Balanced allocation on hypergraphs

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 7-t(t) = {B1, B2, ... , Bst }, and each edge B & ISIN; 7-t(t) contains at least a logarithmic number of bins. Given d 2, our d-choice algorithm chooses an edge B & ISIN; 7-t(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 7-t(1), ..., 7-t(m) satisfy a balancedness condition and have low pair visibility, then after allocating m = O(n) balls, the maximum load of any bin is at most logd log n + 0(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.& COPY; 2023 Elsevier Inc. All rights reserved.