Some pairs problems

A common form of MapReduce application involves discovering relationships between certain pairs of inputs. Similarity joins serve as a good example of this type of problem, which we call a "some-pairs" problem. In the framework of [4], algorithms are measured by the tradeoff between reducer size (maximum number of inputs a reducer can handle) and the replication rate (average number of reducers to which an input must be sent. There are two obvious approaches to solving some-pairs problems in general. We show that no general-purpose MapReduce algorithm can beat both of these two algorithms in the worst case. We then explore a recursive algorithm for solving some-pairs problems and heuristics for beating the lower bound on common instances of the some-pairs class of problems.