## CDAM: Computational, Discrete and Applicable Mathematics@LSE | |

## CDAM Research Report, LSE-CDAM-2007-34December 2007 |

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

** Tugkan Batu, Petra Berenbrink,
and Colin Cooper **

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 *

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