[Paper Review] Parallel and distributed Gröbner bases computation in JAS

Heinz Kredel|arXiv (Cornell University)|Jul 30, 2010

Polynomial and algebraic computation24 references3 citations

TL;DR

This paper presents a distributed hybrid algorithm for parallel and distributed Gröbner bases computation in the Java Algebra System (JAS), combining shared memory parallelism on multi-core nodes with distributed memory communication across nodes using a single control channel. The approach achieves significant speedup on regular examples by asynchronously transporting polynomials via a separate distributed data structure, with performance validated on a grid-cluster using InfiniBand networking.

ABSTRACT

This paper considers parallel Gröbner bases algorithms on distributed memory parallel computers with multi-core compute nodes. We summarize three different Gröbner bases implementations: shared memory parallel, pure distributed memory parallel and distributed memory combined with shared memory parallelism. The last algorithm, called distributed hybrid, uses only one control communication channel between the master node and the worker nodes and keeps polynomials in shared memory on a node. The polynomials are transported asynchronous to the control-flow of the algorithm in a separate distributed data structure. The implementation is generic and works for all implemented (exact) fields. We present new performance measurements and discuss the performance of the algorithms.

Motivation & Objective

• To design and implement scalable, type-safe parallel Gröbner bases algorithms in a modern object-oriented language like Java.
• To address the challenges of load imbalance and communication overhead in distributed Gröbner bases computation on HPC clusters.
• To evaluate the performance of three parallelization strategies: shared memory, pure distributed memory, and hybrid (shared + distributed) approaches.
• To optimize the algorithm for real-world HPC environments using high-performance networking and efficient memory management.
• To provide a generic, extensible, and thread-safe implementation supporting all exact fields in JAS.

Proposed method

• The distributed hybrid algorithm uses a single control channel between master and worker nodes, with polynomials stored in shared memory per node to reduce communication overhead.
• Polynomial transport is decoupled from control flow using a separate distributed data structure, enabling asynchronous communication and improved load balancing.
• The system leverages Java’s generic types to support exact fields and ensures type safety and thread safety across all algebraic operations.
• Three parallelization strategies are implemented: pure shared-memory threading, pure distributed memory with inter-node communication, and the hybrid model combining both.
• Termination detection, workload balancing, and selection strategies for critical pairs are carefully designed to maintain correctness and performance.
• Performance evaluation is conducted on a grid-cluster with InfiniBand and Ethernet, using real-world Gröbner basis examples to measure speedup and scalability.

Experimental results

Research questions

• RQ1 How does the hybrid parallelization model compare to pure shared-memory and pure distributed-memory approaches in terms of performance and scalability?
• RQ2 To what extent does the workload paradox—where some workers receive more work than others—limit the effectiveness of parallel Gröbner bases computation?
• RQ3 Can asynchronous polynomial transport via a distributed data structure improve performance and reduce communication bottlenecks in distributed memory systems?
• RQ4 How do different selection strategies for critical pairs affect the convergence and efficiency of the parallel Buchberger algorithm?
• RQ5 What is the impact of high-speed networking (e.g., InfiniBand) on the performance of distributed Gröbner bases computation?

Key findings

• The distributed hybrid algorithm achieves notable speedup on regular examples, particularly when using optimal node and CPU configurations.
• Performance is significantly better on InfiniBand networks compared to standard Ethernet, highlighting the importance of low-latency communication.
• The workload paradox limits scalability in unfavorable cases, with polynomial counts increasing from 275 to 564 and reductions from 1577 to 1717, reducing parallel efficiency.
• The first selection strategy—choosing the first finished reduction result—performed better and more robustly than the sequence-based strategy in tested workloads.
• The implementation is type-safe, generic, and supports all exact fields, demonstrating strong software engineering foundations for future extensions.
• The system’s design allows for future optimization, including direct InfiniBand communication, fault tolerance, and higher-level parallelization via multiple term orderings.