# Balanced allocations: balls-into-bins revisited and chains-into-bins

Batu, Tugkan ORCID: 0000-0003-3914-4645, Berenbrink, Petra and Cooper, Colin (2007) Balanced allocations: balls-into-bins revisited and chains-into-bins. LSE-CDAM-2007-34. London School of Economics and Political Science, London, UK.

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## Abstract

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<0.97677 n and d=4.

Item Type: Monograph (Report) http://www.cdam.lse.ac.uk/Reports/ © 2007 London School of Economis and Political Science Mathematics H Social Sciences > H Social Sciences (General) 09 Jul 2008 08:30 16 May 2024 13:12 http://eprints.lse.ac.uk/id/eprint/6722