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74
CacheOblivious Algorithms
, 1999
"... This thesis presents "cacheoblivious" 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 si ..."
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Cited by 83 (1 self)
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This thesis presents "cacheoblivious" 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 cacheline 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 Jacobistyle 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 cacheline length L, where Z =# (L 2 ), the number of cache misses for an m &times; n matrix transpose is #(1 + mn=L). The number of cache misses for either an npoint FFT or the sorting of n numbers is #(1 + (n=L)(1 + log Z n)). The cache complexity of computing n ...
Cache Oblivious Search Trees via Binary Trees of Small Height
 In Proc. ACMSIAM Symp. on Discrete Algorithms
, 2002
"... We propose a version of cache oblivious search trees which is simpler than the previous proposal of Bender, Demaine and FarachColton and has the same complexity bounds. In particular, our data structure avoids the use of weight balanced Btrees, and can be implemented as just a single array of ..."
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Cited by 66 (10 self)
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We propose a version of cache oblivious search trees which is simpler than the previous proposal of Bender, Demaine and FarachColton and has the same complexity bounds. In particular, our data structure avoids the use of weight balanced Btrees, and can be implemented as just a single array of data elements, without the use of pointers. The structure also improves space utilization.
Synthesizing transformations for locality enhancement of imperfectlynested loop nests
 In Proceedings of the 2000 ACM International Conference on Supercomputing
, 2000
"... We present an approach for synthesizing transformations to enhance locality in imperfectlynested loops. The key idea is to embed the iteration space of every statement in a loop nest into a special iteration space called the product space. The product space can be viewed as a perfectlynested loop ..."
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Cited by 59 (3 self)
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We present an approach for synthesizing transformations to enhance locality in imperfectlynested loops. The key idea is to embed the iteration space of every statement in a loop nest into a special iteration space called the product space. The product space can be viewed as a perfectlynested loop nest, so embedding generalizes techniques like code sinking and loop fusion that are used in ad hoc ways in current compilers to produce perfectlynested loops from imperfectlynested ones. In contrast to these ad hoc techniques however, our embeddings are chosen carefully to enhance locality. The product space is then transformed further to enhance locality, after which fully permutable loops are tiled, and code is generated. We evaluate the effectiveness of this approach for dense numerical linear algebra benchmarks, relaxation codes, and the tomcatv code from the SPEC benchmarks. 1. BACKGROUND AND PREVIOUSWORK Sophisticated algorithms based on polyhedral algebra have been developed for determining good sequences of linear loop transformations (permutation, skewing, reversal and scaling) for enhancing locality in perfectlynested loops 1. Highlights of this technology are the following. The iterations of the loop nest are modeled as points in an integer lattice, and linear loop transformations are modeled as nonsingular matrices mapping one lattice to another. A sequence of loop transformations is modeled by the product of matrices representing the individual transformations; since the set of nonsingular matrices is closed under matrix product, this means that a sequence of linear loop transformations can be represented by a nonsingular matrix. The problem of finding an optimal sequence of linear loop transformations is thus reduced to the problem of finding an integer matrix that satisfies some desired property, permitting the full machinery of matrix methods and lattice theory to ¢ This work was supported by NSF grants CCR9720211, EIA9726388, ACI9870687,EIA9972853. £ A perfectlynested loop is a set of loops in which all assignment statements are contained in the innermost loop.
Towards a theory of cacheefficient algorithms
 PROCEEDINGS OF THE SYMPOSIUM ON DISCRETE
, 2000
"... We present a model that enables us to analyze the running time of an algorithm on a computer with a memory hierarchy with limited associativity, in terms of various cache parameters. Our cache model, an extension of Aggarwal and Vitter’s I/O model, enables us to establish useful relationships betw ..."
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Cited by 54 (3 self)
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We present a model that enables us to analyze the running time of an algorithm on a computer with a memory hierarchy with limited associativity, in terms of various cache parameters. Our cache model, an extension of Aggarwal and Vitter’s I/O model, enables us to establish useful relationships between the cache complexity and the I/O complexity of computations. As a corollary, we obtain cacheefficient algorithms in the singlelevel cache model for fundamental problems like sorting, FFT, and an important subclass of permutations. We also analyze the averagecase cache behavior of mergesort, show that ignoring associativity concerns could lead to inferior performance, and present supporting experimental evidence. We further extend our model to multiple levels of cache with limited associativity and present optimal algorithms for matrix transpose and sorting. Our techniques may be used for systematic
Data Cache Locking for Higher Program Predictability
, 2003
"... Caches have become increasingly important with the widening gap between main memory and processor speeds. However, they are a source of unpredictability due to their characteristics, resulting in programs behaving in a different way than expected. Cache locking ..."
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Cited by 41 (3 self)
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Caches have become increasingly important with the widening gap between main memory and processor speeds. However, they are a source of unpredictability due to their characteristics, resulting in programs behaving in a different way than expected. Cache locking
Let’s Study WholeProgram Cache Behaviour Analytically
 In Proceedings of International Symposium on HighPerformance Computer Architecture (HPCA 8
, 2002
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Tiling Imperfectlynested Loop Nests
 In Proc. of SC 2000
, 2000
"... Tiling is one of the more important transformations for enhancing locality of reference in programs. Tiling of perfectlynested loop nests (which are loop nests in which all assignment statements are contained in the innermost loop) is well understood. In practice, most loop nests are imperfectlyne ..."
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Cited by 34 (0 self)
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Tiling is one of the more important transformations for enhancing locality of reference in programs. Tiling of perfectlynested loop nests (which are loop nests in which all assignment statements are contained in the innermost loop) is well understood. In practice, most loop nests are imperfectlynested, so existing compilers heuristically try to find a sequence of transformations that convert such loop nests into perfectlynested ones but not always succeed. In this paper, we propose a novel approach to tiling imperfectlynested loop nests. The key idea is to embed the iteration space of every statement in the imperfectlynested loop nest into a special space called the product space. The set of possible embeddings is constrained so that the resulting product space can be legally tiled. From this set we choose embeddings that enhance data reuse. We evaluate the effectiveness of this approach for dense numerical linear algebra benchmarks, relaxation codes, and the tomcatv code from the...
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 33 (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 ...
Transforming Loops to Recursion for MultiLevel Memory Hierarchies
 In Proceedings of the SIGPLAN ’00 Conference on Programming Language Design and Implementation
, 2000
"... Recently, there have been several experimental and theoretical results showing significant performance benefits of recursive algorithms on both multilevel memory hierarchies and on sharedmemory systems. In particular, such algorithms have the data reuse characteristics of a blocked algorithm that ..."
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Cited by 32 (4 self)
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Recently, there have been several experimental and theoretical results showing significant performance benefits of recursive algorithms on both multilevel memory hierarchies and on sharedmemory systems. In particular, such algorithms have the data reuse characteristics of a blocked algorithm that is simultaneously blocked at many different levels. Most existing applications, however, are written using ordinary loops. We present a new compiler transformation that can be used to convert loop nests into recursive form automatically. We show that the algorithm is fast and effective, handling loop nests with arbitrary nesting and control flow. The transformation achieves substantial performance improvements for several linear algebra codes even on a current system with a two level cache hierarchy. As a sideeffect of this work, we also develop an improved algorithm for transitive dependence analysis (a powerful technique used in the recursion transformation and other loop transformations) that ...
Generating Cache Hints for Improved Program Efficiency
 JOURNAL OF SYSTEMS ARCHITECTURE
, 2004
"... One of the new extensions in EPIC architectures are cache hints. On each memory instruction, two kinds of hints can be attached: a source cache hint and a target cache hint. The source hint indicates the true latency of the instruction, which is used by the compiler to improve the instruction schedu ..."
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Cited by 31 (4 self)
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One of the new extensions in EPIC architectures are cache hints. On each memory instruction, two kinds of hints can be attached: a source cache hint and a target cache hint. The source hint indicates the true latency of the instruction, which is used by the compiler to improve the instruction schedule. The target hint indicates at which cache levels it is profitable to retain data, allowing to improve cache replacement decisions at run time. A compiletime method is presented which calculates appropriate cache hints. Both kind of hints are based on the locality of the instruction, measured by the reuse distance metric. Two