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 l...