Efficient sparse polynomial factoring using the Funnel heap

TL;DR

This paper demonstrates that the Funnel Heap significantly improves the efficiency of sparse polynomial factoring by optimizing insertions during the polytope method, with empirical evidence showing its superiority over traditional heaps and merge structures.

Contribution

It introduces an adaptive Funnel Heap tailored for the polytope method, reducing complexity and outperforming existing merging structures in polynomial factoring.

Findings

Funnel Heap outperforms Binary Heap in external memory scenarios.

Funnel Heap is more efficient than cache-oblivious k-merger for sums of products.

Empirical results confirm the superiority of the overlapping approach with Funnel Heap.

Abstract

This work is a comprehensive extension of Abu-Salem et al. (2015) that investigates the prowess of the Funnel Heap for implementing sums of products in the polytope method for factoring polynomials, when the polynomials are in sparse distributed representation. We exploit that the work and cache complexity of an Insert operation using Funnel Heap can be refined to de- pend on the rank of the inserted monomial product, where rank corresponds to its lifetime in Funnel Heap. By optimising on the pattern by which insertions and extractions occur during the Hensel lifting phase of the polytope method, we are able to obtain an adaptive Funnel Heap that minimises all of the work, cache, and space complexity of this phase. Additionally, we conduct a detailed empirical study confirming the superiority of Funnel Heap over the generic Binary Heap once swaps to external memory begin to take place. We demonstrate that Funnel Heap is a more efficient merger than the cache oblivious k-merger, which fails to achieve its optimal (and amortised) cache complexity when used for performing sums of products. This provides an empirical proof of concept that the overlapping approach for perform- ing sums of products using one global Funnel Heap is more suited than the serialised approach, even when the latter uses the best merging structures available.