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When Are Cache-Oblivious Algorithms Cache Adaptive? A Case Study of Matrix Multiplication and Sorting

Authors Arghya Bhattacharya, Abiyaz Chowdhury, Helen Xu, Rathish Das, Rezaul A. Chowdhury, Rob Johnson, Rishab Nithyanand, Michael A. Bender

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Author Details

Arghya Bhattacharya
  • Stony Brook University, NY, USA
Abiyaz Chowdhury
  • Stony Brook University, NY, USA
Helen Xu
  • Lawrence Berkeley National Laboratory, CA, USA
Rathish Das
  • University of Waterloo, Canada
Rezaul A. Chowdhury
  • Stony Brook University, NY, USA
Rob Johnson
  • VMware Research, Palo Alto, CA, USA
Rishab Nithyanand
  • The University of Iowa, Iowa City, IA, USA
Michael A. Bender
  • Stony Brook University, NY, USA

Funding

This research in part was funded by NSF grant CCF-1617618, CCF-1439084, CCF-1725543, the Canada Research Chairs Program, NSERC Discovery Grants, NSF grant CNS-1553510. This research is funded in part by the Advanced Scientific Computing Research (ASCR) program within the Office of Science of the DOE under contract number DE-AC02-05CH11231, the Exascale Computing Project (17-SC-20-SC), a collaborative effort of the U.S. Department of Energy Office of Science and the National Nuclear Security Administration. This research was sponsored in part by the United States Air Force Research Laboratory and the United States Air Force Artificial Intelligence Accelerator and was accomplished under Cooperative Agreement Number FA8750-19-2-1000. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the United States Air Force or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.

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Arghya Bhattacharya, Abiyaz Chowdhury, Helen Xu, Rathish Das, Rezaul A. Chowdhury, Rob Johnson, Rishab Nithyanand, and Michael A. Bender. When Are Cache-Oblivious Algorithms Cache Adaptive? A Case Study of Matrix Multiplication and Sorting. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 16:1-16:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ESA.2022.16

Abstract

Cache-adaptive algorithms are a class of algorithms that achieve optimal utilization of dynamically changing memory. These memory fluctuations are the norm in today’s multi-threaded shared-memory machines and time-sharing caches.
Bender et al. [Bender et al., 2014] proved that many cache-oblivious algorithms are optimally cache-adaptive, but that some cache-oblivious algorithms can be relatively far from optimally cache-adaptive on worst-case memory fluctuations. This worst-case gap between cache obliviousness and cache adaptivity depends on a highly-structured, adversarial memory profile. Existing cache-adaptive analysis does not predict the relative performance of cache-oblivious and cache-adaptive algorithms on non-adversarial profiles. Does the worst-case gap appear in practice, or is it an artifact of an unrealistically powerful adversary?
This paper sheds light on the question of whether cache-oblivious algorithms can effectively adapt to realistically fluctuating memory sizes; the paper focuses on matrix multiplication and sorting. The two matrix-multiplication algorithms in this paper are canonical examples of "(a, b, c)-regular" cache-oblivious algorithms, which underlie much of the existing theory on cache-adaptivity. Both algorithms have the same asymptotic I/O performance when the memory size remains fixed, but one is optimally cache-adaptive, and the other is not. In our experiments, we generate both adversarial and non-adversarial memory workloads. The performance gap between the algorithms for matrix multiplication grows with problem size (up to 3.8×) on the adversarial profiles, but the gap does not grow with problem size (stays at 2×) on non-adversarial profiles. The sorting algorithms in this paper are not "(a, b, c)-regular," but they have been well-studied in the classical external-memory model when the memory size does not fluctuate. The relative performance of a non-oblivious (cache-aware) sorting algorithm degrades with the problem size: it incurs up to 6 × the number of disk I/Os compared to an oblivious adaptive algorithm on both adversarial and non-adversarial profiles. 
To summarize, in all our experiments, the cache-oblivious matrix-multiplication and sorting algorithms that we tested empirically adapt well to memory fluctuations. We conjecture that cache-obliviousness will empirically help achieve adaptivity for other problems with similar structures.

Subject Classification

ACM Subject Classification
  • Theory of computation → Shared memory algorithms
Keywords
  • Cache-adaptive algorithms
  • cache-oblivious algorithms

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