian search; fault tolerance; parallel computing; randomized algorithms; distributed computing
• ISSN: 0004-5411; 1557-735X
• DOI: 10.1145/3304111

Coordinating the actions of agents (e.g., volunteers analyzing radio signals in SETI@home) yields efficient search algorithms. However, such an efficiency is often at the cost of implementing complex coordination mechanisms which may be expensive in terms of communication and/or computation overheads. Instead, non-coordinating algorithms, in which each agent operates independently from the others, are typically very simple, and easy to implement. They are also inherently robust to slight misbehaviors, or even crashes of agents. In this article, we investigate the “price of non-coordinating,” in terms of search performance, and we show that this price is actually quite small. Specifically, we consider a parallel version of a classical Bayesian search problem, where set of k≥1 searchers are looking for a treasure placed in one of the boxes indexed by positive integers, according to some distribution p. Each searcher can open a random box at each step, and the objective is to find the treasure in a minimum number of steps. We show that there is a very simple non-coordinating algorithm which has expected running time at most 4(1−1/k+1)2 OPT+10, where OPT is the expected running time of the best fully coordinated algorithm. Our algorithm does not even use the precise description of the distribution p, but only the relative likelihood of the boxes. We prove that, under this restriction, our algorithm has the best possible competitive ratio with respect to OPT. For the case where a complete description of the distribution p is given to the search algorithm, we describe an optimal non-coordinating algorithm for Bayesian search. This latter algorithm can be twice as fast as our former algorithm in practical scenarios such as uniform distributions. All these results provide a complete characterization of non-coordinating Bayesian search. The take-away message is that, for their simplicity and robustness, non-coordinating algorithms are viable alternatives to complex coordinating mechanisms subject to significant overheads. Most of these results apply as well to linear search, in which the indices of the boxes reflect their relative importance, and where important boxes must be visited first.