It's 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--{\em 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.