Sparse Matrix Code Dependence Analysis Simplification at Compile Time

TL;DR

This paper introduces a compile-time dependence analysis method for sparse matrix codes that leverages index array properties and SMT solvers to reduce runtime analysis, improving efficiency in parallelization and verification tasks.

Contribution

It formulates array dependence analysis using index array properties as constraints and develops an algorithm to simplify dependences, enabling more effective compile-time analysis.

Findings

Reduces runtime dependence analysis in sparse matrix kernels.

Uses SMT solvers to determine unsatisfiable dependences.

Simplifies remaining dependences through equality and subset relationship detection.

Abstract

Analyzing array-based computations to determine data dependences is useful for many applications including automatic parallelization, race detection, computation and communication overlap, verification, and shape analysis. For sparse matrix codes, array data dependence analysis is made more difficult by the use of index arrays that make it possible to store only the nonzero entries of the matrix (e.g., in A[B[i]], B is an index array). Here, dependence analysis is often stymied by such indirect array accesses due to the values of the index array not being available at compile time. Consequently, many dependences cannot be proven unsatisfiable or determined until runtime. Nonetheless, index arrays in sparse matrix codes often have properties such as monotonicity of index array elements that can be exploited to reduce the amount of runtime analysis needed. In this paper, we contribute a formulation of array data dependence analysis that includes encoding index array properties as universally quantified constraints. This makes it possible to leverage existing SMT solvers to determine whether such dependences are unsatisfiable and significantly reduces the number of dependences that require runtime analysis in a set of eight sparse matrix kernels. Another contribution is an algorithm for simplifying the remaining satisfiable data dependences by discovering equalities and/or subset relationships. These simplifications are essential to make a runtime-inspection-based approach feasible.