A Recursive Algebraic Coloring Technique for Hardware-efficient Symmetric Sparse Matrix-vector Multiplication

Christie Alappat, Achim Basermann, Alan R. Bishop, Holger Fehske, Georg Hager, Olaf Schenk, Jonas Thies, Gerhard Wellein

Research output: Contribution to journalArticleScientificpeer-review

4 Citations (Scopus)


The symmetric sparse matrix-vector multiplication (SymmSpMV) is an important building block for many numerical linear algebra kernel operations or graph traversal applications. Parallelizing SymmSpMV on today's multicore platforms with up to 100 cores is difficult due to the need to manage conflicting updates on the result vector. Coloring approaches can be used to solve this problem without data duplication, but existing coloring algorithms do not take load balancing and deep memory hierarchies into account, hampering scalability and full-chip performance. In this work, we propose the recursive algebraic coloring engine (RACE), a novel coloring algorithm and open-source library implementation that eliminates the shortcomings of previous coloring methods in terms of hardware efficiency and parallelization overhead. We describe the level construction, distance-k coloring, and load balancing steps in RACE, use it to parallelize SymmSpMV, and compare its performance on 31 sparse matrices with other state-of-the-art coloring techniques and Intel MKL on two modern multicore processors. RACE outperforms all other approaches substantially. By means of a parameterized roofline model, we analyze the SymmSpMV performance in detail and discuss outliers. While we focus on SymmSpMV in this article, our algorithm and software are applicable to any sparse matrix operation with data dependencies that can be resolved by distance-k coloring.

Original languageEnglish
Article number19
Issue number3
Publication statusPublished - Aug 2020
Externally publishedYes


  • graph algorithms
  • graph coloring
  • memory hierarchies
  • scheduling
  • Sparse matrix
  • sparse symmetric matrix-vector multiplication

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