Somnath Ghosh, Margaret Martonosi, and Sharad Malik. "Cache Miss Equations: A Compiler Framework for Analyzing and Tuning Memory Behavior". ACM Trans. Program. Lang. Syst., 21(4):703--746, 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....is taken by the compiler community [WL91, KCRB98, RT98a] They developed schemes to apply data locality optimizations automatically within a compiler pass. The optimizations within the compiler are guided by cache capacity estimates [FST91, GJG88, TFJ94] and cache miss prediction techniques [GMM99, GMM00] Available implementations of these techniques, however, are limited to research compilers up to now. Furthermore, these data locality optimization techniques cannot be applied to complex programs as for example multigrid methods since the data dependences within the algorithm are ....

S. Ghosh, M. Martonosi, and S. Malik. Cache Miss Equations: A Compiler Framework for Analyzing and Tuning Memory Behavior. ACM Transactions on Programming Languages and Systems, 21(4):703--746, July 1999.

....these and other work [33, 32, 25, 18, 26, 36, 19, 1, 30] have focused on improving parallel scalability, whereas we address uniprocessor tuning exclusively here. For dense algorithms, a variety of sophisticated static models for selecting transformations and tuning parameters have been developed [12, 17, 28, 11, 42]. However, it is di#cult to apply these analyses directly to sparse matrix kernels due to the presence of indirect and irregular memory access patterns. Nevertheless, there have been a number of notable modeling attempts in the sparse case. Temam and Jalby [39] Heras, et al. 20] and Fraguela, ....

S. Ghosh, M. Martonosi, and S. Malik. Cache miss equations: a compiler framework for analyzing and tuning memory behavior. ACM Transactions on Programming Languages and Systems, 21(4):703--746, 1999.

....Hence, now the problem to be solved is, which variables to be clustered or merged and in what order i.e. from which loop nest onwards so as minimize the cost function. Note that we have to formulate the array merging problem In practice, we can also incorporate a model like cache miss equations [8] which should provide accurate cache miss information to be plugged into this framework. The total number of variables are less than the total number of loop nests. this way because, we can have many tile sizes for each array and there can be different number of arrays alive in different ....

S.Ghosh, M.Martonosi, S.Malik, "Cache Miss Equations : A Compiler Framework for Analyzing and Tuning Memory Behaviour", ACM transactions on Programming Languages and Systems, vol. 21, No. 4, pp. 702-746, July 1999.

....both predictability and good performance. Once the program is transformed, the static analyzer determines the worst case memory performance. It analyzes scalars and array accesses whose indices are a#ne functions of the loop indices. We have implemented Ghosh et al. s Cache Miss Equations (CMEs) [13], extending its applicability following Vera and Xue s work [32] This allows us to analyze very large codes consisting of subroutines, call statements, IF statements and arbitrarily nested loops free of data dependent constructs. We have extended this analysis in such a way that it takes in ....

....and the particular cache architecture we are analyzing. In order to get the best performance from the cache, we should try to lock it as few times as possible. Besides, each locked region should be as small as possible. Thus, the more constructs we can analyze statically, the better. CMEs [13] are mathematical formulas that provide a precise characterization of the cache behavior for perfectly nested loops consisting of straight line assignments. Based on the description of reuse given by reuse vectors, some equations are set up that describe those iteration points where the reuse is ....

[Article contains additional citation context not shown here]

S. Ghosh, M. Martonosi, and S. Malik. Cache miss equations: a compiler framework for analyzing and tuning memory behavior. ACM Transactions on Programming Languages and Systems, 21(4):703--746, 1999.

....cost or resource complexities, but the majority of it is on time analysis, e.g. 17, 24, 18] Analysis of live heap space is di erent because it involves explicit analysis of the graph structure of the data. Most of the work related to analysis of space is on analysis of cache behavior, e.g. [28, 9], much of which is at a lower language level and does not consider liveness. Live heap analysis is a rst step towards analysis of cache behavior in the presence of garbage collection. Persson s work on live memory analysis [22] requires programmers to give annotations, including numerical bounds ....

S. Ghosh, M. Martonosi, and S. Malik. Cache miss equations: A compiler framework for analyzing and tuning memory behavior. ACM Trans. Program. Lang. Syst., 21(4):703-746, July 1999.

....us understand the causes behind these misses. These models can then be employed to guide various optimisations to reduce cache misses in a systematic manner. In the last few years, several compile time analytical methods have been proposed to statically predict the cache behaviour of a program [4, 6, 11, 13, 14, 26]. At this early stage, all these research e orts have focused on loop oriented programs operating on arrays. Such a method consists of (a) a procedure for setting up mathematical formulas to characterise the cache misses in a program and (b) an algorithm for nding cache misses (and their causes, ....

....misses in a program and (b) an algorithm for nding cache misses (and their causes, if required) from these formulas. These formulas describe the relationships among loop indices, array sizes, base addresses and the cache parameters for cache misses in the program. The Cache Miss Equations (CMEs) [13] make use of Wolf and Lam s reuse vectors [30] to characterise the cache misses in a program using a set of Diophantine equations (consisting of actually equalities and inequalities) This seminal work demonstrates the possibility of choosing desirable tile and pad sizes by reasoning about these ....

[Article contains additional citation context not shown here]

S. Ghosh, M. Martonosi, and S. Malik. Cache miss equations: a compiler framework for analyzing and tuning memory behavior. ACM Transactions on Programming Languages and Systems, 21(4):703-746, 1999.

....especially for dense matrix computations. However, the success of loop tiling depends on the tile size and shape selection. Many algorithms have been provided to find suitable approximations for this selection. Ghosh, Martonosi and Malik proposed the use of CME [3] to select the tile size [29] but they did not propose a general algorithm to do it. Their technique consists on maximizing the tile size for every self interference equation, obtaining a tile that has no replacement misses for the given equation. They do not give details about how to combine the different tile sizes obtained ....

Somnath Ghosh, Margaret Martonosi, and Sharad Malik. Cache miss equations: a compiler framework for analyzing and tuning memory behavior. In ACM Transactions, 1998.

....of the polyhedra. However, this important speed up is still insufficient to solve CMEs for huge iteration spaces in a reasonable amount of time. To further reduce the computation cost, we use sampling techniques to study a subset of the iteration space instead of the whole iteration space [5] [9]. The subset of points is selected using Simple Random Sampling [10] We model the number of misses of each reference using a Discrete Random Variable. This random variable follows a Binomial distribution that models phenomena consisting of n different and independent experiments that fol We ....

Somnath Ghosh, Margaret Martonosi, and Sharad Malik. Cache miss equations: a compiler framework for analyzing and tuning memory behavior. In ACM Transactions on Programming Languages and Systems, November 1999.

....predictability and good performance. Once the program is transformed, the static analyzer determines the worst case memory performance. It analyzes scalars and array accesses whose subscripts are a#ne functions of the loop indices. We have implemented Ghosh et al. s Cache Miss Equations (CMEs) [12], extending its applicability following our previous work [32] This allows us to analyze very large codes consisting of subroutines, call statements, IF statements and arbitrarily nested loops free of data dependent constructs. We have extended this analysis in such a way that it takes into ....

....and the particular cache architecture we are analyzing. In order to get the best performance from the cache, we should try to lock it as few times as possible. Besides, each locked region should be as small as possible. Thus, the more constructs we can analyze statically, the better. CMEs [12] are mathematical formulas that provide a precise characterization of the cache behavior for perfectly nested loops consisting of straight line assignments. Based on the description of reuse given by reuse vectors, some equations are set up that describe those iteration points where the reuse is ....

[Article contains additional citation context not shown here]

S. Ghosh, M. Martonosi, and S. Malik. Cache miss equations: a compiler framework for analyzing and tuning memory behavior. ACM Transactions on Programming Languages and Systems, 21(4):703--746, 1999.

....instruction may access multiple memory locations (such as those that implement array or pointer accesses) 1.1 Coyote Project This project addresses the problem of analyzing cache memory behavior at compile time and its possible applications. It builds upon the Cache Miss Equations (CMEs) [7], which describe the cache behavior in a very accurate way. Unfortunately, a direct solution of the CMEs is computationally intractable. Even though the computation cost of generating CMEs is a linear function of the number of references, to solve them is an NP Hard problem and thus trying to ....

S. Ghosh, M. Martonosi, and S. Malik. Cache miss equations: a compiler framework for analyzing and tuning memory behavior. ACM Transactions on Programming Languages and Systems, 21(4):703--746, 1999.

....form. However, the WPS compression is only applicable to scalar data, while our approach addresses compact representations for array accesses and even dynamically allocated objects. Other efforts concentrate on access modeling based on whole program traces [2, 14] using cache miss equations [11] or symbolic reference analysis at the source level based on Presburger formulas [4] These approaches involve linear solvers with response times on the order of several minutes up to over an hour. We concentrate our efforts on providing feedback to a programmer quickly. A number of approaches ....

S. Ghosh, M. Martonosi, and S. Malik. Cache miss equations: a compiler framework for analyzing and tuning memory behavior. ACM Transactions on Programming Languages and Systems, 21(4):703--746, 1999.

....alternatives. In this paper, we propose an automatic approach to perform both inter and intra variable padding in numeric codes, targeting any kind of multi level caches. It is based on a very accurate technique to analyze the locality of a program that is known as Cache Miss Equations (CMEs) [6] and a genetic algorithm in order to search the solution space. Earlier, we have proposed techniques to estimate the locality of a possible solution in a very few seconds [2, 21] in spite of the fact that a direct solution to the CMEs is an NP problem. The proposed genetic algorithm converges ....

....techniques for multi level caches. Figure 8 shows the miss penalty for UltraSparc I and Pentium 4 cache architectures. Our method improves the miss penalty, compared to Rivera et al. s method, by 8.1 and 8.9 for UltraSparc I and Pentium 4 architectures respectively. Ghosh, Martonosi and Malik [6] propose a padding technique for direct mapped caches based on using the CMEs for conflicting arrays that have the same column size. Their technique finds the optimal padding if there is a padding such that the total number of replacement misses after padding is zero. However, if such a padding ....

S. Ghosh, M. Martonosi, and S. Malik. Cache miss equations: a compiler framework for analyzing and tuning memory behavior. ACM Transactions on Programming Languages and Systems, 21(4):703 746, 1999.

....related work on dense matrices, cache and memory behavior have been well studied. A variety of sophisticated static models have been developed, each with the goal of providing a compiler with su#ciently precise models for selecting memory hierarchy transformations and parameters such as tile sizes [8, 11, 22, 7, 32]. However, it is di#cult to apply these analyses directly to sparse matrix kernels due to the presence of indirect and irregular memory access patterns. Despite the di#culty of analysis in the sparse case, there have been a number of notable attempts. Temam and Jalby [29] Heras, et al. 15] and ....

S. Ghosh, M. Martonosi, and S. Malik. Cache miss equations: a compiler framework for analyzing and tuning memory behavior. ACM Transactions on Programming Languages and Systems, 21(4):703--746, 1999.

....at compile time is useful in guiding compiler locality optimisations and improving cache simulation performance. In the past few years, some progress has been made in the development of compile time analytical methods for predicting cache behaviour. These include the Cache Miss Equations (CMEs) [11], the probabilistic method described in [9] and the Presburger formulas based method described in [4] The underlying idea is to set up mathematical formulas to provide a precise characterisation of the cache behaviour of a program in the hope that, if these formulas can be solved or manipulated ....

....precise characterisation of the cache behaviour of a program in the hope that, if these formulas can be solved or manipulated e ciently, then the information gathered such as the number of cache misses and their causes can be exploited for various performance enhancing purposes. However, the CMEs [11] and the probabilistic method [9] are limited to analysing perfect loop nests with straight line assignments. The Presburger formulas based method [4] which is capable of handling multiple nests and IF conditionals, has been applied only to loop nests of small problem sizes with a few references. ....

[Article contains additional citation context not shown here]

S. Ghosh, M. Martonosi, and S. Malik. Cache miss equations: a compiler framework for analyzing and tuning memory behavior. ACM Transactions on Programming Languages and Systems, 21(4):703-746, 1999.

....from different parts of a single array, or from different arrays. The miss is called a self interference miss in the former case and a cross interference miss in the latter case [8] 1. 2 An analysis framework Our general model for counting cache misses follows the framework used in previous work [5], with one significant difference. We first explain the common framework, then highlight the key difference in our version of the problem that necessitates entirely new solution techniques. The program fragment whose cache behavior we are trying to analyze is a perfectly nested normalized loop ....

....number of cache sets to which they map. By using these regularities to tame the potential combinatorial explosion of cases, we will in fact demonstrate algorithms that accurately compute the number of cache misses for the matrx multiplication example in O(max(log n; log(C=B) time. Previous work [5] at this point introduces two additional constraints to make the problem tractable. First, it assumes that the layout functions are row or column major, which is affine in the array co ordinates. We will subsequently use the term canonical layout to refer to these two layout functions. Second, it ....

S. Ghosh, M. Martonosi, and S. Malik. Cache miss equations: A compiler framework for analyzing and tuning memory behavior. ACM Trans. Prog. Lang. Syst., 21(4):703--746, July 1999.

....loops. During height reduction they not only optimize for low reuse vectors, but also for high, non negative dependency vectors. Finally they optimize the tiled loops with height reduction. This algorithm does not choose tile sizes or pad sizes with respect to con ict misses. Gosh et al. GMM97, GMM99] develop a system to estimate all cache misses in a loop nest for scienti c programs. Their system represents a loop nest by several Diophantine equations. These equations are called Cache Miss Equations (CMEs) Each solution of the CMEs is a cache miss of the loop nest. CMEs consider both, the ....

Somnath Gosh, Margaret Martonosi, and Sharad Malik. Cache miss equations: A compiler framework for analyzing and tuning memory behavior. ACM Transactions on Programming Languages and Systems, 21(4):703-746, July

....reuses data that is mapped in a different cache line. These polyhedra are defined over R m 1 , where m is the dimension of the iteration space. A new variable z is introduced for linearity reasons [Cla96] In fact, there is a version of the cache miss equations that ignores this variable [GMM98a], but we focus on the more precise model that includes it. The equations have the following form: CMB) f 1 i 1 f 2 i 2 Delta Delta Delta f m i m Gamma Lz LB 1 f 1 i 1 f 2 i 2 Delta Delta Delta f m i m Gamma Lz LB 2 f 1 i 1 f 2 i 2 Delta Delta Delta f m i m ....

Somnath Ghosh, Margaret Martonosi, and Sharad Malik. Cache miss equations: a compiler framework for analyzing and tuning memory behavior. In ACM Transactions, 1998.

....that affect performance; our plan is that these methods and tools, once perfected, can be applied to the prediction of larger scientific applications. 2. Other Work This work diverges from most work by other researchers in that previous work either developed very detailed models for performance[6,7,9,15,21], or concentrated on tool development[10,11] or was very specific to a given application domain[12,13,14] or focused on predicting scalability[18] Also, some previous work by Worley[23] was in the domain of traditional evaluation of specific machines via benchmarking. What distinguishes this ....

S. Ghosh, M. Martonosi and S. Malik, "Caches Miss Equations: A Compiler Framework for Analyzing and Tuning Memory Behavior", ACM Transactions on Programming Languages and Systems, vol. 21, no. 4, pg. 703-746, July, 1999.

....[18, 24, 26, 19] Stack space and heap allocation analysis [27] is similar to time analysis [19] Analysis of live heap space is di erent because it involves explicit analysis of the graph structure of the data. Most of the work related to analysis of space is on analysis of cache behavior, e.g. [28, 8], much of which is at a lower language level, for compiler generated code, while our analyses are at source level and can serve many purposes, as discussed in Section 1. Live heap space analysis is also a rst step towards analyzing cache behavior in the presence of garbage collection. Persson s ....

S. Ghosh, M. Martonosi, and S. Malik. Cache miss equations: A compiler framework for analyzing and 9 tuning memory behavior. ACM Trans. Program. Lang. Syst., 21(4):703-746, July 1999.

....as we did for stack space and heap allocation analysis. Analysis of live heap space has an important di erence from all these other analyses: it involves explicit analysis of the graph structure of the data. Most of the work related to analysis of space is on analysis of cache behavior, e.g. [10, 38, 54, 14, 33], much of which is at a lower language level, for compiler generated code, and much of which is for facilitating time analysis. Our analyses bound stack space and heap space and are completely at the source level; they can serve many more purposes in understanding and optimizing programs, as ....

S. Ghosh, M. Martonosi, and S. Malik. Cache miss equations: A compiler framework for analyzing and tuning memory behavior. ACM Trans. Program. Lang. Syst., 21(4):703-746, July 1999.

....form. However, the WPS compression is only applicable to scalar data, while our approach addresses compact representations for array accesses and even dynamically allocated objects. Other e orts concentrate on access modeling based on whole program traces [3, 14] using cache miss equations [11] or symbolic reference analysis at the source level based on Presburger formulas [5] These approaches involve linear solvers with response times on the order of several minutes up to over an hour. We concentrate our e orts on providing feedback to a programmer quickly. A number of approaches ....

Somnath Ghosh, Margaret Martonosi, and Sharad Malik. Cache miss equations: a compiler framework for analyzing and tuning memory behavior. ACM Transactions on Programming Languages and Systems, 21(4):703-746, 1999.

....behaviour at compile time is useful in guiding compiler optimisations and improving cache simulation performance. In the past few years, some progress has been made in the development of compile time analytical methods for predicting cache behaviour. These include the Cache Miss Equations (CMEs) [9], the probabilistic method described in [7] and the Presburger formulas based method described in [2] The underlying idea is to set up mathematical formulas to provide a precise characterisation of the cache behaviour of a program in the hope that, if these formulas can be solved or manipulated ....

....a precise characterisation of the cache behaviour of a program in the hope that, if these formulas can be solved or manipulated eciently, then the information gathered such as the number or causes of cache misses can be exploited for various performance enhancing purposes. However, the CMEs [9] and the probabilistic method [7] are limited to analysing perfect loop nests with straight line assignments. The Presburger formulas based method [2] which is capable of handling multiple nests and IF conditionals, has been applied only to loop nests of small problem sizes with a few references. ....

[Article contains additional citation context not shown here]

S. Ghosh, M. Martonosi, and S. Malik. Cache miss equations: a compiler framework for analyzing and tuning memory behavior. ACM Transactions on Programming Languages and Systems, 21(4):703-746, 1999.

....time and space consuming and do not provide insights about the causes of cache misses. Hardware counters [1] although fast and accurate, are architecture dependent and do not usually provide information about the causes of cache misses. Analytical methods such as the Cache Miss Equations (CMEs) [11] attempt to set up mathematical formulas to provide a precise characterisation of the number and causes of cache misses in a program. These formulas can be potentially exploited to guide a range of memory optimisations and improve the simulation times of tools like cache simulators and pro lers. ....

....to set up mathematical formulas to provide a precise characterisation of the number and causes of cache misses in a program. These formulas can be potentially exploited to guide a range of memory optimisations and improve the simulation times of tools like cache simulators and pro lers. The CMEs [11] represent an analytical method for analysing the cache behaviour of loop oriented programs. These programs typically spend a considerable amount of time operating on arrays in loop nests. The CMEs describe the relationships among loop indices, array sizes, base addresses and the cache parameters ....

[Article contains additional citation context not shown here]

Somnath Ghosh, Margaret Martonosi, and Sharad Malik. Cache miss equations: a compiler framework for analyzing and tuning memory behavior. ACM Transactions on Programming Languages and Systems, 21(4):703-746, 1999.

....where TX is the time that process X takes to complete, X:Y denotes the sequence of two processes, XjY denotes parallel execution of two processes. Forward computation of TX is possible (within reason) using analytical techniques and rigorous mathematics like that developed in [1] 18] and [15] 4] [14]. The segregation of memory access makes this prediction significantly more simple (and computationally cheaper) especially for the inner loops that are often the most important. This model assumes that processes A and B can 148 D. May et al. Effective Caching for Multithreaded Processors ....

Ghosh Somnath, Margaret Matonosi, and Sharad Malik. Cache miss equations: A compiler framework for analyzing and tuning memory behaviour. In ACM Transactions on Programming Languages and Systems, pages 703--746, July 1999.

.... techniques havebeen proposed to help guide data locality optimizations [9, 33] These techniques can also be enhanced to take into account limited cache associativity [8, 30] More recently, Ghosh et al. developed symbolic cache representation which are highly accurate in predicting cache misses [11, 12, 13]. Their cache miss equations can be used to predict the number of cache misses for a computation, and also be used to guide compiler transformations such as tiling [14] A number of researchers have investigated tiling as 11 a means of exploiting reuse. Tiling was first proposed by Irigoin and ....

S. Ghosh, M. Martonosi, and S. Malik. Cache miss equations: A compiler framework for analyzing and tuning memory behavior. ACM Transactions on Programming Languages and Systems, 21(4):703--746, July 1999.

....and memory widens, the study of cache memory is increasingly important. Through the effort of many researchers, formal theory for cache behavior of arrays in numeric programs have become well established; numerous code and memory layout transformation methods have been proposed and implemented [12, 6]. For pointer data structures, however, the success has been limited. These structures are dynamically allocated at run time, and their irregular data placement and pointer chasing characteristics pose a big barrier for formalized methods. Nonetheless, some sub problems, such as hardware and ....

.... show success [11, 15] Other efforts have examined making malloc and garbage collectors cache conscious [2, 4] reorganizing data layout, or even transforming code dynamically to change data access order to optimize program cache behavior [5, 3] This study is a first step towards extending CMEs [6] to pointer based programs. The goal is to obtain some formal mathematical method to characterize the cache behavior of pointer data structures (PDSs) An analytical model of cache misses for PDS will ultimately help us devise general and formal means to optimize PDS cache behavior. We studied ....

[Article contains additional citation context not shown here]

S. Ghosh, M. Martonosi, and S. Malik. Cache miss equations: A compiler framework for analyzing and tuning memory behavior. ACM Transactions on Programming Languages and Systems, July 1999.

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Somnath Ghosh, Margaret Martonosi, and Sharad Malik. "Cache Miss Equations: A Compiler Framework for Analyzing and Tuning Memory Behavior". ACM Trans. Program. Lang. Syst., 21(4):703--746, 1999.

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S. Ghosh, M. Martonosi, and S. Malik. Cache miss equations: a compiler framework for analyzing and tuning memory behavior. ACM Transactions on Programming Languages and Systems, 21(4):703--746, 1999.

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S. Ghosh, M. Martonosi, and S. Malik. Cache miss equations: a compiler framework for analyzing and tuning memory behavior. ACM Transactions on Programming Languages and Systems, 21(4):703--746, July 1999. 4.7

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S. Ghosh, M. Martonosi, and S. Malik. Cache miss equations: a compiler framework for analyzing and tuning memory behavior. ACM Transactions on Programming Languages and Systems, 21(4):703--746, 1999.

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S. Ghosh, M. Martonosi, and S. Malik. Cache miss equations: a compiler framework for analyzing and tuning memory behavior. ACM Transactions on Programming Languages and Systems, 21(4):703--746, 1999.

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Ghosh S, Martonosi M, Malik S. Cache miss equations: A compiler framework for analyzing and tuning memory behavior. ACM Transactions on Programming Languages and Systems 1999; 21(4):703--746.

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Somnath Ghosh, Margaret Martonosi, and Sharad Malik. Cache miss equations: a compiler framework for analyzing and tuning memory behavior. In ACM Transactions, 1998.

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Somnath Ghosh , Margaret Martonosi , Sharad Malik, "Cache miss equations: a compiler framework for analyzing and tuning memory behavior", ACM Transactions on Programming Languages and Systems (TOPLAS), v.21 n.4, p.703-746, July 1999

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S. Ghosh, M. Martonosi, and S. Malik. Cache miss equations: a compiler framework for analyzing and tuning memory behavior. ACM Transactions on Programming Languages and Systems, 21(4):703--746, 1999.

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S. Ghosh, M. Martonosi, S. Malik, Cache miss equations: A compiler framework for analyzing and tuning memory behavior, ACM Transactions on Programming Languages and Systems 21 (4) (1999) 702--745.

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Ghosh, S., Martonosi, M., Malik, S.: Cache Miss Equations: A Compiler Framework for Analyzing and Tuning Memory Behavior. ACM Transactions on Programming Languages and Systems 21 (1999) 702--745

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S. Ghosh, M. Martonosi, and S. Malik. Cache miss equations: a compiler framework for analyzing and tuning memory behavior. ACM Transactions on Programming Languages and Systems, 21(4):703--746, 1999.

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Somnath Ghosh, Margaret Martonosi, and Sharad Malik. Cache miss equations: a compiler framework for analyzing and tuning memory behavior. ACM Transactions on Programming Languages and Systems, 21(4):703--746, 1999.

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Somnath Ghosh, Margaret Martonosi, and Sharad Malik. Cache miss equations: a compiler framework for analyzing and tuning memory behavior. In ACM Transactions, 1998.

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S. Ghosh, M. Martonosi, and S. Malik. Cache miss equations: a compiler framework for analyzing and tuning memory behavior. ACM Transactions on Programming Languages and Systems (TOPLAS), 21(4):703--746, 1999.

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S. Ghosh, M. Martonosi, and S. Malik. Cache miss equations: A compiler framework for analyzing and tuning memory behavior. ACM Transactions on Programming Languages and Systems (TOPLAS), 21(4):703--746, July 1999. 24

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S. Ghosh, M. Martonosi, and S. Malik. Cache miss equations: A compiler framework for analyzing and tuning memory behavior. To appear in ACM Transactions on Programming Languages and Systems (TOPLAS), 21(4):703--746, July 1999.

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S. GHOSH, M. MARTONOSI, and S. MALIK. Cache miss equations: A compiler framework for analyzing and tuning memory behavior. ACM Transactions on Programming Languages and Systems, 21(4):703--746, Jul 1999.

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Ghosh, S., Martonosi, M., Malik, S.: Cache miss equations: A compiler framework for analyzing and tuning memory behavior. ACM Transactions on Programming Languages and Systems 21 (1999) 703--746

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S. Ghosh, M. Martonosi, and S. Malik. Cache miss equations: a compiler framework for analyzing and tuning memory behavior. ACM Transactions on Programming Languages and Systems, 21(4):703--746, July 1999. 4.7

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S. Ghosh, M. Martonosi, and S. Malik. Cache miss equations: a compiler framework for analyzing and tuning memory behavior. ACM Transactions on Programming Languages and Systems, 21(4):703--746, 1999.

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S. Ghosh, M. Martonosi, and S. Malik. Cache miss equations: a compiler framework for analyzing and tuning memory behavior. ACM Transactions on Programming Languages and Systems, 21(4):703--746, 1999.

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S. Ghosh, M. Martonosi, and S. Malik. Cache miss equations: a compiler framework for analyzing and tuning memory behavior. ACM Transactions on Programming Languages and Systems, 21(4):703--746, 1999.

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Somnath Ghosh, Margaret Martonosi, and Sharad Malik. Cache miss equations: a compiler framework for analyzing and tuning memory behavior. In ACM Transactions, 1998.

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