Problem
Research questions and friction points this paper is trying to address.
Linear Algebra
Conjunctive Logic
Query Optimization
Enumeration and updates for conjunctive linear algebra queries through expressibility
🤖 AI Summary
This paper investigates classes of linear algebra programs that support efficient enumeration and dynamic updates. Specifically, it characterizes—within the MATLANG language—the precise fragment admitting O(n) preprocessing, O(1) constant-delay enumeration, and O(1) constant-time updates.
Method: The core approach establishes a semantic correspondence between MATLANG and free-connex and q-hierarchical conjunctive queries in relational algebra, then generalizes Boolean complexity bounds to arbitrary semiring-annotated models. It leverages semiring-annotated relational algebra modeling, conjunctive query complexity analysis, and expressiveness equivalence proofs.
Contribution/Results: The paper provides necessary and sufficient conditions for linear algebra programs to admit efficient dynamic evaluation. It proves that free-connex and q-hierarchical structures retain optimal complexity bounds under semiring semantics. These results establish a theoretical foundation and practical complexity boundary for relational optimization of linear algebraic computations.
📝 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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Linear Algebra Efficiency
Conjunctive Logic Optimization
Semi-ring Algebra Extension
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