Distributed cooperation in the absence of communication (brief announcement)

This work studies a distributed cooperation problem under an extreme assumption that no two processors may be able to communicate during a prolonged period of time. For problems where the quality of distributed decision-making depends on, and can be traded for, communication, the solution space needs to consider the possibility of no communication. Notably, this is the case in the load-balancing setting introduced by Papadimitriou and Yanakakis [PY] and studied by Georgiades, Mavronicolas and Spirakis [GMS]. The distributed cooperation problem that we consider here is defined for n processors in terms of t tasks that need to be performed efficiently and that are known to all processors. The fact that the tasks are initially known makes it possible for the problem to be solved in the absence of communication. The efficiency requirement and the possibility of eventual availability of communication make it desirable to structure the work of the processors so that when eventually some processors are able to communicate, the amount of wasted (redundant) work they have collectively performed prior to that time is controlled. We model solutions to the problem as sets of n lists of distinct tasks from {1,… ,t}. We call such lists schedules. We define and study the notion of k-waste that, for a set of n schedules, measures the maximum number of redundant task identifiers contained in any subset of k (≤ n) schedules. We are interested in expressing k-waste as a function of the length of schedules. Our goal is to construct n schedules of length t such that k-waste is controlled for any prefixes of the schedules.Dolev et al. [DSS] developed a solution to this problem where each processor performs up to T(3√n) tasks and at most one redundant task is performed for any pair of processors (i.e., 2-waste is equal to 1). In our work we establish a lower bound for the class of schedules considered in [DSS]. Assuming the most challenging case n = t, we show that for schedules longer than √n the number of redundant tasks for two (or more) processors must be at least two. We also show that for schedules of length r, 2-waste must be O(r2/n), i.e., the redundant work must grow quadratically in the length of schedules. We present a deterministic construction of schedules of length √n, more precisely of length √n - 3/4 + 1/2, such that exactly one redundant task is performed for any pair of processors. This result exhibits an interesting connection between design theory and the distributed problem we consider. Design theory has already been applied to derive solutions to several problems in computing, e.g., see Karp and Wigderson [KW]. However in our case design theory offers little insight on how to extend schedules into longer schedules in which waste is increased in a controlled fashion. We show that longer schedules with controlled waste can be constructed in time linear in the length of the schedule. Our novel deterministic construction yields schedules of length 4/9n such that pairwise wasted work increases gradually as processors progress through their schedules.For each pair of processors p1, p2, the overlap of the first t1 tasks of processor p1 and the first t2 tasks of processor p2 is bounded by O (t1t2/n + √n). The quadratic growth in the overlap is in fact anticipated by our lower bound. The schedules are constructed by each processor independently such that the order of tasks is precomputed in √n-size segments and each segment is computed in O(√n) time. Thus the overall construction takes linear time and, except for the first √n tasks, the cost of constructing the schedule is completely amortized. Finally we explore the behavior of schedules constructed at random.For the case of pairwise waste, we show that with high probability these random schedules enjoy two satisfying properties: (i) for each pair of processors p1, p2, the overlap of the first t1 tasks of processor p1 and the first t2 tasks of processor p2 is no more than t1t2/t + O(log n + √(t1t2/t)log n), (ii) all but a vanishing fraction of the pairs of processors experience no more than a single redundant task in the first √t tasks of their schedules. The quadratic growth observed in property (i) above is anticipated by our lower bound.