Randomized allocation processes

Abstract

Many dynamic resource allocation and on-line load balancing problems can be modeled by processes that sequentially allocate balls into bins. The balls arrive one by one and are to be placed into bins on-line without using a centralized controller. If n balls are sequentially placed into n bins by placing each ball in a randomly chosen bin, then it is widely known that the maximum load in bins is ln n/ ln ln n · 1 + o 1 with high probability. Azar, Broder, Karlin, and Upfal extended this scheme, so that each ball is placed sequentially into the least full of d randomly chosen bins. They showed that the maximum load of the bins reduces exponentially and is ln ln n/ ln d + 1 with high probability, provided d ≥ 2. In this paper we investigate various extensions of these schemes that arise in applications in dynamic resource allocation and on-line load balancing. Traditionally, the main aim of allocation processes is to place balls into bins to minimize the maximum load in bins. However, in many applications it is equally important to minimize the number of choices performed (the allocation time). We study adaptive allocation schemes that achieve optimal tradeoffs between the maximum load, the maximum allocation time, and the average allocation time. We also investigate allocation processes that may reallocate the balls. We provide a tight analysis of a natural class of processes that each time a ball is placed in one of d randomly chosen bins may move balls among these d bins arbitrarily. Finally, we provide a tight analysis of the maximum load of the off-line process in which each ball may be placed into one of d randomly chosen bins. We apply this result to competitive analysis of on-line load balancing processes.

Multiple-Choice Balanced Allocation in (Almost) Parallel

“…For example, if one allocates sequentially n balls into n bins by choosing d ≥ 2 random bins for each ball and then places the ball into the lesser loaded of the chosen bins (such a scheme will be denoted by GREEDY [d]), then no bin will have load greater than ln ln n ln d + O(1) with high probability (w.h.p. 5 ) [3], which compares favorably to the allocation of the balls performed i.u.r. (independently and uniformly at random), where the maximum load is Θ( ln n ln ln n ), w.h.p.…”

“…The classical processes allocating balls into random bins (the single-choice schemes) have been surveyed, e.g., in [8,10], and used in many areas of mathematics, computer science, and engineering. The multiple-choice schemes have been used in these areas and in various settings, e.g., in adaptive load sharing [7], PRAM simulations [9], load balancing [3], and numerous follow-up papers, e.g., [1,4,5,14].…”

Self-stabilizing repeated balls-into-bins

“…Finally, [35] provided a general framework to study the stabilization time of asynchronous balls-into-bins processes, and [36] investigated the generalization of the 2-choices balls-into-bins process in which at each round a ball is re-launched by selecting d bins u.a.r. and placing the ball in the one with the smallest load.…”

Tight bounds for parallel randomized load balancing

“…3 Given that contacted bins are chosen u.i.r., they prove that this bound is stochastically optimal in the sense that any other strategy to assign the balls majorizes 4 their approach. The expected number of bins each ball queries during the execution of the algorithm was later improved to 1 + ε (for any constant ε > 0) by Czumaj and Stemann [9]. This is achieved by placing each ball immediately if the load of an inspected bin is not too large, rather than always querying d bins.…”