In this paper, we study the communication complexity for the problem of computing a conjunctive query on a large database in a parallel setting with p servers. In contrast to previous work, where upper and lower bounds on the communication were specified for particular structures of data (either data without skew, or data with specific types of skew), in this work we focus on worst-case analysis of the communication cost. The goal is to find worst-case optimal parallel algorithms, similar to

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... work of [17] for sequential algorithms. We first show that for a single round we can obtain an optimal worst-case algorithm. The optimal load for a conjunctive query q when all relations have size equal to M is O(M/p 1/ψ *), where ψ * is a new query-related quantity called the edge quasi-packing number, which is different from both the edge packing number and edge cover number of the query hypergraph. For multiple rounds, we present algorithms that are optimal for several classes of queries. Finally, we show a surprising connection to the external memory model, which allows us to translate parallel algorithms to external memory algorithms. This technique allows us to recover (within a polylogarithmic factor) several recent results on the I/O complexity for computing join queries, and also obtain optimal algorithms for other classes of queries. 1 Introduction The last decade has seen the development and widespread use of massively parallel systems that perform data analytics tasks over big data: examples of such systems are MapReduce [7], Dremel [16], Spark [21] and Myria [10]. In contrast to traditional database systems, where the computational complexity is dominated by the disk access time, the data now typically fits in main memory, and the dominant cost becomes that of communicating data and synchronizing among the servers in the cluster. In this paper, we present a worst-case analysis of algorithms for processing of conjunctive queries (multiway join queries) on such massively parallel systems. Our analysis is based on the Massively Parallel Computation model, or MPC [4, 5]. MPC is a theoretical model where the computational complexity of an algorithm is characterized by both the number of *