GYM: A Multiround Distributed Join Algorithm

Multiround algorithms are now commonly used in distributed data processing systems, yet the extent to which algorithms can benefit from running more rounds is not well understood. This paper answers this question for several rounds for the problem of computing the equijoin of n relations. Given any query Q with width w, intersection width iw, input size IN, output size OUT, and a cluster of machines with M=Omega(IN frac{1}{epsilon}) memory available per machine, where epsilon > 1 and w ge 1 are constants, we show that: 1. Q can be computed in O(n) rounds with O(n(INw + OUT)2/M) communication cost with high probability. Q can be computed in O(log(n)) rounds with O(n(INmax(w, 3iw) + OUT)2/M) communication cost with high probability. Intersection width is a new notion we introduce for queries and generalized hypertree decompositions (GHDs) of queries that captures how connected the adjacent components of the GHDs are. We achieve our first result by introducing a distributed and generalized version of Yannakakis\u27s algorithm, called GYM. GYM takes as input any GHD of Q with width w and depth d, and computes Q in O(d + log(n)) rounds and O(n (INw + OUT)2/M) communication cost. We achieve our second result by showing how to construct GHDs of Q with width max(w, 3iw) and depth O(log(n)). We describe another technique to construct GHDs with longer widths and lower depths, demonstrating other tradeoffs one can make between communication and the number of rounds

It\u27s All a Matter of Degree: Using Degree Information to Optimize Multiway Joins

We optimize multiway equijoins on relational tables using degree information. We give a new bound that uses degree information to more tightly bound the maximum output size of a query. On real data, our bound on the number of triangles in a social network can be up to 95 times tighter than existing worst case bounds. We show that using only a constant amount of degree information, we are able to obtain join algorithms with a running time that has a smaller exponent than existing algorithms - for any database instance. We also show that this degree information can be obtained in nearly linear time, which yields asymptotically faster algorithms in the serial setting and lower communication algorithms in the MapReduce setting. In the serial setting, the data complexity of join processing can be expressed as a function O(IN^x + OUT) in terms of input size IN and output size OUT in which x depends on the query. An upper bound for x is given by fractional hypertreewidth. We are interested in situations in which we can get algorithms for which x is strictly smaller than the fractional hypertreewidth. We say that a join can be processed in subquadratic time if x < 2. Building on the AYZ algorithm for processing cycle joins in quadratic time, for a restricted class of joins which we call 1-series-parallel graphs, we obtain a complete decision procedure for identifying subquadratic solvability (subject to the 3-SUM problem requiring quadratic time). Our 3-SUM based quadratic lower bound is tight, making it the only known tight bound for joins that does not require any assumption about the matrix multiplication exponent omega. We also give a MapReduce algorithm that meets our improved communication bound and handles essentially optimal parallelism