Non-linear memory layout transformations and data prefetching ...

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xiiCONTENTS
List of Figures2.1 The Memory Hierarchy pyramid . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Virtual Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.1 Row-major indexing of a 4×4 matrix, analogous Morton indexing and Mortonindexing of the order-4 quadtree . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2 The 4-dimensional array in 4D layout . . . . . . . . . . . . . . . . . . . . . . . . 363.3 The tiled array and indexing according to Morton layout . . . . . . . . . . . . . . 383.4 The 4-dimensional array in Morton layout in their actual storage order: paddingelements are not canonically placed on the borders of the array, but mixed withuseful elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.5 ZZ-transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.6 NZ-transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.7 NN-transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.8 ZN-transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.9 A 2-dimensional array converted to 1-dimensional array in ZZ-transformation . . 473.10 Conversion of the linear values of row and column indices to dilated ones throughthe use of masks. This is an 8 × 8 array with 4 × 4 element tiles . . . . . . . . . 473.11 Matrix multiplication: C[i, j]+ = A[i, k] ∗ B[k, j]: Oval boxes show the form ofrow and column binary masks. Shifting from element to element inside a tile takesplace by changing the four least signicant digits of the binary representation ofthe element position. Switching from tile to tile takes place by changing the 2most signicant digits of the binary representation of the element position. . . . 483.12 Flow Chart of the proposed optimization algorithm: it guides optimal nested loopordering, which is being matched with respective storage transformation . . . . . 514.1 Reuse of array elements in the matrix multiplication code . . . . . . . . . . . . . 574.2 Alignment of arrays A, B, C, when N 2 ≤ C L1 . . . . . . . . . . . . . . . . . . . . 584.3 Alignment of arrays A, B, C, when C L1 = N 2 . . . . . . . . . . . . . . . . . . . . 59
Page 1: NATIONAL TECHNICAL UNIVERSITY OF AT
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Page 10 and 11: xCONTENTS2.7 Iteration Space . . .
Page 14 and 15: xivLIST OF FIGURES4.4 Alignment of
Page 16 and 17: xviLIST OF FIGURES
Page 18 and 19: xviiiLIST OF TABLES
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Page 22 and 23: 2 Introductionthe processor die its
Page 24 and 25: 4 Introductionthe instruction strea
Page 26 and 27: 6 IntroductionUnfortunately, the pe
Page 28 and 29: 8 Introduction• A study of the ee
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Page 32 and 33: 12 Basic ConceptsPart of the On-chi
Page 34 and 35: 14 Basic Conceptsapplication execut
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Page 38 and 39: 18 Basic Concepts• Least-Recently
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Page 48 and 49: 28 Basic Conceptsfor (i = 0; i
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42 Fast Indexing for Blocked Array
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56 A Tile Size Selection Analysis[M
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58 A Tile Size Selection Analysismi
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60 A Tile Size Selection Analysismi
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62 A Tile Size Selection Analysisd.
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70 A Tile Size Selection AnalysisWe
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72 A Tile Size Selection AnalysisTL
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74 A Tile Size Selection Analysis4.
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76 Simultaneous MultithreadingAlong
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78 Simultaneous Multithreadingas be
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80 Simultaneous MultithreadingFor c
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82 Simultaneous Multithreading871th
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84 Simultaneous MultithreadingExami
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86 Experimental Results3500MBalttil
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88 Experimental Results2000MBaLttil
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90 Experimental Results140MBaLtMBaL
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92 Experimental Resultsnumber of mi
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94 Experimental Resultsnumber of mi
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96 Experimental ResultsTotal penalt
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98 Experimental Resultsso that dier
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100 Experimental Results10STRMMSSYR
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102 Experimental ResultsNorm. Perfo
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104 Experimental Resultstool. Figur
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106 Experimental Results
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108 Conclusionsdata are now stored
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APPENDIXATable of SymbolsExplanatio
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APPENDIXBHardware ArchitectureUltra
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APPENDIXCProgram CodesIn the follow
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C.4 SSYMM: Symmetric Matrix-Matrix
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Bibliography[AAKK05]Evangelia Athan
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BIBLIOGRAPHY 121ference on Programm
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BIBLIOGRAPHY 123[KPCM99][KRC97][KRC
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BIBLIOGRAPHY 125[RS01][RT98a][RT98b
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BIBLIOGRAPHY 127[WM95][WMC96]Wm. A.
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