Output-Optimal Algorithms for Join-Aggregate Queries

One of the most celebrated results of computing join-aggregate queries defined over commutative semi-rings is the classic Yannakakis algorithm proposed in 1981. It is known that the runtime of the Yannakakis algorithm is O(N + OUT) for any free-connex query, where N is the input size of the database and ØUT is the output size of the query result. This is already output-optimal. However, only an upper bound O(N • OUT) on the runtime is known for the large remaining class of acyclic but non-free-connex queries. Alternatively, one can convert a non-free-connex query into a free-connex one using tree decomposition techniques and then run the Yannakakis algorithm. This approach takes O(N #fn-subw + OUT) time, where #fn-subw is the free-connex sub-modular width of the query. But, none of these results is known to be output-optimal. In this paper, we show a matching lower and upper bound Θ(N • OUT 1 - 1/(fn-fhtw) + OUT) for computing general acyclic join-aggregate queries by semiring algorithms, where fn-fhtw is the free-connex fractional hypertree width of the query. For example, fn-fhtw = 1 for free-connex queries, fn-fhtw = 2 for line queries (a.k.a. chain matrix multiplication), and fn-fhtw = k for star queries (a.k.a. star matrix multiplication) with k relations. Although free-connex fractional hypertree width is a natural and well-established measure of how far a join-aggregate query is from being free-connex, we demonstrate that it precisely captures the output-optimal complexity of these queries. To our knowledge, this has been the first polynomial improvement over the Yannakakis algorithm in the last 40 years and completely resolves the open question of computing acyclic join-aggregate queries in an output-optimal way. As a by-product, our output-optimal algorithm for acyclic queries also yields new output-sensitive algorithms for cyclic queries via tree decomposition techniques.