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 hypergraph H(t)={B1,B2,…,Bst}, and edge B∈H(t) contains at least logarithmic number Given d⩾2, our d-choice algorithm chooses an B∈H(t), uniformly random, then set D d random bins from the selected B. The is allocated to least-loaded bin D, ties are broken randomly. prove if hypergraphs H(1),…,H(m) satisfy balancedness condition have low pair visibility, after allocating m=Θ(n) balls, maximum any bin, called load, most logd⁡log⁡n+O(1), high probability. Moreover, establish lower bound for load attained by balanced allocation sequence in terms showing relevance visibility parameter load.