The Gaussian Elimination Paradigm (GEP) was introduced by the authors in [Chowdhury-Ramachandran, SODA'06] to represent the triply-nested loop computation that occurs in several important algorithms including Gaussian elimination without pivoting and Floyd-Warshall's all-pairs shortest paths algorithm. An efficient cache-oblivious algorithm for these instances of GEP was presented in [Chowdhury-Ramachandran, SODA'06]. In this paper we establish several important properties of this cache-oblivious framework, and extend the framework to solve GEP in its full generality within the same time and I/O bounds. We then analyze a parallel implementation of the framework and its caching performance for both shared and distributed caches. We present extensive experimental results for both in-core and out-of-core performance of our algorithms. We consider both sequential and parallel implementations of our algorithms, and compare them with finely-tuned cache-aware BLAS code for matrix multiplication and Gaussian elimination without pivoting.
Our results indicate that cache-oblivious GEP offers an attractive trade-off between efficiency and portability.