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29
Cacheoblivious Btrees
, 2000
"... Abstract. This paper presents two dynamic search trees attaining nearoptimal performance on any hierarchical memory. The data structures are independent of the parameters of the memory hierarchy, e.g., the number of memory levels, the blocktransfer size at each level, and the relative speeds of me ..."
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Cited by 135 (22 self)
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Abstract. This paper presents two dynamic search trees attaining nearoptimal performance on any hierarchical memory. The data structures are independent of the parameters of the memory hierarchy, e.g., the number of memory levels, the blocktransfer size at each level, and the relative speeds of memory levels. The performance is analyzed in terms of the number of memory transfers between two memory levels with an arbitrary blocktransfer size of B; this analysis can then be applied to every adjacent pair of levels in a multilevel memory hierarchy. Both search trees match the optimal search bound of Θ(1+logB+1 N) memory transfers. This bound is also achieved by the classic Btree data structure on a twolevel memory hierarchy with a known blocktransfer size B. The first search tree supports insertions and deletions in Θ(1 + logB+1 N) amortized memory transfers, which matches the Btree’s worstcase bounds. The second search tree supports scanning S consecutive elements optimally in Θ(1 + S/B) memory transfers and supports insertions and deletions in Θ(1 + logB+1 N + log2 N) amortized memory transfers, matching the performance of the Btree for B = B Ω(log N log log N).
Nonlinear Array Layouts for Hierarchical Memory Systems
, 1999
"... Programming languages that provide multidimensional arrays and a flat linear model of memory must implement a mapping between these two domains to order array elements in memory. This layout function is fixed at language definition time and constitutes an invisible, nonprogrammable array attribute. ..."
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Cited by 72 (5 self)
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Programming languages that provide multidimensional arrays and a flat linear model of memory must implement a mapping between these two domains to order array elements in memory. This layout function is fixed at language definition time and constitutes an invisible, nonprogrammable array attribute. In reality, modern memory systems are architecturally hierarchical rather than flat, with substantial differences in performance among different levels of the hierarchy. This mismatch between the model and the true architecture of memory systems can result in low locality of reference and poor performance. Some of this loss in performance can be recovered by reordering computations using transformations such as loop tiling. We explore nonlinear array layout functions as an additional means of improving locality of reference. For a benchmark suite composed of dense matrix kernels, we show by timing and simulation that two specific layouts (4D and Morton) have low implementation costs (25% of total running time) and high performance benefits (reducing execution time by factors of 1.12.5); that they have smooth performance curves, both across a wide range of problem sizes and over representative cache architectures; and that recursionbased control structures may be needed to fully exploit their potential.
A memory model for scientific algorithms on graphics processors
 in Proc. of the ACM/IEEE Conference on Supercomputing (SC’06
, 2006
"... We present a memory model to analyze and improve the performance of scientific algorithms on graphics processing units (GPUs). Our memory model is based on texturing hardware, which uses a 2D blockbased array representation to perform the underlying computations. We incorporate many characteristics ..."
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Cited by 50 (3 self)
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We present a memory model to analyze and improve the performance of scientific algorithms on graphics processing units (GPUs). Our memory model is based on texturing hardware, which uses a 2D blockbased array representation to perform the underlying computations. We incorporate many characteristics of GPU architectures including smaller cache sizes, 2D block representations, and use the 3C’s model to analyze the cache misses. Moreover, we present techniques to improve the performance of nested loops on GPUs. In order to demonstrate the effectiveness of our model, we highlight its performance on three memoryintensive scientific applications – sorting, fast Fourier transform and dense matrixmultiplication. In practice, our cacheefficient algorithms for these applications are able to achieve memory throughput of 30–50 GB/s on a NVIDIA 7900 GTX GPU. We also compare our results with prior GPUbased and CPUbased implementations on highend processors. In practice, we are able to achieve 2–5× performance improvement.
Cacheoblivious mesh layouts
 ACM Trans. Graph
, 2005
"... ACM acknowledges that this contribution was authored or coauthored by a contractor of affiliate of the U.S. Government. As such, the Government retains a nonexclusive, royaltyfree right to publish or reproduce this article, or to allow others to do so, for Government purposes only. ..."
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Cited by 38 (7 self)
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ACM acknowledges that this contribution was authored or coauthored by a contractor of affiliate of the U.S. Government. As such, the Government retains a nonexclusive, royaltyfree right to publish or reproduce this article, or to allow others to do so, for Government purposes only.
Scanning and traversing: maintaining data for traversals in a memory hierarchy
 In Proceedings of the 10th Annual European Symposium on Algorithms
, 2002
"... Abstract. We study the problem of maintaining a dynamic ordered set subject to insertions, deletions, and traversals of k consecutive elements. This problem is trivially solved on a RAM and on a simple twolevel memory hierarchy. We explore this traversal problem on more realistic memory models: the ..."
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Cited by 32 (11 self)
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Abstract. We study the problem of maintaining a dynamic ordered set subject to insertions, deletions, and traversals of k consecutive elements. This problem is trivially solved on a RAM and on a simple twolevel memory hierarchy. We explore this traversal problem on more realistic memory models: the cacheoblivious model, which applies to unknown and multilevel memory hierarchies, and sequentialaccess models, where sequential block transfers are less expensive than random block transfers. 1
Recursive Array Layouts and Fast Matrix Multiplication
, 1999
"... The performance of both serial and parallel implementations of matrix multiplication is highly sensitive to memory system behavior. False sharing and cache conflicts cause traditional columnmajor or rowmajor array layouts to incur high variability in memory system performance as matrix size var ..."
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Cited by 31 (0 self)
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The performance of both serial and parallel implementations of matrix multiplication is highly sensitive to memory system behavior. False sharing and cache conflicts cause traditional columnmajor or rowmajor array layouts to incur high variability in memory system performance as matrix size varies. This paper investigates the use of recursive array layouts to improve performance and reduce variability. Previous work on recursive matrix multiplication is extended to examine several recursive array layouts and three recursive algorithms: standard matrix multiplication, and the more complex algorithms of Strassen and Winograd. While recursive layouts significantly outperform traditional layouts (reducing execution times by a factor of 1.22.5) for the standard algorithm, they offer little improvement for Strassen's and Winograd's algorithms. For a purely sequential implementation, it is possible to reorder computation to conserve memory space and improve performance between ...
Optimizing Graph Algorithms for Improved Cache Performance
 In Proc. Int’l Parallel and Distributed Processing Symp. (IPDPS 2002), Fort Lauderdale, FL
, 2002
"... In this paper, we develop algorithmic optimizations to improve the cache performance of four fundamental graph algorithms. We present a cacheoblivious implementation of the FloydWarshall Algorithm for the fundamental graph problem of allpairs shortest paths by relaxing some dependencies in the it ..."
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Cited by 28 (0 self)
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In this paper, we develop algorithmic optimizations to improve the cache performance of four fundamental graph algorithms. We present a cacheoblivious implementation of the FloydWarshall Algorithm for the fundamental graph problem of allpairs shortest paths by relaxing some dependencies in the iterative version. We show that this implementation achieves the lower bound on processormemory traffic of Ω(N 3 / C), where N and C are the problem size and cache size respectively. Experimental results show that this cacheoblivious implementation shows more than 6× improvement in real execution time over that of the iterative implementation with the usual row major data layout, on three stateoftheart architectures. Secondly, we address Dijkstra's algorithm for the singlesource shortest paths problem and Prim's algorithm for Minimum Spanning Tree. For these algorithms, we demonstrate up to 2 × improvement in real execution time by using a simple cachefriendly graph representation, namely adjacency arrays. Finally, we address the matching algorithm for bipartite graphs. We show performance improvements of 2 × 3 × in real execution time by using the technique of making the algorithm initially work on subproblems to generate a suboptimal solution and then solving the whole problem using the suboptimal solution as a starting point.
CacheEfficient Matrix Transposition
"... We investigate the memory system performance of several algorithms for transposing an N N matrix inplace, where N is large. Specifically, we investigate the relative contributions of the data cache, the translation lookaside buffer, register tiling, and the array layout function to the overall runn ..."
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Cited by 23 (1 self)
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We investigate the memory system performance of several algorithms for transposing an N N matrix inplace, where N is large. Specifically, we investigate the relative contributions of the data cache, the translation lookaside buffer, register tiling, and the array layout function to the overall running time of the algorithms. We use various memory models to capture and analyze the effect of various facets of cache memory architecture that guide the choice of a particular algorithm, and attempt to experimentally validate the predictions of the model. Our major conclusions are as follows: limited associativity in the mapping from main memory addresses to cache sets can significantly degrade running time; the limited number of TLB entries can easily lead to thrashing; the fanciest optimal algorithms are not competitive on real machines even at fairly large problem sizes unless cache miss penalties are quite high; lowlevel performance tuning “hacks”, such as register tiling and array alignment, can significantly distort the effects of improved algorithms; and hierarchical nonlinear layouts are inherently superior to the standard canonical layouts (such as row or columnmajor) for
this problem.
Efficient Sorting Using Registers and Caches
 in Proceedings of the 4th Workshop on Algorithm Engineering (WAE 2000
, 2000
"... Modern computer systems have increasingly complex memory systems.Common machine models for algorithm analysis do not reflect many of the features... ..."
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Cited by 19 (5 self)
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Modern computer systems have increasingly complex memory systems.Common machine models for algorithm analysis do not reflect many of the features...
The cost of cacheoblivious searching
 IN PROC. 44TH ANN. SYMP. ON FOUNDATIONS OF COMPUTER SCIENCE (FOCS
, 2003
"... This paper gives tight bounds on the cost of cacheoblivious searching. The paper shows that no cacheoblivious search structure can guarantee a search performance of fewer than lgelog B N memory transfers between any two levels of the memory hierarchy. This lower bound holds even if all of the bloc ..."
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Cited by 18 (8 self)
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This paper gives tight bounds on the cost of cacheoblivious searching. The paper shows that no cacheoblivious search structure can guarantee a search performance of fewer than lgelog B N memory transfers between any two levels of the memory hierarchy. This lower bound holds even if all of the block sizes are limited to be powers of 2. The paper gives modified versions of the van Emde Boas layout, where the expected number of memory transfers between any two levels of the memory hierarchy is arbitrarily close to [lge+O(lglgB/lgB)]log B N +O(1). This factor approaches lge ≈ 1.443 as B increases. The expectation is taken over the random placement in memory of the first element of the structure. Because searching in the diskaccess machine (DAM) model can be performed in log B N+O(1) block transfers, thisresultestablishes aseparation between the (2level) DAM model and cacheoblivious model. The DAM model naturally extends to k levels. The paper also shows that as k grows, the search costs of the optimal klevel DAM search structure and the optimal cacheoblivious search structure rapidly converge. This result demonstrates that for a multilevel memory hierarchy, a simple cacheoblivious structure almost replicates the performance of an optimal parameterized klevel DAM structure.