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

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. [8] 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 worstcase 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 cacheobliviousness will empirically help achieve adaptivity for other problems with similar structures.