Cache-oblivious Algorithms Cache-oblivious Algorithms Acknowledgments

This thesis presents “cache-oblivious” algorithms that use asymptotically optimal amounts of work, and move data asymptotically optimally among multiple levels of cache. An algorithm is cache oblivious if no program variables dependent on hardware configuration parameters, such as cache size and cache-line length need to be tuned to minimize the number of cache misses. We show that the ordinary algorithms for matrix transposition, matrix multiplication, sorting, and Jacobi-style multipass filtering are not cache optimal. We present algorithms for rectangular matrix transposition, FFT, sorting, and multipass filters, which are asymptotically optimal on computers with multiple levels of caches. For a cache with size Z and cache-line length L, where Z = Ω(L2), the number of cache misses for an m n matrix transpose is Θ(1 + mn=L). The number of cache misses for either an n-point FFT or the sorting of n numbers is Θ(1 + (n=L)(1 + logZn)). The cache complexity of computing n time steps of a Jacobi-style multipass filter on an array of size n is Θ(1 + n=L + n2=ZL). We also give an Θ(mnp)-work algorithm to multiply an m n matrix by an n p matrix that incurs Θ(m+ n+ p+ (mn+ np+mp)=L+mnp=LpZ) cache misses. We introduce an “ideal-cache” model to analyze our algorithms, and we prove that an optimal cache-oblivious algorithm designed for two levels of memory is also optimal for multiple levels. We further prove that any optimal cache-oblivious algorithm is also optimal in the previously studied HMM and SUMHmodels. Algorithms developed for these earlier models are perforce cache-aware: their behavior varies as a function of hardware-dependent parameters which must be tuned to attain optimality. Our cache-oblivious algorithms achieve the same asymptotic optimality on all these models, but without any tuning. Thesis Supervisor: Charles E. Leiserson Title: Professor of Computer Science and Engineering