Abstract

We present the design and implementation of a parallel algorithm for computing Gröbner bases on distributed memory multiprocessors. The parallel algorithm is irregular both in space and time: the data structures are dynamic pointer-based structures and the computations on the structures have unpredictable duration. The algorithm is presented as a series of refinements on a transition rule program, in which computation proceeds by nondeterministic invocations of guarded commands. Two key data structures, a set and a priority queue, are distributed across processors in the parallel algorithm. The data structures are designed for high throughput and latency tolerance, as appropriate for distributed memory machines. The programming style represents a compromise between shared-memory and message-passing models. The distributed nature of the data structures shows through their interface in that the semantics are weaker than with shared atomic objects, but they still provide a shared abstraction that can be used for reasoning about program correctness. In the data structure design there is a classic trade-off between locality and load balance. We argue that this is best solved by designing scheduling structures in tandem with the state data structures, since the decision to replicate or partition state affects the overhead of dynamically moving tasks.

Citations