2 research outputs found
Balanced Allocation on Hypergraphs
We consider a variation of balls-into-bins which randomly allocates balls
into bins. Following Godfrey's model (SODA, 2008), we assume that each ball
, , comes with a hypergraph
, and each edge
contains at least a logarithmic number of bins. Given
, our -choice algorithm chooses an edge ,
uniformly at random, and then chooses a set of random bins from the
selected edge . The ball is allocated to a least-loaded bin from , with
ties are broken randomly. We prove that if the hypergraphs
satisfy a \emph{balancedness}
condition and have low \emph{pair visibility}, then after allocating
balls, the maximum number of balls at any bin, called the
\emph{maximum load}, is at most , with high probability. The
balancedness condition enforces that bins appear almost uniformly within the
hyperedges of , , while the pair visibility
condition measures how frequently a pair of bins is chosen during the
allocation of balls. Moreover, we establish a lower bound for the maximum load
attained by the balanced allocation for a sequence of hypergraphs in terms of
pair visibility, showing the relevance of the visibility parameter to the
maximum load. In Godfrey's model, each ball is forced to probe all bins in a
randomly selected hyperedge and the ball is then allocated in a least-loaded
bin. Godfrey showed that if each , , is
balanced and , then the maximum load is at most one, with high
probability. However, we apply the power of choices paradigm, and only
query the load information of random bins per ball, while achieving very
slow growth in the maximum load