Balanced allocation on hypergraphs

Abstract

We consider a variation of balls-into-bins which randomly allocates m balls into n bins. Following Godfrey's model (SODA, 2008), we assume that each ball t, 1⩽t⩽m, comes with a hypergraph H(t)={B1,B2,…,Bst}, and each edge B∈H(t) contains at least a logarithmic number of bins. Given d⩾2, our d-choice algorithm chooses an edge B∈H(t), uniformly at random, and then chooses a set D of d random bins from the selected edge B. The ball is allocated to a least-loaded bin from D. We prove that if the hypergraphs H(1),…,H(m) satisfy a balancedness condition and have low pair visibility, then after allocating m=Θ(n) balls, the maximum load of any bin is at most logd⁡log⁡n+O(1), with high probability. 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.