Balls into bins with related random choices

• Authors: Petra Berenbrink;André Brinkmann;Tom Friedetzky;Lars Nagel
• Venue: Proceedings of the twenty-second annual ACM symposium on Parallelism in algorithms and architectures
• Year: 2010

We consider a variation of classical ball-into-bins games. We randomly allocate m balls into ◊n bins. Following Godfrey's model [6], we assume that each ball i comes with a β-balanced set of clusters of bins Βi = Βi,...Βsi}. The condition of β-balancedness essentially enforces a uniform-like selection of bins, where the parameter β governs the deviation from uniformity. We use a more relaxed notion of balancedness than [6], and also generalise the concept to deterministic balancedness. Each ball i=1,...,m, in turn, runs the following protocol: (i) it i.u.r. (independently and uniformly at random) chooses a cluster of bins Βi ∈ Βi, and (ii) i.u.r. chooses one of the empty bins in Βi and allocates itself to it. Should the cluster not contain at least a single empty bin then the protocol fails. If the protocol terminates successfully, that is, every ball has indeed been able to find at least one empty bin in its chosen cluster, then this will obviously result in a maximum load of one. The main goal is to find a tight bound on the maximum number of balls, m, so that the protocol terminates successfully (with high probability). We improve on Godfrey's result and show m = n ‾ Θ(β). This upper bound holds for all mentioned types of balancedness. It even holds when we generalise the model by allowing runs. In this extended model, motivated by P2P networks, each ball i tosses a coin, and with constant probability pi (0 pi ≤ 1) it runs the protocol as described above, but with the remaining probability it copies the previous ball's choice Βi_1, that is, it re-uses the previous cluster of bins.