Parallel sparse polynomial division using heaps

Abstract

We present a parallel algorithm for exact division of sparse distributed polynomials on a multicore processor. This is a problem with significant data dependencies, so our solution requires fine-grained parallelism. Our algorithm manages to avoid waiting for each term of the quotient to be computed, and it achieves superlinear speedup over the fastest known sequential method. We present benchmarks comparing the performance of our C implementation of sparse polynomial division to the routines of other computer algebra systems.

Sparse polynomial multiplication and division in Maple 14

“…This is important so that performance is good for both sparse and dense problems. In (11) we presented an algorithm for dividing polynomials, also using a heap, that achieves the same complexity -O(mn log min(m, n)) monomial comparisons. But the overall performance of an algorithm will also depend on the data structure that is used for representing polynomials.…”

POLY: A New Polynomial Data Structure for Maple 17

“…The first improvement (compare Maple 13 and Maple 16) is due to our improvements to polynomial multiplication and division in [14,15,16] which we reported at ISSAC 2010 in [13]. The speedup for factorization is due to the speedup in polynomial multiplication and division.…”

“…In Maple 16, the larger multiplications and divisions are done by our external library. This includes our software for parallel polynomial multiplication and parallel polynomial division from [15,16]. Polynomials are converted from the old sum-of-products representation into our new POLY dag, and back.…”

“…For our Toeplitz matrices, the polynomial multiplications in (2) are all of the form variable × polynomial. For example, to multiply f = 9xy 3 z − 4y 3 z 2 − 6xy 2 z − 8x 3 − 5 by y, we add the monomial representation for y, namely 1010 = 2 48 + 2 16 to each monomial in f , namely the integer encodings of 5131 , 5032 , 4121 , 3300 , 0000 . Notice that result will already be sorted in the graded ordering.…”

Sparse Polynomial Powering Using Heaps

“…The complexity of powering is discussed in Section 2.1. Section 2.2 describes our approach to parallelization which we also used successfully for sparse polynomial division in [15]. Section 3 compares the performance of the algorithms on benchmark problems.…”