Cache-oblivious algorithm

Abstract: Blocking is a well-known optimization technique for improving the effectiveness of memory hierarchies. Instead of operating on entire rows or columns of an array, blocked algorithms operate on submatrices or blocks, so that data loaded into the faster levels of the memory hierarchy are reused. This paper presents cache performance data for blocked programs and evaluates several optimization to improve this performance. The data is obtained by a theoretical model of data conflicts in the cache, which has been validated by large amounts of simulation. We show that the degree of cache interference is highly sensitive to the stride of data accesses and the size of the blocks, and can cause wide variations in machine performance for different matrix sizes. The conventional wisdom of frying to use the entire cache, or even a fixed fraction of the cache, is incorrect. If a fixed block size is used for a given cache size, the block size that minimizes the expected number of cache misses is very small. Tailoring the block size according to the matrix size and cache parameters can improve the average performance and reduce the variance in performance for different matrix sizes. Finally, whenever possible, it is beneficial to copy non-contiguous reused data into consecutive locations.

Abstract: This paper presents asymptotically optimal algorithms for rectangular matrix transpose, FFT, and sorting on computers with multiple levels of caching. Unlike previous optimal algorithms, these algorithms are cache oblivious: no variables dependent on hardware parameters, such as cache size and cache-line length, need to be tuned to achieve optimality. Nevertheless, these algorithms use an optimal amount of work and move data optimally among multiple levels of cache. For a cache with size Z and cache-line length L where Z=/spl Omega/(L/sup 2/) the number of cache misses for an m/spl times/n matrix transpose is /spl Theta/(1+mn/L). The number of cache misses for either an n-point FFT or the sorting of n numbers is /spl Theta/(1+(n/L)(1+log/sub Z/n)). We also give an /spl Theta/(mnp)-work algorithm to multiply an m/spl times/n matrix by an n/spl times/p matrix that incurs /spl Theta/(1+(mn+np+mp)/L+mnp/L/spl radic/Z) cache faults. We introduce an "ideal-cache" model to analyze our algorithms. We prove that an optimal cache-oblivious algorithm designed for two levels of memory is also optimal for multiple levels and that the assumption of optimal replacement in the ideal-cache model. Can be simulated efficiently by LRU replacement. We also provide preliminary empirical results on the effectiveness of cache-oblivious algorithms in practice.

Abstract: Growing wire delays will force substantive changes in the designs of large caches. Traditional cache architectures assume that each level in the cache hierarchy has a single, uniform access time. Increases in on-chip communication delays will make the hit time of large on-chip caches a function of a line's physical location within the cache. Consequently, cache access times will become a continuum of latencies rather than a single discrete latency. This non-uniformity can be exploited to provide faster access to cache lines in the portions of the cache that reside closer to the processor. In this paper, we evaluate a series of cache designs that provides fast hits to multi-megabyte cache memories. We first propose physical designs for these Non-Uniform Cache Architectures (NUCAs). We extend these physical designs with logical policies that allow important data to migrate toward the processor within the same level of the cache. We show that, for multi-megabyte level-two caches, an adaptive, dynamic NUCA design achieves 1.5 times the IPC of a Uniform Cache Architecture of any size, outperforms the best static NUCA scheme by 11%, outperforms the best three-level hierarchy--while using less silicon area--by 13%, and comes within 13% of an ideal minimal hit latency solution.

Abstract: The commonly used LRU replacement policy is susceptible to thrashing for memory-intensive workloads that have a working set greater than the available cache size. For such applications, the majority of lines traverse from the MRU position to the LRU position without receiving any cache hits, resulting in inefficient use of cache space. Cache performance can be improved if some fraction of the working set is retained in the cache so that at least that fraction of the working set can contribute to cache hits.We show that simple changes to the insertion policy can significantly reduce cache misses for memory-intensive workloads. We propose the LRU Insertion Policy (LIP) which places the incoming line in the LRU position instead of the MRU position. LIP protects the cache from thrashing and results in close to optimal hitrate for applications that have a cyclic reference pattern. We also propose the Bimodal Insertion Policy (BIP) as an enhancement of LIP that adapts to changes in the working set while maintaining the thrashing protection of LIP. We finally propose a Dynamic Insertion Policy (DIP) to choose between BIP and the traditional LRU policy depending on which policy incurs fewer misses. The proposed insertion policies do not require any change to the existing cache structure, are trivial to implement, and have a storage requirement of less than two bytes. We show that DIP reduces the average MPKI of the baseline 1MB 16-way L2 cache by 21%, bridging two-thirds of the gap between LRU and OPT.