Balanced Allocations: Balls-into-Bins Revisited and Chains-into-Bins

The study of balls-into-bins games or occupancy problems has a long history since these processes can be used to translate realistic problems into mathematical ones in a natural way. In general, the goal of a balls-into-bins game is to allocate a set of independent objects (tasks, jobs, balls) to a set of resources (servers, bins, urns) and, thereby, to minimize the maximum load. In this paper we show two results. First, we analyse the maximum load for the chains-into-bins problem where we have n bins and the balls are connected in n/` chains of length `. In this process, the balls of one chain have to be allocated to ` consecutive bins. We allow each chain d i.u.r. bin choices. The chain is allocated using the rule that the maximum load of any bin receiving a ball of that chain is minimized. We show that, for d ≥ 2, the maximum load is (ln ln(n/`))/ ln d + O(1) with probability 1 − O(1/ ln ln(n/`)). This shows that the maximum load is decreasing with increasing chain length. Secondly, we analyse for which number of random choices d and which number of balls m < n, the maximum load of an off-line assignment can be upper bounded by one. This holds, for example, for m < 0.97677 · n and d = 4. Department of Mathematics, London School of Economics, London WC2A 2AE, UK. Email: [email protected]. School of Computing Science, Simon Fraser University, Burnaby, BC V5A 1S6, Canada. Email: [email protected]. Department of Computer Science, King’s College, University of London, London WC2R 2LS, UK. Email: [email protected].