Efficient sparse polynomial factoring using the Funnel heap

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 depend 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. This, in turn, maximises the chances of having all polynomial arithmetic performed in the innermost levels of the memory hierarchy, and observes nearly optimal spatial locality. We provide proofs of results introduced in Abu-Salem et al. (2015) pertaining to properties of Funnel Heap, several of which are of independent worth extending beyond Hensel lifting. 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 support the theoretical analysis of the cache and space complexity in Abu-Salem et al. (2015) using accounts of cache misses and memory consumption, and compare the run-time results appearing there against adaptive Funnel Heap. We ∗Corresponding Author Email addresses: [email protected] (Fatima K. Abu Salem), [email protected] (Khalil El-Harake), [email protected] (Karl Gemayel) Preprint submitted to Journal of Symbolic Computation December 19, 2016 further 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 performing 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. Our main conclusion is that Funnel Heap will outperform Binary Heap for performing sums of products, whether data fits in in-core memory or not.