Balls into Bins Made Faster

Balls-into-bins games describe in an abstract setting several multiple-choice scenarios, and allow for a systematic and unified theoretical treatment. In the process that we consider, there are n bins, initially empty, and m = bcnc balls. Each ball chooses independently and uniformly at random k ≥ 3 bins. We are looking for an allocation such that each ball is placed into one of its chosen bins and no bin has load greater than 1. How quickly can we find such an allocation? We present a simple and novel algorithm that finds such an allocation (if it exists) and runs in linear time with high probability. Our algorithm finds applications in finding perfect matchings in a special class of sparse random bipartite graphs, orientation of random hypergraphs, load balancing and hashing.