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/l chains of length l. In this process, the balls of one chain have to be allocated to l 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/l))/ln d +O(1) with probability 1-O(1/lnln(n/l)). 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, the maximum load of an off-line assignment can be upper bounded by one. This holds, for example, for m