CDAM Research Report, LSE-CDAM-2007-34
Balanced Allocations: Balls-into-Bins Revisited and Chains-into-Bins
Tugkan Batu, Petra Berenbrink, and Colin CooperThe 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
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