Topology Dependent Bounds For FAQs
Abstract
In this paper, we prove topology dependent bounds on the number of rounds needed to compute Functional Aggregate Queries (FAQs) studied by Abo Khamis et al. [PODS 2016] in a synchronous distributed network under the model considered by Chattopadhyay et al. [FOCS 2014, SODA 2017]. Unlike the recent work on computing database queries in the Massively Parallel Computation model, in the model of Chattopadhyay et al., nodes can communicate only via private pointtopoint channels and we are interested in bounds that work over an {\em arbitrary} communication topology. This is the first work to consider more practically motivated problems in this distributed model. For the sake of exposition, we focus on two special problems in this paper: Boolean Conjunctive Query (BCQ) and computing variable/factor marginals in Probabilistic Graphical Models (PGMs). We obtain tight bounds on the number of rounds needed to compute such queries as long as the underlying hypergraph of the query is $O(1)$degenerate and has $O(1)$arity. In particular, the $O(1)$degeneracy condition covers most wellstudied queries that are efficiently computable in the centralized computation model like queries with constant treewidth. These tight bounds depend on a new notion of `width' (namely internalnodewidth) for Generalized Hypertree Decompositions (GHDs) of acyclic hypergraphs, which minimizes the number of internal nodes in a subclass of GHDs. To the best of our knowledge, this width has not been studied explicitly in the theoretical database literature. Finally, we consider the problem of computing the product of a vector with a chain of matrices and prove tight bounds on its round complexity (over the finite field of two elements) using a novel minentropy based argument.
 Publication:

arXiv eprints
 Pub Date:
 March 2020
 arXiv:
 arXiv:2003.05575
 Bibcode:
 2020arXiv200305575L
 Keywords:

 Computer Science  Distributed;
 Parallel;
 and Cluster Computing;
 Computer Science  Computational Complexity;
 Computer Science  Databases
 EPrint:
 A conference version was presented at PODS 2019