Skew in parallel query processing

We study the problem of computing a conjunctive query q in parallel, using p of servers, on a large database. We consider algorithms with one round of communication, and study the complexity of the communication. We are especially interested in the case where the data is skewed, which is a major challenge for scalable parallel query processing. We establish a tight connection between the fractional edge packing of the query and the amount of communication in two cases. First, in the case when the only statistics on the database are the cardinalities of the input relations, and the data is skew-free, we provide matching upper and lower bounds (up to a polylogarithmic factor of p) expressed in terms of fractional edge packings of the query q. Second, in the case when the relations are skewed and the heavy hitters and their frequencies are known, we provide upper and lower bounds expressed in terms of packings of residual queries obtained by specializing the query to a heavy hitter. All our lower bounds are expressed in the strongest form, as number of bits needed to be communicated between processors with unlimited computational power. Our results generalize prior results on uniform databases (where each relation is a matching) [4], and lower bounds for the MapReduce model [1].