Computing Tropical Prevarieties in Parallel

TL;DR

This paper presents a parallel algorithm for computing tropical prevarieties, enabling faster solutions for polynomial systems and applications in pipelined solvers, with implementations demonstrating efficiency on complex problems.

Contribution

Introduces a parallel implementation of dynamic enumeration for tropical prevariety computation, utilizing shared memory work stealing and exact arithmetic libraries.

Findings

Achieved good speedups with distributed memory implementation.

Reduced variability in execution times using work stealing.

Successfully computed tropical prevarieties for complex problems like cyclic 16-roots.

Abstract

The computation of the tropical prevariety is the first step in the application of polyhedral methods to compute positive dimensional solution sets of polynomial systems. In particular, pretropisms are candidate leading exponents for the power series developments of the solutions. The computation of the power series may start as soon as one pretropism is available, so our parallel computation of the tropical prevariety has an application in a pipelined solver. We present a parallel implementation of dynamic enumeration. Our first distributed memory implementation with forked processes achieved good speedups, but quite often resulted in large variations in the execution times of the processes. The shared memory multithreaded version applies work stealing to reduce the variability of the run time. Our implementation applies the thread safe Parma Polyhedral Library (PPL), in exact arithmetic with the GNU Multiprecision Arithmetic Library (GMP), aided by the fast memory allocations of TCMalloc. Our parallel implementation is capable of computing the tropical prevariety of the cyclic 16-roots problem. We also report on computational experiments on the $n$-body and $n$-vortex problems; our computational results compare favorably with Gfan.