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The tensor programming abstraction is a foundational paradigm which allows users to write high performance programs via a high-level imperative interface. Recent work onsparse tensor compilershas extended this paradigm to sparse tensors (i.e., tensors where most entries are not explicitly represented). With these systems, users define the semantics of the program and the algorithmic decisions in a concise language that can be compiled to efficient low-level code. However, these systems still require users to make complex decisions about program structure and memory layouts to write efficient programs. This work presents.Galley, a system for declarative tensor programming that allows users to write efficient tensor programs without making complex algorithmic decisions. Galley is the first system to perform cost based lowering of sparse tensor algebra to the imperative language of sparse tensor compilers, and the first to optimize arbitrary operators beyond Σ and *. First, it decomposes the input program into a sequence of aggregation steps through a novel extension of the FAQ framework. Second, Galley optimizes and converts each aggregation step to a concrete program, which is compiled and executed with a sparse tensor compiler. We show that Galley produces programs that are 1-300x faster than competing methods for machine learning over joins and 5-20x faster than a state-of-the-art relational database for subgraph counting workloads with a minimal optimization overhead. 

In the last decade, various works have used statistics on relations to improve both the theory and practice of conjunctive query execution. Starting with the AGM bound which took advantage of relation sizes, later works incorporated statistics like functional dependencies and degree constraints. Each new statistic prompted work along two lines; bounding the size of conjunctive query outputs and worst-case optimal join algorithms. In this work, we continue in this vein by introducing a new statistic called a partition constraint. This statistic captures latent structure within relations by partitioning them into sub-relations which each have much tighter degree constraints. We show that this approach can both refine existing cardinality bounds and improve existing worst-case optimal join algorithms. 

From FORTRAN to NumPy, tensors have revolutionized how we express computation. However, tensors in these, and almost all prominent systems, can only handle dense rectilinear integer grids. Real world tensors often contain underlying structure, such as sparsity, runs of repeated values, or symmetry. Support for structured data is fragmented and incomplete. Existing frameworks limit the tensor structures and program control flow they support to better simplify the problem. In this work, we propose a new programming language, Finch, which supports both flexible control flow and diverse data structures. Finch facilitates a programming model which resolves the challenges of computing over structured tensors by combining control flow and data structures into a common representation where they can be co-optimized. Finch automatically specializes control flow to data so that performance engineers can focus on experimenting with many algorithms. Finch supports a familiar programming language of loops, statements, ifs, breaks, etc., over a wide variety of tensor structures, such as sparsity, run-length-encoding, symmetry, triangles, padding, or blocks. Finch reliably utilizes the key properties of structure, such as structural zeros, repeated values, or clustered non-zeros. We show that this leads to dramatic speedups in operations such as SpMV and SpGEMM, image processing, and graph analytics. 

Cardinality Estimation is to estimate the size of the output of a query without computing it, by using only statistics on the input relations. Existing estimators try to return an unbiased estimate of the cardinality: this is notoriously difficult. A new class of estimators have been proposed recently, called pessimistic estimators, which compute a guaranteed upper bound on the query output. Two recent advances have made pessimistic estimators practical. The first is the recent observation that degree sequences of the input relations can be used to compute query upper bounds. The second is a long line of theoretical results that have developed the use of information theoretic inequalities for query upper bounds. This paper is a short overview of pessimistic cardinality estimators, contrasting them with traditional estimators. 

Recent work has reemphasized the importance of cardinality estimates for query optimization. While new techniques have continuously improved in accuracy over time, they still generally allow for under-estimates which often lead optimizers to make overly optimistic decisions. This can be very costly for expensive queries. An alternative approach to estimation is cardinality bounding, also called pessimistic cardinality estimation, where the cardinality estimator provides guaranteed upper bounds of the true cardinality. By never underestimating, this approach allows the optimizer to avoid potentially inefficient plans. However, existing pessimistic cardinality estimators are not yet practical: they use very limited statistics on the data, and cannot handle predicates. In this paper, we introduce SafeBound, the first practical system for generating cardinality bounds. SafeBound builds on a recent theoretical work that uses degree sequences on join attributes to compute cardinality bounds, extends this framework with predicates, introduces a practical compression method for the degree sequences, and implements an efficient inference algorithm. Across four workloads, SafeBound achieves up to 80% lower end-to-end runtimes than PostgreSQL, and is on par or better than state of the art ML-based estimators and pessimistic cardinality estimators, by improving the runtime of the expensive queries. It also saves up to 500x in query planning time, and uses up to 6.8x less space compared to state of the art cardinality estimation methods.