Cache oblivious sparse polynomial factoring using the funnel heap

Abstract

In [2] we demonstrated that overlapping sums of products arising in the Hensel lifting phase of the polytope factoring method using a Max priority queue reduces expression swell and achieves asymptotic reductions in the Hensel lifting phase. In this paper, we propose to implement the priority queue as a Funnel Heap, when polynomials are in sparse distributed representation. Funnel Heap is a cache oblivious priority queue with optimal cache complexity, and we additionally tailor several of its features to the polynomial arithmetic required. Funnel Heap is able to identify equal order monomials "for free" whilst it re-organises itself over sufficiently many updates. We adopt a batched mode for chaining equal order monomials that gets overlapped with Funnel Heap's mechanism for emptying its in-core components. We also develop a customised analysis of performance that captures the overhead due to chaining in terms of the fraction of reduction and replication observed in the queue, and get that batched chaining is sensitive to the number of distinct monomials residing in the queue, as opposed to the number of replicas chained. For sufficiently large input size with respect to the cache-line length, batched chaining that is "search free" leads to an implementation of Hensel lifting that exhibits optimal cache complexity in the number of replicas found in the queue. Additionally, we obtain an order of magnitude reduction in space, as well as a reduction in the logarithmic factor in work and cache complexity, when comparing our adaptation against [2]. Also, the resulting Hensel lifting process is cache-oblivious. Our benchmarks of the polytope method using Funnel Heap with chaining demonstrate dramatic improvements over the regular binary heap as well as MAGMA, where the latter fails to