Enumeration and updates for conjunctive linear algebra queries through expressibility

TL;DR

This paper explores the connection between linear algebra programs and conjunctive query fragments, identifying which linear algebra computations can be efficiently enumerated and updated using relational query optimization techniques.

Contribution

It characterizes the fragments of MATLANG corresponding to free-connex and q-hierarchical conjunctive queries, enabling efficient evaluation and updates of linear algebra programs.

Findings

Identified MATLANG fragments corresponding to efficient query evaluation.

Extended relational algebra correspondences to semiring-annotated relations.

Generalized complexity bounds for various semirings.

Abstract

Due to the importance of linear algebra and matrix operations in data analytics, there is significant interest in using relational query optimization and processing techniques for evaluating (sparse) linear algebra programs. In particular, in recent years close connections have been established between linear algebra programs and relational algebra that allow transferring optimization techniques of the latter to the former. In this paper, we ask ourselves which linear algebra programs in MATLANG correspond to the free-connex and q-hierarchical fragments of conjunctive first-order logic. Both fragments have desirable query processing properties: free-connex conjunctive queries support constant-delay enumeration after a linear-time preprocessing phase, and q-hierarchical conjunctive queries further allow constant-time updates. By characterizing the corresponding fragments of MATLANG, we hence identify the fragments of linear algebra programs that one can evaluate with constant-delay enumeration after linear-time preprocessing and with constant-time updates. To derive our results, we improve and generalize previous correspondences between MATLANG and relational algebra evaluated over semiring-annotated relations. In addition, we identify properties on semirings that allow to generalize the complexity bounds for free-connex and q-hierarchical conjunctive queries from Boolean annotations to general semirings.