The Design of Maple's Sum-of-Products and POLY Data Structures for Representing Mathematical Objects

Abstract

The principal data structure in Maple used to represent polynomials and general mathematical expressions involving functions like √ x, sin x, e 2x , y (x) etc., is known to the Maple developers as the sum-of-products data structure. Gaston Gonnet, as the primary author of the Maple kernel, designed and implemented this data structure in the early 80s. As part of the process of simplifying a mathematical formula, he represented every Maple object and every sub-object uniquely in memory. This makes testing for equality, which is used in many operations, very fast. In this article, on occasion of Gaston's retirement, we present details of his design, its pros and cons, and changes we have made to it over the years. One of the cons is the sum-of-products data structure is not nearly as efficient for multiplying multivariate polynomials as other special purpose computer algebra systems. We describe the new data structure called POLY which we added to Maple 17 (released 2013) to improve performance for polynomials in Maple, and recent work done for Maple 18 (released 2014).

Algorithms and Data Structures for Sparse Polynomial Arithmetic

“…The most simple and common scheme for sparsely representing a polynomial is a linked list, or some similar variation of data blocks linked together by pointers [6,29,30]. This representation allows for very easy manipulation of terms using simple pointer manipulation.…”

“…However, the indirection created by pointers can lead to poor locality while the pointers themselves must occupy memory, resulting in memory wasted to encode the structure rather than the data itself. More efficient sparse data structures have been explored by Gastineau and Laskar [29], where burst tries store monomials in the TRIP computer algebra system, and Monagan and Peace [30], where the so-called POLY data structure for MAPLE closely stores monomials in a dense array. In both cases, the multi-precision coefficients corresponding to those monomials are stored in a secondary structure and accessed by either indices stored alongside the monomials (in the case of TRIP) or pointers (in the case of MAPLE).…”

“…Our implementation thus removes one level of indirection compared to these other schemes. We do note, however, that the data structure described in [6,30] includes an additional feature which automatically makes use of machine-precision integers stored directly in the data structure, rather than GMP integers, if coefficients are small enough.…”

Nearly Optimal Sparse Polynomial Multiplication

“…Researchers have tried to obtain algorithms that beat the O(n log n)-time bound, when the two polynomials are sparse, i.e. the number of non-zero terms in each polynomial is at most k. Interestingly, some ideas from the relevant literature have found applications in computer algebra packages such as Maple, Mathematica and Singular, including ways to represent and store polynomials [Maz01,MP14,MP15,GR16].…”

“…A data structure for carefully allocating and de-allocating memory has been designed in [Yan98], trying to tackle the problem of memory handling can be the main bottleneck in complexity of sparse multiplication in practical scenarios. The aforementioned algorithm is based on a heap, an idea which also lead to implementations developed in [MP07,MP09,MP14,MP15]. The authors in [MP09] develop a parallel algorithm for multiplying sparse distributed polynomials, where each core uses a heap of pointers to multiply parts of polynomials, exploiting its L3 cache.…”