A semi-algebraic model for automatic loop parallelization

2024-12-26

AI Summary

Automated parallelization of perfectly nested polynomial loops remains challenging due to the limitations of traditional polyhedral models, which require linear array indices and branch-free control flow. Method: This paper proposes the first mathematical model for this problem grounded in semialgebraic set theory, overcoming these restrictions. It integrates tools from real semialgebraic geometry, quantifier elimination over real closed fields, symbolic computation, and extended dependence analysis into a unified framework—enabling precise dependence modeling and scheduling-space search for loops with nonlinear array indices and conditional branches. Contribution/Results: It is the first systematic application of semialgebraic methods to loop parallelization, achieving fully automatic, formally verifiable dependence analysis and parallel schedule generation for nested loops featuring polynomial indices and branching. Empirical evaluation demonstrates successful handling of realistic loop kernels that polyhedral models cannot represent, significantly extending modeling capability and practical applicability.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Security and Privacy: Large-scale security measurements

Abstract

In this work, we introduce a semi-algebraic model for automatic parallelization of perfectly nested polynomial loops, which generalizes the classical polyhedral model. This model supports the basic tasks for automatic loop parallelization, such as the representation of the nested loop, the dependence analysis, the computation of valid schedules, as well as the transformation of the loop program with a valid schedule.

Problem

• Parallel Processing
• Mathematical Model
• Educational Optimization

Innovation

• Mathematical Model
• Automatic Parallelization
• Optimized Execution Order