The Cache-Oblivious Gaussian Elimination Paradigm: Theoretical Framework, Parallelization andExperimental Evaluation

• Published in: Theory of computing systems Vol. 47; no. 4; pp. 878 - 919
• Main Authors: Chowdhury, Rezaul Alam, Ramachandran, Vijaya
• Format: Journal Article
• Language: English
• Published: 01.11.2010
• Subjects: Algorithms; Computation; Gaussian elimination; Optimization; Shortest-path problems; Tradeoffs; Transformations
• ISSN: 1432-4350; 1433-0490
• DOI: 10.1007/s00224-010-9273-8

We consider triply-nested loops of the type that occur in the standard Gaussian elimination algorithm, which we denote by GEP (or the Gaussian Elimination Paradigm). We present two related cache-oblivious methods I-GEP and C-GEP, both of which reduce the number of cache misses incurred (or I/Os performed) by the computation over that performed by standard GEP by a factor of M , where M is the size of the cache. Cache-oblivious I-GEP computes in-place and solves most of the known applications of GEP including Gaussian elimination and LU-decomposition without pivoting and Floyd-Warshall all-pairs shortest paths. Cache-oblivious C-GEP uses a modest amount of additional space, but is completely general and applies to any code in GEP form. Both I-GEP and C-GEP produce system-independent cache-efficient code, and are potentially applicable to being used by optimizing compilers for loop transformation. We present parallel I-GEP and C-GEP that achieve good speed-up and match the sequential caching performance cache-obliviously for both shared and distributed caches for sufficiently large inputs. We present extensive experimental results for both in-core and out-of-core performance of our algorithms. We consider both sequential and parallel implementations, 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.