# Problem 17: The Massive, Unordered, Distributed-Data Model

The Massive, Unordered, Distributed-data (MUD) model was recently introduced by Feldman et al. [FeldmanMSSS-06] as an abstraction of part of the infrastructure used at Google. It is related to the MapReduce framework presented in [DeanG-04]. In the multi-round, multi-key MUD model, $n$ data records are distributed arbitrarily between $M$ machines. Each machine maps each record to (key, value) pairs. All pairs corresponding to the same key are then “reduced” to a single record. This reduction is performed by an $O(\operatorname{polylog} n)$-space streaming computation. The process repeats for a total of $l$ rounds.
The model is very powerful and it was proven that any EREW-PRAM algorithm can be simulated in the multi-round, multi-key MUD model if the number of keys and rounds is sufficiently large [FeldmanMSSS-06]. In practice we are primarily interested in computing with a small number of keys and rounds. What can be computed given $k$ keys and $l$ rounds?