1 | %----------------------------------------------------------------------------- |
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2 | % |
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3 | % asplos094-cameron.tex |
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4 | % Robert D. Cameron and Dan Lin |
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5 | % |
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6 | % Based on sigplanconf-template.tex (2005-02-15), by Paul C. Anagnostopoulos |
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7 | % |
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8 | %----------------------------------------------------------------------------- |
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9 | \input epsf |
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10 | |
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11 | %\documentclass[preprint,natbib,10pt]{sigplanconf} |
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12 | %\documentclass[natbib,10pt]{sigplanconf} |
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13 | \documentclass[10pt]{sigplanconf} |
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14 | |
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15 | \usepackage{amsmath} |
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16 | \usepackage{graphicx} |
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17 | |
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18 | \begin{document} |
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19 | |
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20 | \conferenceinfo{ASPLOS'09,} {March 7--11, 2009, Washington, DC, USA.} |
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21 | \CopyrightYear{2009} |
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22 | \copyrightdata{978-1-60558-215-3/09/03} |
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23 | |
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24 | |
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25 | \titlebanner{banner above paper title} % These are ignored unless |
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26 | \preprintfooter{short description of paper} % 'preprint' option specified. |
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27 | |
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28 | \title{Architectural Support for SWAR Text Processing with Parallel Bit |
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29 | Streams: The Inductive Doubling Principle} |
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30 | %\subtitle{Subtitle Text, if any} |
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31 | |
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32 | \authorinfo{Robert D. Cameron \and Dan Lin} |
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33 | {School of Computing Science, Simon Fraser University} |
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34 | {\{cameron, lindanl\}@cs.sfu.ca} |
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35 | |
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36 | |
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37 | \maketitle |
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38 | |
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39 | \begin{abstract} |
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40 | Parallel bit stream algorithms exploit the SWAR (SIMD within a |
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41 | register) capabilities of commodity |
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42 | processors in high-performance text processing applications such as |
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43 | UTF-8 to UTF-16 transcoding, XML parsing, string search and regular expression |
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44 | matching. Direct architectural support for these algorithms in future SIMD |
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45 | instruction sets could further increase performance as well as simplifying the |
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46 | programming task. A set of simple SWAR instruction set extensions are proposed |
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47 | for this purpose based on the principle of systematic support for inductive |
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48 | doubling as an algorithmic technique. These extensions are shown to |
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49 | significantly reduce instruction count in core parallel bit stream algorithms, |
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50 | often providing a 3X or better improvement. The extensions are also shown to be useful |
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51 | for SIMD programming in other application areas, including providing a |
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52 | systematic treatment for horizontal operations. An implementation model for |
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53 | these extensions |
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54 | involves relatively simple circuitry added to the operand fetch components |
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55 | in a pipelined processor. |
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56 | \end{abstract} |
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57 | |
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58 | \category{CR-number}{subcategory}{third-level} |
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59 | |
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60 | \terms |
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61 | term1, term2 |
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62 | |
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63 | \keywords |
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64 | keyword1, keyword2 |
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65 | |
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66 | |
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67 | |
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68 | \section{Introduction} |
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69 | |
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70 | In the landscape of parallel computing research, finding ways to |
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71 | exploit intrachip (multicore) and intraregister (SWAR) parallelism |
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72 | for text processing and other non-numeric applications is particularly |
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73 | challenging. Indeed, in documenting this landscape, a widely cited Berkeley |
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74 | study \cite{Landscape} identifies the finite-state machine algorithms associated |
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75 | with text processing to be the hardest of the thirteen ``dwarves'' |
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76 | to parallelize, concluding that nothing seems to help. Indeed, |
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77 | the study even speculates that applications in this area may simply be |
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78 | ``embarrasingly sequential,'' easy to tackle for traditional |
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79 | sequential processing approaches suitable for uniprocessors, |
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80 | but perhaps fundamentally unsuited to parallel methods. |
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81 | |
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82 | One approach that shows some promise, however, is the |
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83 | method of parallel bit streams, recently applied to |
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84 | UTF-8 to UTF-16 transcoding \cite{PPoPP08}, XML parsing \cite{CASCON08} |
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85 | and amino acid sequencing\cite{Green}. |
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86 | In this method, byte-oriented character data is first transposed to eight |
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87 | parallel bit streams, one for each bit position within the character code |
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88 | units (bytes). Loading bit stream data into 128-bit SIMD registers, |
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89 | then, allows data from 128 consecutive code units to be represented and |
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90 | processed at once. Bitwise logic and shift operations, bit scans, |
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91 | population counts and other bit-based operations are then used to carry |
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92 | out the work. |
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93 | |
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94 | The cited study of UTF-8 to UTF-16 transcoding reports a 3X to 25X |
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95 | speed-up in using parallel bit stream techniques on SIMD-capable uniprocessors |
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96 | employing the SSE or Altivec instruction sets. A full implementation |
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97 | for each of these platforms is documented using literate programming techniques |
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98 | and available as an open source code base \cite{u8u16}. |
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99 | Although this transcoding task represents but a single text processing kernel, |
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100 | it is widely cited as a critical bottleneck in XML parsing, accounting for 30\% or more |
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101 | of processing time\cite{NicolaJohn03, Perkins05, Psaila06}. The example is also interesting in that |
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102 | it illustrates a text processing task that can largely |
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103 | be carried out using SIMD operations even though |
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104 | UTF-8 character sequences are variable length. Indeed, one weakness of the |
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105 | actual implementation is that it reverts to byte-at-a-time processing |
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106 | at block boundaries, employing a block shortening strategy to |
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107 | reduce block lengths to as low as 125 bytes from 128. |
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108 | With more capable SIMD instruction set architectures providing |
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109 | better facilities for multiregister shifts and larger register |
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110 | files, a solution employing SIMD techniques for virtually all |
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111 | of the main work would maintain better data alignment, avoid |
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112 | problems of register pressure and be easier to parallelize across |
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113 | multiple cores. It should also naturally scale to 256-bit SIMD technology |
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114 | such as the expected AVX technology of Intel. |
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115 | |
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116 | The report addressing the broader problem of XML parsing is perhaps more |
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117 | interesting, demonstrating the utility of parallel bit stream techniques |
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118 | in delivering performance benefits through a significant portion of the |
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119 | web technology stack. In an XML statistics gathering application, |
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120 | including the implementation of XML well-formedness checking, an |
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121 | overall 3X to 10X performance improvement is achieved in using |
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122 | the parallel bit stream methods in comparison with a similarly |
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123 | coded application using such well known parsers as Expat and Xerces. |
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124 | Although still a work in progress, the parser has functional |
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125 | coverage of XML and related specifications comparable to, but somewhat |
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126 | beyond Expat. The study also identifies promising further |
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127 | work in extending the parallel bit stream methods to parallel |
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128 | hash value computation and parallel regular expression matching |
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129 | for the purpose of validating XML datatype declarations in |
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130 | accord with XML Schema. |
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131 | |
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132 | Given these promising initial results in the application of |
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133 | parallel bit stream methods, what role might architectural support play in |
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134 | further enhancing this route to parallelization of text processing? |
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135 | This paper addresses this question through presentation and |
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136 | analysis of a constructive proposal: a set of SIMD instruction set |
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137 | features based on the principle of systematic support for |
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138 | inductive doubling algorithms. Inductive doubling refers |
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139 | to a general property of certain kinds of algorithm that |
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140 | systematically double the values of field widths or other |
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141 | data attributes with each iteration. In essence, the goal |
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142 | of the proposed features is to support such algorithms |
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143 | with specific facilities to transition between successive power-of-2 field |
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144 | widths. These transitions are quite frequent in several critical |
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145 | algorithms for parallel bit streams. These transitions also |
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146 | occur in other applications as well. In related work, |
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147 | efficient automatic interleaving based on power-of-2 strides has been |
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148 | reported quite useful for a number of SIMD kernels \cite{Nuzman}. |
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149 | The specific features presented herein will be referred to |
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150 | as IDISA: inductive doubling instruction set architecture. |
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151 | |
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152 | To assess the benefits of IDISA features in comparison to |
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153 | those found in existing SIMD support on commodity |
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154 | processors, we will focus on an idealized three-register |
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155 | model of SIMD instruction sets and evaluation of |
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156 | parallel bit stream and other computational kernels with |
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157 | respect to these features. The three-register model is |
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158 | based on the general approach to binary SIMD operations |
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159 | as ones that operate on the contents of two source operand |
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160 | registers to produce a value to be written back to a |
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161 | single destination operand register. This three-register |
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162 | model is directly used by the Altivec SIMD instructions |
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163 | of the Power PC, for example. On the Intel SSE platform, |
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164 | the three-register model is used as a programmer's |
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165 | interface for the C-language intrinsics, while the |
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166 | underlying instructions use a two-register model with |
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167 | destructive updating. The computational kernels we study consist |
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168 | of 100\% branch-free code operating on registers, without |
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169 | memory access For such kernels, we use straight-line |
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170 | instruction count as the performance metric of interest, |
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171 | assuming that pipelined processors are well-designed |
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172 | to handle latencies. |
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173 | |
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174 | |
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175 | The remainder of this paper is organized as follows. |
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176 | |
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177 | The second section of this paper introduces IDISA and the |
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178 | SIMD notation used throughout this paper. A brief comparison of |
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179 | IDISA features with existing SIMD instruction |
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180 | sets of commodity processors such as the Intel SSE |
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181 | instruction set and the Power PC Altivec instruction set |
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182 | is also made. |
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183 | |
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184 | The third section provides a short first example of |
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185 | the inductive doubling principle in action through |
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186 | the case of population count. Although this operation |
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187 | is not a strong determinant of performance for parallel bit |
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188 | stream applications, it is nevertheless an operation needed |
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189 | frequently enough in the general computing milieux to find |
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190 | its way into some instruction set architectures, typically |
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191 | at one particular field width. By way of comparison, the |
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192 | inductive doubling architecture sacrifices some |
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193 | performance at the chosen field width, while offering a more |
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194 | general solution with frequently better performance at |
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195 | other field widths. |
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196 | |
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197 | The fourth section then moves on to consider the performance-critical |
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198 | and key task of conversion between serial byte streams and parallel |
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199 | bit streams. A first algorithm that uses the existing SIMD |
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200 | operations common to SSE and Altivec is shown, requiring 72 |
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201 | operations to transform 128 bytes of data using the three-register |
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202 | instruction form. We then move on to consider how the task may |
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203 | be simplified using IDISA to |
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204 | require a mere 24 operations. As well as providing a 3X speed-up, |
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205 | it is also argued that the version using the inductive doubling |
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206 | architecture is considerably simpler and easier to program. |
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207 | |
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208 | The fifth section then briefly considers the inverse transposition |
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209 | process, converting parallel bit stream data back into byte |
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210 | streams. Again, an algorithm to carry out this task requires |
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211 | 72 straight-line SIMD operations in the Altivec three-register |
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212 | form, but is reduced to a simpler 24 operations using IDISA. |
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213 | |
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214 | The sixth section then goes on to consider the problem of |
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215 | parallel bit deletion. This operation is performance-critical |
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216 | to any applications that require filtering or |
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217 | editing operations on strings using the parallel bit stream |
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218 | algorithms. For example, it is fundamental to the |
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219 | high-speed UTF-8 to UTF-16 transcoding algorithm that is |
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220 | often a critical component in XML parsing. In this |
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221 | section, an inductive doubling algorithm based on the |
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222 | concept of a central deletion result is described and |
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223 | shown to have much better performance than a parallel-prefix |
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224 | alternative. |
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225 | |
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226 | The seventh section considers the issue of |
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227 | horizontal SIMD operations, that is, operations for combining |
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228 | sets of adjacent fields within individual SIMD registers rather than |
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229 | corresponding fields within sets of registers. While existing |
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230 | SIMD instruction set architectures tend to only support a few |
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231 | ad hoc horizontal combinations, IDISA is shown to provide a systematic |
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232 | means for efficient horizontal combinations of any kind. |
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233 | |
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234 | An implementation model for IDISA is then considered |
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235 | in section 8 of the paper, focusing on a pipelined |
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236 | SIMD architecture featuring a modified operand fetch stage. |
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237 | A gate-count analysis of one feasible implementation is |
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238 | provided as well as a discussion of the implementation |
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239 | of required extensions to handle 2-bit and 4-bit fields not |
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240 | commonly supported on existing SIMD architectures. Design |
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241 | tradeoffs are also considered focusing the potential removal of |
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242 | various {\em ad hoc} instructions on existing processors in favor of |
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243 | more general alternatives provided through IDISA. |
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244 | |
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245 | The ninth section concludes the paper with a summary of results |
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246 | and discussion of areas for future work. |
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247 | |
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248 | |
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249 | \section{Inductive Doubling Architecture} |
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250 | |
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251 | This section presents an idealized model for a single-instruction |
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252 | multiple-data (SIMD) instruction set architecture designed |
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253 | specifically to support inductive doubling algorithms in the |
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254 | domain of parallel bit stream programming. The architecture is idealized |
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255 | in the sense that we concentrate on only the necessary features |
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256 | for our purpose, without enumerating the additional |
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257 | operations that would be required for |
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258 | SIMD applications in other domains. The goal is to focus |
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259 | on the principles of inductive doubling support in a way |
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260 | that can accommodate a variety of realizations as other |
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261 | design constraints are brought to bear on the overall instruction set |
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262 | design. First we introduce a simple model and notation for |
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263 | SIMD operations in general and then present the four key |
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264 | features of an idealized architecture in support of parallel |
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265 | bit stream programming. |
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266 | |
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267 | The idealized architecture supports typical SIMD integer |
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268 | operations common to existing commodity architectures such as SSE |
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269 | and Altivec. The architecture is defined to support SIMD |
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270 | operations on registers of size $N=2^K$ bits, for some integer $K$. |
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271 | Typical values of $K$ for commodity processors include $K=6$ for |
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272 | the 64-bit registers of Intel MMX and Sun VIS technology, $K=7$ for |
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273 | the 128-bit registers of SSE and Altivec technology and $K=8$ for |
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274 | the upcoming Intel AVX technology. The registers may be |
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275 | partitioned into $N/n$ fields of width $n=2^k$ bits for some values |
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276 | of $k \leq K$. Typical values of $k$ used on commodity processors |
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277 | include $k = 3$ for SIMD operations on 8-bit fields (bytes), |
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278 | $k = 4$ for operations on 16-bit fields and $k = 5$ for operations |
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279 | on 32-bit fields. Whenever a register $r$ is partitioned into $n$-bit |
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280 | fields, the fields are indexed $r_n[0]$ through $r_n[N/n-1]$. |
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281 | Field $r_n[i]$ consists of bits $i \times n$ through $(i+1) \times n -1$ of |
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282 | register $r$, using big-endian numbering. |
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283 | |
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284 | Let \verb:simd<n>: represent the class of SIMD operations defined |
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285 | on fields of size $n$ using C++ template syntax. Given a |
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286 | binary function $F_n$ on $n$-bit fields, we denote the SIMD |
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287 | version of this operation as \verb#simd<n>::F#. Given two |
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288 | SIMD registers \verb:a: and \verb:b: holding values $a$ and $b$, |
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289 | respectively, the operation \verb#r=simd<n>::F(a,b)# stores |
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290 | the value $r$ in the register \verb:r: as determined by |
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291 | the simultaneous calculation of individual field values in |
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292 | accord with the following equation. |
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293 | \[ r_i = F_n(a_i, b_i) \] |
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294 | |
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295 | For example, addition(\verb:add:), subtraction (\verb:sub:) and shift left |
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296 | logical (\verb:sll:) may be defined as binary functions on $n$-bit unsigned |
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297 | integers as follows. |
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298 | %\singlespace |
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299 | \begin{eqnarray} |
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300 | \mbox{add}_n(a,b) & = & (a+b) \bmod 2^n \\ |
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301 | \mbox{sub}_n(a,b) & = & (a-b+2^n) \bmod 2^n \\ |
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302 | \mbox{sll}_n(a,b) & = & a \times 2^{b \bmod n} \bmod 2^n |
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303 | \end{eqnarray} |
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304 | %\doublespace |
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305 | The SSE and Altivec instruction sets support |
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306 | each of the addition and subtraction operations in SIMD form |
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307 | for 8, 16 and 32-bit fields, while the SSE set also includes |
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308 | the 64-bit forms. For example, \verb#simd<8>::add# in our |
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309 | nomenclature is provided by the operation \verb:paddb: |
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310 | on SSE and the operation \verb:vaddubm: on Altivec. |
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311 | The equivalents of \verb#simd<8>::sll#, \verb#simd<16>::sll#, and |
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312 | \verb#simd<32>::sll# are avilable on Altivec; the SSE facilities are |
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313 | more constrained, requiring that all field shifts by the same amount. |
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314 | |
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315 | Given these definitions and notation, we now present |
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316 | the four key elements of an inductive doubling architecture. |
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317 | The first is a definition of a core set of binary functions |
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318 | on $n$-bit fields for all field widths $n=2^k$ for $0 \leq k \leq K$. |
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319 | The second is a set of {\em half-operand modifiers} that allow |
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320 | the inductive processing of fields of size $2n$ in terms of |
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321 | combinations of $n$-bit values selected from the fields. |
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322 | The third is the definition of packing operations that compress |
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323 | two consecutive registers of $n$-bit values into a single |
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324 | register of $n/2$-bit values. The fourth is the definition |
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325 | of merging operations that produce a set of $2n$ bit fields |
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326 | by concatenating corresponding $n$-bit fields from two |
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327 | parallel registers. Each of these features is described below. |
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328 | |
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329 | For the purpose of direct and efficient support for inductive |
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330 | doubling algorithms on bit streams, the provision of |
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331 | a core set of operations at field widths of 2 and 4 as |
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332 | well as the more traditional field witdhs of 8, 16 and 32 |
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333 | is critical for elegant and efficient expression of the |
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334 | algorithms. In essence, inductive doubling algorithms |
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335 | work by establishing some base property at either single |
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336 | or 2-bit fields. Each iteration of the algorithm then |
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337 | goes on to establish the property for the power-of-2 |
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338 | field width. In order for this inductive step to be |
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339 | most conveniently and efficiently expressed, the |
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340 | core operations needed for the step should be available |
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341 | at each field width. In the case of work with parallel |
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342 | bit streams, the operations \verb:add:, \verb:sub:, |
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343 | \verb:sll:, \verb:srl: (shift right logical), and \verb:rotl: |
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344 | (rotate left) comprise the core. In other domains, |
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345 | additional operations may be usefully included in the |
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346 | core depending on the work that needs to be performed |
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347 | at each inductive doubling level. |
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348 | |
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349 | Note that the definition of field widths $n=2^k$ for $0 \leq k \leq K$ |
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350 | also includes fields of width 1. These are included for |
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351 | logical consistency, but are easily implemented by mapping |
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352 | directly to appropriate bitwise logic operations, which |
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353 | we assume are also available. For example, |
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354 | \verb#simd<1>::add# is equivalent to \verb:simd_xor:, the |
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355 | bitwise exclusive-or operation on SIMD registers. |
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356 | |
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357 | The second key facility of the inductive doubling architecture |
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358 | is the potential application of half-operand modifiers to |
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359 | the fields of either or both of the operands of a SIMD |
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360 | operation. These modifiers select either the |
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361 | low $n/2$ |
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362 | bits of each $n$-bit field (modifier ``\verb:l:'') or the |
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363 | high $n/2$ bits (modifier ``\verb:h:''). When required, |
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364 | the modifier ``\verb:x:'' means that the full $n$ bits |
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365 | should be used, unmodified. The semantics of these |
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366 | modifiers are given by the following equations. |
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367 | %\singlespace |
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368 | \begin{eqnarray} |
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369 | l(r_n) & = & r_n \bmod 2^{n/2} \\ |
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370 | h(r_n) & = & r_n / 2^{n/2} \\ |
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371 | x(r_n) & = & r_n |
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372 | \end{eqnarray} |
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373 | %\doublespace |
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374 | In our notation, the half-operand modifiers are |
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375 | specified as optional template (compile-time) parameters |
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376 | for each of the binary functions. Thus, |
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377 | \verb#simd<4>::add<l,h>(a,b)# is an operation which adds |
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378 | the 2-bit quantity found in the low 2-bits of each 4-bit field |
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379 | of its first operand (\verb:a:) |
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380 | together with the corresponding 2-bit quantity found in the |
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381 | high 2-bits of its second operand (\verb:b:). |
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382 | In general, the purpose of the half-operand modifiers |
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383 | in support of inductive doubling is to allow the processing |
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384 | of $n$-bit fields to easily expressed in terms of |
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385 | combination of the results determined by processing |
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386 | $n/2$ bit fields. |
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387 | |
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388 | The third facility of the inductive doubling architecture |
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389 | is a set of pack operations at each field width $n=2^k$ for $1 \leq k \leq K$. |
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390 | The field values of \verb#r=simd<n>::pack(a,b)# are |
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391 | defined by the following equations. |
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392 | %\singlespace |
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393 | \begin{eqnarray} |
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394 | r_{n/2}[i] & = & \mbox{conv}(a_n[i], n/2), \textrm{for } i < N/n \\ |
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395 | r_{n/2}[i] & = & \mbox{conv}(b_n[i - N/n], n/2), \textrm{for } i \geq N/n |
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396 | \end{eqnarray} |
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397 | %\doublespace |
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398 | Here conv is a function which performs conversion of an $n$-bit |
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399 | value to an $n/2$ bit value by signed saturation (although |
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400 | conversion by unsigned saturation would also suit our purpose). |
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401 | |
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402 | Half-operand modifiers may also be used with the pack |
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403 | operations. Thus packing with conversion by masking off all |
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404 | but the low $n/2$ bits of each field may be |
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405 | be performed using the operation \verb#simd<n>::pack<l,l># |
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406 | |
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407 | |
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408 | The final facility of the inductive doubling architecture is |
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409 | a set of merging operations \verb#simd<n>::mergeh# and |
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410 | \verb#simd<n>::mergel# that produce $2n$-bit fields |
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411 | by concatenating corresponding $n$-bit fields from the |
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412 | operand registers. The respective |
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413 | operations \verb#r=simd<n>::mergeh(a,b)# and |
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414 | \verb#s=simd<n>::mergel(a,b)# |
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415 | are defined by the following equations. |
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416 | %\singlespace |
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417 | \begin{eqnarray} |
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418 | r_{2n}[i] & = & a[i] \times 2^n + b[i] \\ |
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419 | s_{2n}[i] & = & a[i+N/(2n)] \times 2^n + b[i+N/(2n)] |
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420 | \end{eqnarray} |
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421 | %\doublespace |
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422 | Both SSE and Altivec provide versions of pack and merge operations |
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423 | for certain field widths. The pack operations are provided |
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424 | with operands having 16-bit or 32-bit fields on each platform, although |
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425 | with some variation in how conversion is carried out. |
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426 | The merge operations are provided at 8-bit, 16-bit and 32-bit |
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427 | field widths on both architectures and also at the 64-bit level |
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428 | on SSE. |
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429 | |
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430 | This completes the description of the proposed inductive |
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431 | doubling architecture. As described, many of the features |
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432 | are already available with the SIMD facilities of |
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433 | existing commodity processors. The extensions enumerated |
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434 | here are relatively straightforward. The innovation |
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435 | is to specifically tackle the design of facilities to |
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436 | offer systematic support for transitions between power-of-2 field widths. |
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437 | As we shall show in the remainder of this paper, these facilities |
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438 | can dramatically reduce instruction count in core parallel bit |
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439 | stream algorithms, with a factor of 3 reduction being typical. |
---|
440 | The next section goes on to illustrate a first simple example |
---|
441 | for the task of population count. |
---|
442 | |
---|
443 | \section{Population Count} |
---|
444 | |
---|
445 | \begin{figure} |
---|
446 | \begin{center}\small |
---|
447 | \begin{verbatim} |
---|
448 | c = (x & 0x55555555) + ((x >> 1) & 0x55555555); |
---|
449 | c = (c & 0x33333333) + ((c >> 2) & 0x33333333); |
---|
450 | c = (c & 0x0F0F0F0F) + ((c >> 4) & 0x0F0F0F0F); |
---|
451 | c = (c & 0x00FF00FF) + ((c >> 8) & 0x00FF00FF); |
---|
452 | c = (c & 0x0000FFFF) + ((c >>16) & 0x0000FFFF); |
---|
453 | \end{verbatim} |
---|
454 | \end{center} |
---|
455 | \caption{Population Count Reference Algorithm} |
---|
456 | \label{HD-pop} |
---|
457 | \end{figure} |
---|
458 | |
---|
459 | \begin{figure} |
---|
460 | \begin{center}\small |
---|
461 | \begin{verbatim} |
---|
462 | c = simd<2>::add<l,h>(x, x); |
---|
463 | c = simd<4>::add<l,h>(c, c); |
---|
464 | c = simd<8>::add<l,h>(c, c); |
---|
465 | c = simd<16>::add<l,h>(c, c); |
---|
466 | c = simd<32>::add<l,h>(c, c); |
---|
467 | \end{verbatim} |
---|
468 | \end{center} |
---|
469 | \caption{Ideal SIMD Population Count} |
---|
470 | \label{inductivepopcount} |
---|
471 | \end{figure} |
---|
472 | |
---|
473 | As an initial example to illustrate the principle of inductive doubling |
---|
474 | in practice, consider the problem of {\em population count}: determining |
---|
475 | the number of one bits within a particular bit field. It is important |
---|
476 | enough for such operations as |
---|
477 | calculating Hamming distance to be included as a built-in instruction |
---|
478 | on some processors. For example, the SPU of the Cell Broadband Engine |
---|
479 | has a SIMD population count instruction \verb:si_cntb: for simultaneously |
---|
480 | determining the |
---|
481 | number of 1 bits within each byte of a 16-byte register. |
---|
482 | In text processing with parallel bit streams, population count has direct |
---|
483 | application to keeping track of line numbers for error reporting, for example. |
---|
484 | Given a bit block identifying the positions of newline characters within |
---|
485 | a block of characters being processed, the population count of the |
---|
486 | bit block can be used to efficiently and conveniently be used to update |
---|
487 | the line number upon completion of block processing. |
---|
488 | |
---|
489 | Figure \ref{HD-pop} presents a traditional divide-and-conquer |
---|
490 | implementation for a 32-bit integer {\tt x} adapted from |
---|
491 | Warren \cite{HackersDelight}, while |
---|
492 | Figure \ref{inductivepopcount} shows the corresponding SWAR |
---|
493 | implementation for a vector of 32-bit fields using the inductive doubling |
---|
494 | instruction set architecture. Each implementation employs |
---|
495 | five steps of inductive doubling to produce population counts |
---|
496 | within 32 bit fields. The traditional implementation employs |
---|
497 | explicit masking and shifting operations, while these |
---|
498 | operations are implicit within the semantics of the inductive |
---|
499 | doubling instructions used in Figure \ref{inductivepopcount}. |
---|
500 | In each implementation, the first step determines the |
---|
501 | the population counts within 2-bit fields |
---|
502 | by adding the high bit of each such field to the low bit |
---|
503 | to produce a set of 2-bit counts in {\tt c}. |
---|
504 | In the second step, the counts within 4-bit fields of {\tt c} are determined |
---|
505 | by adding the counts of the corresponding high and low 2-bit subfields. |
---|
506 | Continuing in this fashion, |
---|
507 | the final population counts within 32-bit fields are determined in five steps. |
---|
508 | |
---|
509 | With the substitution of longer mask constants replicated for four |
---|
510 | 32-bit fields, the implementation of Figure \ref{HD-pop} can be |
---|
511 | easily adapted to SWAR processing using 128-bit registers. |
---|
512 | Such an implementation requires 10 operations to load or generate |
---|
513 | mask constants, 10 bitwise-and operations, 5 shifts and 5 adds for a |
---|
514 | total of 30 operations to complete the task in comparison |
---|
515 | to a mere However, Warren further refines the |
---|
516 | implementation to an optimized version requiring only 20 operations, |
---|
517 | 5 of which are required to generate mask constants. At the cost |
---|
518 | of register pressure, it is possible that these constants |
---|
519 | could be kept preloaded. In any case, the IDISA implementation |
---|
520 | offers a 3X to 4X improvement in instruction count as well as |
---|
521 | a reduction in register pressure. |
---|
522 | |
---|
523 | |
---|
524 | |
---|
525 | |
---|
526 | \section{Transposition to Parallel Bit Streams} |
---|
527 | |
---|
528 | In this section, we consider the first major |
---|
529 | application of the inductive doubling architecture: |
---|
530 | transposition of byte stream data to parallel bit stream |
---|
531 | form. Of course, this operation is critical to the |
---|
532 | method of parallel bit streams and all applications |
---|
533 | of the method can benefit from a highly efficient |
---|
534 | transposition process. Before considering how |
---|
535 | the inductive doubling architecture supports this |
---|
536 | transposition process, however, we first consider |
---|
537 | algorithms on existing architectures. Two algorithms |
---|
538 | are presented; the best of these requires 72 |
---|
539 | SIMD operations in the three-register model to perform |
---|
540 | transposition of eight serial registers of byte stream data into |
---|
541 | eight parallel registers of bit stream data. |
---|
542 | |
---|
543 | We then go on to show how the transposition problem |
---|
544 | can be solved using the inductive doubling architecture |
---|
545 | with a mere 24 three-register SIMD operations. We also prove |
---|
546 | that this is optimal for any three-register instruction set model. |
---|
547 | |
---|
548 | |
---|
549 | \begin{figure}[tbh] |
---|
550 | \begin{center} |
---|
551 | \includegraphics[width=120mm, trim= 0 90 0 50]{S2P_IO.pdf} |
---|
552 | \caption{Input/Output Model for Serial to Parallel Transposition} |
---|
553 | \label{s2p-spec} |
---|
554 | \end{center} |
---|
555 | |
---|
556 | \end{figure} |
---|
557 | Figure \ref{s2p-spec} illustrates the input-output requirements of |
---|
558 | the transposition problem. We assume that inputs and |
---|
559 | outputs are each SIMD registers of size $N=2^K$ bits. |
---|
560 | The input consists of $N$ bytes of serial byte data, |
---|
561 | stored consecutively in eight SIMD registers each holding |
---|
562 | $N/8$ bytes. The output consists of eight parallel |
---|
563 | registers, one each for the eight individual bit positions |
---|
564 | within a byte. Upon completion of the transposition process, |
---|
565 | each output register is to hold the $N$ bits corresponding |
---|
566 | to the selected bit position in the sequence of $N$ input |
---|
567 | bytes. |
---|
568 | |
---|
569 | \subsection{Bit Gathering Algorithm} |
---|
570 | |
---|
571 | \begin{figure}[tbh] |
---|
572 | \begin{center} |
---|
573 | \includegraphics[width=150mm, trim= 0 150 0 50]{S2P.pdf} |
---|
574 | \caption{Serial to Parallel Transposition Using Bit-Gathering} |
---|
575 | \label{gather} |
---|
576 | \end{center} |
---|
577 | \end{figure} |
---|
578 | One straightforward algorithm for implementing the transposition process |
---|
579 | takes advantage of SIMD bit gathering operations that exist |
---|
580 | on some architectures. This operation gathers one bit per byte |
---|
581 | from a particular position within each byte of a SIMD register. |
---|
582 | For example, the {\tt pmovmskb} operation of the Intel MMX and |
---|
583 | SSE instruction sets forms an 8-bit (MMX) or 16-bit (SSE) mask |
---|
584 | consisting of the high bit of each byte. Similarly, the |
---|
585 | {\tt \verb:si_gbb:} operation of the synergistic processing units of the |
---|
586 | Cell Broadband Engine gathers together the low bit of each byte |
---|
587 | from the SIMD register. Figure \ref{gather} illustrates the |
---|
588 | bit gathering process. |
---|
589 | |
---|
590 | For each bit stream of output, the bit gather algorithm requires |
---|
591 | one gather operation for each of the 8 input registers, |
---|
592 | giving 64 bit gather operations in all. In addition, for seven |
---|
593 | of the eight bit positions, it is necessary to shift the bits |
---|
594 | of each input register into the conventional position for |
---|
595 | gathering. A total of 56 shift operations are required for this |
---|
596 | task. Finally, the result of each bit gather operation must |
---|
597 | be properly inserted into the output stream. If the first |
---|
598 | gather operation for a stream can be used to also initialize |
---|
599 | the output register, there will then need to be 7 insert |
---|
600 | operations for the results of the remaining gather operations |
---|
601 | for this stream, making 56 insert operations in all. |
---|
602 | The insert step may be more complex than a single operation |
---|
603 | in some cases, but we use one operation per insert as a lower bound. |
---|
604 | Thus, the bit gather algorithm requires |
---|
605 | at least 176 operations to perform the transposition task. |
---|
606 | |
---|
607 | \subsection{BytePack Algorithm} |
---|
608 | |
---|
609 | A much more efficient transposition algorithm on commodity |
---|
610 | SIMD architectures (SSE and Altivec) involves three |
---|
611 | stages of binary division transformation. This is similar |
---|
612 | to the three stage bit matrix inversion described by |
---|
613 | Warren \cite{HackersDelight}, although modified to use SIMD operations. |
---|
614 | In each stage, input streams are divided into two half-length output streams. |
---|
615 | The first stage separates the bits at even numbered positions from those |
---|
616 | at odd number positions. The two output streams from the first |
---|
617 | stage are then further divided in the second stage. |
---|
618 | The stream comprising even numbered bits from the original byte stream |
---|
619 | divides into one stream consisting of bits from positions 0 and 4 of each |
---|
620 | byte in the original stream and a second stream consisting of bits |
---|
621 | from positions 2 and 6 of each original byte. The stream of bits from |
---|
622 | odd positions is similarly divided into streams for bits from Each of the |
---|
623 | positions 1 and 5 and bits from positions 2 and 6. |
---|
624 | Finally, each of the four streams resulting from the second stage are |
---|
625 | divided into the desired individual bit streams in the third stage. |
---|
626 | |
---|
627 | \begin{figure}[tbh] |
---|
628 | \begin{center} |
---|
629 | \begin{verbatim} |
---|
630 | t0 = simd<16>::pack<h,h>(s0, s1); |
---|
631 | t1 = simd<16>::pack<l,l>(s0, s1); |
---|
632 | p0 = simd_if(simd<8>::const(0xC0C0), t0, simd::<16>srli<1>(t1)); |
---|
633 | p1 = simd_if(simd<8>::const(0xC0C0), simd::<16>slli<1>(t0), t1); |
---|
634 | \end{verbatim} |
---|
635 | \end{center} |
---|
636 | \caption{Basic Stage 1 Transposition Step in the BytePack Algorithm} |
---|
637 | \label{s2pstep} |
---|
638 | \end{figure} |
---|
639 | |
---|
640 | The binary division transformations are accomplished in each stage |
---|
641 | using byte packing, shifting and masking. In each stage, a |
---|
642 | transposition step combines each pair of serial input registers to |
---|
643 | produce a pair of parallel output registers. In essence, |
---|
644 | doublebytes from the input registers are packed into bytes |
---|
645 | in the output registers, with the bits from even positions stored |
---|
646 | in the bytes of one output stream and the bits from odd positions |
---|
647 | stored in the bytes of the second output stream. |
---|
648 | Figure \ref{s2pstep} shows a step in stage 1 of the algorithm |
---|
649 | producing two parallel registers \verb:p0: and \verb:p1: from |
---|
650 | two serial registers \verb:s0: and \verb:s1:. This step is applied |
---|
651 | four times in stage 1; stages 2 and 3 also consist of four applications |
---|
652 | of a similar step with different shift and masking constants. |
---|
653 | |
---|
654 | Although we have used the idealized SIMD notation here, each of the |
---|
655 | operations maps to a single operation in the Altivec set and a small number |
---|
656 | of operations in the SSE set. Using the Altivec set, there are |
---|
657 | 6 operations per step for a total of 24 operations per stage. |
---|
658 | The three stages combined required 72 operations to transpose 128 bytes |
---|
659 | to parallel bit stream form. This is the best algorithm known to |
---|
660 | us for existing SIMD architectures. |
---|
661 | |
---|
662 | \subsection{Inductive Halving Algorithm} |
---|
663 | |
---|
664 | Using the inductive doubling architecture, it is possible to design |
---|
665 | a transposition algorithm that is both easier to understand and requires |
---|
666 | many fewer operations than the the bytepack algorithm described above. |
---|
667 | We call it the inductive halving algorithm for serial to parallel |
---|
668 | transposition, because it proceeds by reducing byte streams to |
---|
669 | two sets of nybble streams in a first stage, dividing the nybble |
---|
670 | streams into streams of bitpairs in a second stage and finally |
---|
671 | dividing the bitpair streams into bit streams in the third stage. |
---|
672 | |
---|
673 | |
---|
674 | \begin{figure}[tbh] |
---|
675 | \begin{verbatim} |
---|
676 | p0 = simd<8>::pack<h,h>(s0, s1); |
---|
677 | p1 = simd<8>::pack<l,l>(s0, s1); |
---|
678 | \end{verbatim} |
---|
679 | \caption{Basic Stage 1 Transposition Step in the Inductive Halving Algorithm} |
---|
680 | \label{halvingstep} |
---|
681 | \end{figure} |
---|
682 | |
---|
683 | Figure \ref{halvingstep} shows one step in stage 1 of the inductive |
---|
684 | halving algorithm, comprising just two SIMD operations. |
---|
685 | The \verb#simd<8>::pack<h,h># operation extracts the high nybble of each byte |
---|
686 | from the input registers, while the \verb#simd<8>::pack<l,l># operation extracts |
---|
687 | the low nybble of each byte. As in the bytepack algorithm, this step is |
---|
688 | applied 4 times in stage 1, for a total of 8 operations. |
---|
689 | |
---|
690 | Stage 2 of the inductive halving algorithm reduces nybble streams |
---|
691 | to streams of bit pairs. The basic step in this algorithm consists |
---|
692 | of one \verb#simd<4>::pack<h,h># operation to extract the high pair |
---|
693 | of each nybble and one \verb#simd<4>::pack<l,l># operation to extract the |
---|
694 | low pair of each nybble. Four applications of this step complete stage 2. |
---|
695 | |
---|
696 | Stage 3 similarly uses four applications of a step that uses a |
---|
697 | \verb#simd<2>::pack<h,h># operation to extract the high bit of |
---|
698 | each pair and a \verb#simd<2>::pack<l,l># to extract the low bit of |
---|
699 | each pair. The complete algorithm to transform eight serial |
---|
700 | byte registers s0 through s7 into the eight parallel bit stream |
---|
701 | registers bit0 through bit7 is shown in Figure \ref{halvingalgorithm}. |
---|
702 | |
---|
703 | \begin{figure}[tbh] |
---|
704 | \begin{verbatim} |
---|
705 | hi_nybble0 = simd<8>::pack<h,h>(s0, s1); |
---|
706 | lo_nybble0 = simd<8>::pack<l,l>(s0, s1); |
---|
707 | hi_nybble1 = simd<8>::pack<h,h>(s2, s3); |
---|
708 | lo_nybble1 = simd<8>::pack<l,l>(s2, s3); |
---|
709 | hi_nybble2 = simd<8>::pack<h,h>(s4, s5); |
---|
710 | lo_nybble2 = simd<8>::pack<l,l>(s4, s5); |
---|
711 | hi_nybble3 = simd<8>::pack<h,h>(s6, s7); |
---|
712 | lo_nybble3 = simd<8>::pack<l,l>(s6, s7); |
---|
713 | hh_pair0 = simd<4>::pack<h,h>(hi_nybble0, hi_nybble1); |
---|
714 | hl_pair0 = simd<4>::pack<l,l>(hi_nybble0, hi_nybble1); |
---|
715 | lh_pair0 = simd<4>::pack<h,h>(lo_nybble0, lo_nybble1); |
---|
716 | ll_pair0 = simd<4>::pack<l,l>(lo_nybble0, lo_nybble1); |
---|
717 | hh_pair1 = simd<4>::pack<h,h>(hi_nybble2, hi_nybble3); |
---|
718 | hl_pair1 = simd<4>::pack<l,l>(hi_nybble2, hi_nybble3); |
---|
719 | lh_pair1 = simd<4>::pack<h,h>(lo_nybble2, lo_nybble3); |
---|
720 | ll_pair1 = simd<4>::pack<l,l>(lo_nybble2, lo_nybble3); |
---|
721 | bit0 = simd<2>::pack<h,h>(hh_pair0, hh_pair1); |
---|
722 | bit1 = simd<2>::pack<l,l>(hh_pair0, hh_pair1); |
---|
723 | bit2 = simd<2>::pack<h,h>(hl_pair0, hl_pair1); |
---|
724 | bit3 = simd<2>::pack<l,l>(hl_pair0, hl_pair1); |
---|
725 | bit4 = simd<2>::pack<h,h>(lh_pair0, lh_pair1); |
---|
726 | bit5 = simd<2>::pack<l,l>(lh_pair0, lh_pair1); |
---|
727 | bit6 = simd<2>::pack<h,h>(ll_pair0, ll_pair1); |
---|
728 | bit7 = simd<2>::pack<l,l>(ll_pair0, ll_pair1); |
---|
729 | \end{verbatim} |
---|
730 | \caption{Complete Inductive Halving Algorithm} |
---|
731 | \label{halvingalgorithm} |
---|
732 | \end{figure} |
---|
733 | |
---|
734 | \subsection{Optimality of the Inductive Halving Algorithm} |
---|
735 | |
---|
736 | Here we show that the inductive halving algorithm presented in |
---|
737 | the previous subsection is optimal in the following sense: |
---|
738 | no other algorithm on any 3-register SIMD architecture can use |
---|
739 | fewer than 24 operations to transform eight serial registers |
---|
740 | of byte stream data into eight parallel registers of bit stream data. |
---|
741 | By 3-register SIMD architecture, we refer to any architecture |
---|
742 | that uses SIMD instructions consistent with our overall model of |
---|
743 | binary operations using two input register operands to produce |
---|
744 | one output register value. |
---|
745 | |
---|
746 | Observe that the $N$ data bits from each input register must be |
---|
747 | distributed $N/8$ each to the 8 output registers by virtue of |
---|
748 | the problem definition. Each output register can effectively |
---|
749 | be given a 3-bit address; the partitioning problem can be viewed |
---|
750 | as moving data to the correct address. However, each |
---|
751 | operation can move results into at most one register. |
---|
752 | At most this can result in the assignment of one correct address |
---|
753 | bit for each of the $N$ input bits. As all $8N$ input bits |
---|
754 | need to be moved to a register with a correct 3-bit address, |
---|
755 | a minimum of 24 operations is required. |
---|
756 | |
---|
757 | \section{Parallel to Serial Conversion} |
---|
758 | |
---|
759 | Parallel bit stream applications may apply string editing |
---|
760 | operations in bit space to substitute, delete or insert |
---|
761 | parallel sets of bits at particular positions. In such cases, |
---|
762 | the inverse transform that converts a set of parallel bit |
---|
763 | streams back into byte space is needed. In the example of |
---|
764 | UTF-8 to UTF-16 transcoding, the inverse transform is |
---|
765 | actually used twice for each application of the forward |
---|
766 | transform, to separately compute the high and low byte |
---|
767 | streams of each UTF-16 code unit. Those two byte streams |
---|
768 | are subsequentely merged to form the final result. |
---|
769 | |
---|
770 | Algorithms for performing the inverse transform mirror those |
---|
771 | of the forward transform, employing SIMD merge operations |
---|
772 | in place of pack operations. The best algorithm known |
---|
773 | to us on the commodity SIMD architectures takes advantage |
---|
774 | of versions of the \verb#simd<8>::mergeh# and \verb#simd<8>::mergel# |
---|
775 | operations that are available with each of the SSE and Altivec instruction |
---|
776 | sets. These algorithms take 72 operations to perform the |
---|
777 | inverse transposition of 8 parallel registers of bit stream |
---|
778 | data into 8 serial registers of byte stream data. |
---|
779 | |
---|
780 | \begin{figure}[tbh] |
---|
781 | \begin{center} |
---|
782 | \begin{verbatim} |
---|
783 | bit01_r0 = simd<1>::mergeh(bit0, bit1); |
---|
784 | bit01_r1 = simd<1>::mergel(bit0, bit1); |
---|
785 | bit23_r0 = simd<1>::mergeh(bit2, bit3); |
---|
786 | bit23_r1 = simd<1>::mergel(bit2, bit3); |
---|
787 | bit45_r0 = simd<1>::mergeh(bit4, bit5); |
---|
788 | bit45_r1 = simd<1>::mergel(bit4, bit5); |
---|
789 | bit67_r0 = simd<1>::mergeh(bit6, bit7); |
---|
790 | bit67_r1 = simd<1>::mergel(bit6, bit7); |
---|
791 | bit0123_r0 = simd<2>::mergeh(bit01_r0, bit23_r0); |
---|
792 | bit0123_r1 = simd<2>::mergel(bit01_r0, bit23_r0); |
---|
793 | bit0123_r2 = simd<2>::mergeh(bit01_r1, bit23_r1); |
---|
794 | bit0123_r3 = simd<2>::mergel(bit01_r1, bit23_r1); |
---|
795 | bit4567_r0 = simd<2>::mergeh(bit45_r0, bit67_r0); |
---|
796 | bit4567_r1 = simd<2>::mergel(bit45_r0, bit67_r0); |
---|
797 | bit4567_r2 = simd<2>::mergeh(bit45_r1, bit67_r1); |
---|
798 | bit4567_r3 = simd<2>::mergel(bit45_r1, bit67_r1); |
---|
799 | s0 = simd<4>::mergeh(bit0123_r0, bit4567_r0); |
---|
800 | s1 = simd<4>::mergel(bit0123_r0, bit4567_r0); |
---|
801 | s2 = simd<4>::mergeh(bit0123_r1, bit4567_r1); |
---|
802 | s3 = simd<4>::mergel(bit0123_r1, bit4567_r1); |
---|
803 | s4 = simd<4>::mergeh(bit0123_r2, bit4567_r2); |
---|
804 | s5 = simd<4>::mergel(bit0123_r2, bit4567_r2); |
---|
805 | s6 = simd<4>::mergeh(bit0123_r3, bit4567_r3); |
---|
806 | s7 = simd<4>::mergel(bit0123_r3, bit4567_r3); |
---|
807 | \end{verbatim} |
---|
808 | \end{center} |
---|
809 | \label{p2s-inductive} |
---|
810 | \caption{Parallel to Serial Transposition by Inductive Doubling} |
---|
811 | \end{figure} |
---|
812 | |
---|
813 | An algorithm employing only 24 operations using the |
---|
814 | inductive doubling instruction set architecture is relatively |
---|
815 | straightforward.. In stage 1, parallel registers for individual bit streams |
---|
816 | are first merged with bit-level interleaving |
---|
817 | using \verb#simd<1>::mergeh# and \verb#simd<8>::mergel# |
---|
818 | operations. For each of the four pairs of consecutive |
---|
819 | even/odd bit streams (bit0/bit1, bit2/bit3, bit4/bit5, bit6/bit7), |
---|
820 | two consecutive registers of bitpair data are produced. |
---|
821 | In stage 2, \verb#simd<2>::mergeh# and \verb#simd<2>::mergel# |
---|
822 | are then applied to merge to bitpair streams to produce streams |
---|
823 | of nybbles for the high and low nybble of each byte. Finally, |
---|
824 | the nybble streams are merged in stage 3 to produce the |
---|
825 | desired byte stream data. The full inductive doubling |
---|
826 | algorithm for parallel to serial transposition is shown in Figure |
---|
827 | \ref{p2s-inductive}. |
---|
828 | |
---|
829 | This algorithm is again optimal, requiring the fewest operations |
---|
830 | of any possible algorithm using any 3-register instruction set |
---|
831 | model. The proof directly follows that for the serial to parallel |
---|
832 | transposition. |
---|
833 | |
---|
834 | The existence of high-performance algorithms for transformation of |
---|
835 | character data between byte stream and parallel bit stream form |
---|
836 | in both directions makes it possible to consider applying these |
---|
837 | transformations multiple times during text processing applications. |
---|
838 | Just as the time domain and frequency domain each have their |
---|
839 | use in signal processing applications, the byte stream form and |
---|
840 | parallel bit stream form can then each be used at will in |
---|
841 | character stream applications. |
---|
842 | |
---|
843 | |
---|
844 | |
---|
845 | |
---|
846 | |
---|
847 | \section{Parallel Bit Deletion} |
---|
848 | |
---|
849 | Parallel bit deletion is an important operation that allows string |
---|
850 | editing operations to be carried out while in parallel bit stream |
---|
851 | form. It is also fundamental to UTF-8 to UTF-16 transcoding |
---|
852 | using parallel bit streams, allowing the excess code unit |
---|
853 | positions for UTF-8 two-, three- and four-byte sequences to |
---|
854 | be deleted once the sixteen parallel bit streams of UTF-16 have |
---|
855 | been computed \cite{PPoPP08}. |
---|
856 | |
---|
857 | Parallel bit deletion is specified using a deletion mask. |
---|
858 | A deletion mask is defined as a bit stream consisting of 1s at positions identifying bits |
---|
859 | to be deleted and 0s at positions identifying bits to be retained. |
---|
860 | For example, consider an 8-bit deletion mask \verb:10100010: and two corresponding 8-element parallel |
---|
861 | bit streams \verb:abcdefgh: and \verb:ABCDEFGH:. Parallel deletion of elements from both bit streams in |
---|
862 | accordance with the mask yields two five element streams, i.e., \verb:bdefh: and \verb:BDEFH:. |
---|
863 | |
---|
864 | Bit deletion may be performed using |
---|
865 | the parallel-prefix compress algorithm documented by |
---|
866 | Warren and attributed to Steele \cite{HackersDelight}. This algorithm uses |
---|
867 | only logic and shifts with a constant parameter to carry |
---|
868 | out the deletion process. However, it requires $k^2$ |
---|
869 | preprocessing steps for a final field width parameter |
---|
870 | of size $2^k$, as well as 4 operations per deletion step |
---|
871 | per stream. Using the inductive doubling instruction set architecture |
---|
872 | it is possible to carry out bit deletion much more efficiently. |
---|
873 | |
---|
874 | Deletion within fixed size fields or registers may produce results that are either |
---|
875 | left justified or right-justified. For example, a five-element stream \verb:bdefh: within an |
---|
876 | eight-element field may be represented as either \verb:bdefhxxx: or \verb:xxxbdefh:, with don't |
---|
877 | care positions marked `\verb:x:'. Concatenating an adjacent right-justified result with a |
---|
878 | left-justified result produces an important intermediate form known as a |
---|
879 | {\em central deletion result}. For example, \verb:xxbd: and \verb:efhx: may be respective |
---|
880 | right-justified and left-justified results from the application of the |
---|
881 | 4-bit deletion masks \verb:1010: and \verb:0010: to the two consecutive 4-element |
---|
882 | stream segments \verb:abcd: and \verb:efgh:. Concatenation of \verb:xxbd: and \verb:efhx: produces |
---|
883 | the central result \verb:xxbdefhx:, which may easily be converted to a either a |
---|
884 | left or a right justified 8-element result by an appropriate shift operation. |
---|
885 | |
---|
886 | \begin{figure} |
---|
887 | \begin{center} |
---|
888 | \begin{tabular}{|c||c|c|c|c|c|c|c|c|} |
---|
889 | \hline |
---|
890 | \verb:delmask: & \verb:1001: & \verb:1100: & \verb:0100: & \verb:1111: & \verb:0111: & \verb:0010: & \verb:0011: & \verb:0010: \\ \hline |
---|
891 | \verb:bits: & \verb:0bc0: & \verb:00gh: & \verb:i0kl: & \verb:0000: & \verb:q000: & \verb:uv0x: & \verb:yz00: & \verb:CD0F: \\ \hline |
---|
892 | \verb:rslt_8: & \multicolumn{2}{c|}{\tt 00bcgh00} & \multicolumn{2}{c|}{\tt 0ikl0000} & \multicolumn{2}{c|}{\tt 000quvx0} & \multicolumn{2}{c|}{\tt 00yzCDF0} \\ \hline |
---|
893 | \verb:cts_4: & 2 & 2 & 1 & 4 & 3 & 1 & 2 & 1 \\ \hline |
---|
894 | \verb:rj: & \multicolumn{2}{c|}{6} & \multicolumn{2}{c|}{XX} & \multicolumn{2}{c|}{7} & \multicolumn{2}{c|}{XX} \\ \hline |
---|
895 | \verb:lj: & \multicolumn{2}{c|}{XX} & \multicolumn{2}{c|}{1} & \multicolumn{2}{c|}{XX} & \multicolumn{2}{c|}{2} \\ \hline |
---|
896 | \verb:rot_8: & \multicolumn{2}{c|}{6} & \multicolumn{2}{c|}{1} & \multicolumn{2}{c|}{7} & \multicolumn{2}{c|}{2} \\ \hline |
---|
897 | \verb:rslt_16: & \multicolumn{4}{c|}{\tt 0000bcghikl00000} & \multicolumn{4}{c|}{\tt 0000quvxyzCDF000} \\ \hline |
---|
898 | \end{tabular} |
---|
899 | \end{center} |
---|
900 | \label{centraldelstep} |
---|
901 | \caption{Example $8 \rightarrow 16$ Step in Deletion by Central Result Induction} |
---|
902 | \end{figure} |
---|
903 | |
---|
904 | The observation about how two $n$-bit central deletion results can |
---|
905 | combine to yield a $2n$ central deletion result provides the basis |
---|
906 | for an inductive doubling algorithm. Figure \ref{centraldelstep} |
---|
907 | illustrates the inductive process for the transition from 8-bit central |
---|
908 | deletion results to 16-bit central deletion results. The top row shows |
---|
909 | the original deletion mask, while the second row shows the original |
---|
910 | bit stream to which deletions are to be applied, with deleted bits zeroed out. |
---|
911 | The third row shows the central result for each 8-bit field as the |
---|
912 | result of the previous inductive step. |
---|
913 | |
---|
914 | To perform the $8 \rightarrow 16$ central deletion step, we first form |
---|
915 | the population counts of 4-bit fields of the original deletion mask as |
---|
916 | shown in row 4 of Figure \ref{centraldelstep}. Note that in right-justifying |
---|
917 | an 8-bit central result, we perform a right shift by the population count |
---|
918 | of the low half of the field. Similarly, |
---|
919 | left-justification requires a left-shift by the population count in the |
---|
920 | high half of the field. |
---|
921 | |
---|
922 | The left and right shifts can be performed simultaneously using a rotate |
---|
923 | left instruction. Right justification by shifting an $n$ bit field |
---|
924 | $i$ positions to the right is equivalent to a left rotate of $n-i$ |
---|
925 | positions. These rotation amounts are computed by the operation \newline |
---|
926 | \verb#rj=simd<8>::sub<x,l>(simd<8>::const(8), cts_4)# as shown in row 5, |
---|
927 | except that don't care fields (which won't be subsequently used) |
---|
928 | are marked \verb:XX:. |
---|
929 | |
---|
930 | The left shift amounts are calculated by \verb#lj=simd<8>::srli<4>(cts_4)# |
---|
931 | as shown in row 6, and are combined with the right shift amounts |
---|
932 | by the selection operation \newline \verb#rot_8=simd_if(simd<16>::const(0xFF00), rj, lj)# |
---|
933 | as shown in row 7. Using these computed values, the inductive step |
---|
934 | is completed by application of the operation \newline \verb#rslt_16=simd<8>::rotl(rslt_8, rot_8)# |
---|
935 | as shown in row 8. |
---|
936 | |
---|
937 | At each inductive doubling level, it requires 4 operations to compute the |
---|
938 | required deletion infomation and one operation per bit stream to perform deletion. |
---|
939 | Note that, if deletion is to be applied to a set of eight parallel bit streams, |
---|
940 | the computed deletion information is used for each stream without recomputation, |
---|
941 | thus requiring 12 operations per inductive level. |
---|
942 | |
---|
943 | In comparison to the parallel-prefix compress method, the method of central |
---|
944 | deletion results using the inductive doubling architecture has far fewer operations. |
---|
945 | The total preprocessing cost is $4k$ for $k$ steps of deletion by central result |
---|
946 | induction versus $4k^2$ for the parallel-prefix method. Using the computed |
---|
947 | deletion operation, only a single SIMD rotate operation per bit stream |
---|
948 | per level is needed, in comparison with 4 operations per level for parallel-prefix |
---|
949 | compress. |
---|
950 | |
---|
951 | |
---|
952 | \section{Systematic Support for Horizontal SIMD Operations} |
---|
953 | |
---|
954 | In SIMD parlance, {\em horizontal} operations are |
---|
955 | operations which combine values from two or more fields |
---|
956 | of the same register, in contrast to the normal |
---|
957 | {\em vertical} operations which combine corresponding |
---|
958 | fields of different registers. Horizontal operations |
---|
959 | can be found that combine two (e.g., haddpd on SSE3), |
---|
960 | four (e.g, \verb:si_orx: on SPU), eight (e.g, psadbw on SSE) |
---|
961 | or sixteen values (e.g., vcmpequb on Altivec). Some |
---|
962 | horizontal operations have a vertical component as well. |
---|
963 | For example, psadbw first forms the absolute value of |
---|
964 | the difference of eight corresponding byte fields before |
---|
965 | performing horizontal add of the eight values, while |
---|
966 | vsum4ubs on Altivec performs horizontal add of sets of |
---|
967 | four unsigned 8-bit fields within one register |
---|
968 | and then combines the result horizontally with |
---|
969 | corresponding 32-bit fields of a second registers. |
---|
970 | |
---|
971 | The space of potential horizontal operations thus has |
---|
972 | many dimensions, including not only the particular |
---|
973 | combining operation and the operand field width, but |
---|
974 | also the number of fields being combined, whether a |
---|
975 | vertical combination is applied and whether it is applied |
---|
976 | before or after the horizontal operation and what the |
---|
977 | nature of the vertical combining operation is. |
---|
978 | Within this space, commodity SIMD architectures tend |
---|
979 | to support only a very few combinations, without any |
---|
980 | particular attempt at systematic support for horizontal |
---|
981 | operations in general. |
---|
982 | |
---|
983 | By making use of \verb:<l,h>: half-operand modifier |
---|
984 | combinations, the inductive doubling architecture |
---|
985 | offers systematic support for horizontal operations |
---|
986 | on pairs of adjacent fields. |
---|
987 | For example, \verb#simd<16>::add<l,h># adds values |
---|
988 | in adjacent 8 bit fields to produce 16 bit results, |
---|
989 | while \verb#simd<32>::min<l,h># can produce the |
---|
990 | minimum value of adjacent 16-bit fields. In general, |
---|
991 | \newline \verb#simd<n>::F<l,h># denotes the horizontal |
---|
992 | binary combination of adjacent fields for any |
---|
993 | operator $F$ and field width $n$. |
---|
994 | |
---|
995 | Horizontal combinations of larger numbers of fields |
---|
996 | makes use of the inductive doubling property. |
---|
997 | For example, consider the or-across operation \verb:si_orx: |
---|
998 | of the SPU, that performs a logical or operation |
---|
999 | on four 32-bit fields. This four field combination |
---|
1000 | involves two steps in the inductive doubling approach. |
---|
1001 | %\begin{singlespace} |
---|
1002 | \begin{verbatim} |
---|
1003 | t = simd<64>::or<l,h>(x, x) |
---|
1004 | t = simd<128>::or<l,h>(t, t) |
---|
1005 | \end{verbatim} |
---|
1006 | %\end{singlespace} |
---|
1007 | This example is also interesting in showing a potential |
---|
1008 | value for supporting bitwise logical operations at |
---|
1009 | different field widths, i.e., specifically for use with |
---|
1010 | half-operand modifiers. |
---|
1011 | |
---|
1012 | Similarly, to combine any eight fields simply requires |
---|
1013 | three inductive doubling steps using the desired |
---|
1014 | operator at successive power-of-two field widths, while |
---|
1015 | combining sixteen fields requires four such operations. |
---|
1016 | In this way, the inductive doubling architecture provides |
---|
1017 | systematic support for horizontal operations well beyond |
---|
1018 | the existing facilities of commodity architectures, |
---|
1019 | although lacking some of the special features found in |
---|
1020 | some cases. |
---|
1021 | |
---|
1022 | |
---|
1023 | \section{Implementation} |
---|
1024 | |
---|
1025 | We have constructed libraries that provide |
---|
1026 | simulated implementation of the inductive doubling architecture |
---|
1027 | on each of the MMX, SSE, Altivec, and SPU platforms and have |
---|
1028 | used these libraries in the implementation of each of the |
---|
1029 | parallel bit stream algorithms discussed herein. |
---|
1030 | This implementation work has been successful in validating |
---|
1031 | the basic concepts underlying the inductive doubling instruction |
---|
1032 | set architecture. |
---|
1033 | |
---|
1034 | Implementation of the architecture on chip is beyond the |
---|
1035 | scope of our present resources and capabilities. However, |
---|
1036 | the principal requirements are implementation of the various |
---|
1037 | operations at all power-of-2 field widths and implementation |
---|
1038 | of half-operand modifiers. Implementation of SIMD operations |
---|
1039 | at additional field widths involves design trade-offs |
---|
1040 | with respect to transistor counts, available opcode space, |
---|
1041 | and the potential value of the new operations to SIMD |
---|
1042 | programmers. From the perspective of parallel bit |
---|
1043 | stream programming, the primary need is for SIMD integer, |
---|
1044 | shift, pack and merge operations at field widths of 2, 4 |
---|
1045 | and 8, as well as the field width of 1, where it makes |
---|
1046 | sense (e.g. with merge operations). In support of the |
---|
1047 | general concept of inductive doubling architecture, |
---|
1048 | SIMD operations at large field widths (64, 128) are also |
---|
1049 | called for, but these operations cannot be justified on |
---|
1050 | the basis of parallel bit stream programming. |
---|
1051 | |
---|
1052 | Implementation of half-operand modifiers can logically |
---|
1053 | be carried out with additional circuitry attached to the |
---|
1054 | register fetch units of a pipelined processor. This |
---|
1055 | circuitry would require control signals from the |
---|
1056 | instruction decode unit to identify the field widths |
---|
1057 | of operands and the particular half-operand modifier to be applied, |
---|
1058 | if any. The additional logic required for instruction |
---|
1059 | decode and that required for operand modification |
---|
1060 | as part of the operand fetch process is expected to be |
---|
1061 | reasonably modest. |
---|
1062 | |
---|
1063 | Full assessment of implementation issues is an important |
---|
1064 | area for future work. |
---|
1065 | |
---|
1066 | |
---|
1067 | \section{Conclusions} |
---|
1068 | |
---|
1069 | This paper has considered the issue of architectural support for |
---|
1070 | SIMD text processing using the method of parallel bit streams and has |
---|
1071 | argued that this architectural support can best be provided |
---|
1072 | through a SIMD instruction set architecture that implements |
---|
1073 | features for direct support of inductive doubling algorithms. |
---|
1074 | Four key features of the inductive doubling architecture have |
---|
1075 | been identified include support for operations at all |
---|
1076 | power-of-2 field widths, half-operand modifiers and |
---|
1077 | pack and merge operations. The principle innovation is the |
---|
1078 | notion of half-operand modifiers to support efficient |
---|
1079 | transition between successive power-of-two field widths. |
---|
1080 | |
---|
1081 | Several algorithms key to parallel bit stream methods |
---|
1082 | have been examined and shown to benefit from dramatic |
---|
1083 | reductions in instruction count compared to the best |
---|
1084 | known algorithms on existing architecture. In the case |
---|
1085 | of transposition algorithms to and from parallel bit stream |
---|
1086 | form, the architecture has been shown to make possible |
---|
1087 | straightforward inductive doubling algorithms with the |
---|
1088 | lowest total number of operations that can be achieved by any |
---|
1089 | possible 3-register SIMD architecture. |
---|
1090 | |
---|
1091 | The inductive doubling architecture also has considerable |
---|
1092 | benefits beyond its role in supporting SIMD programming |
---|
1093 | with parallel bit streams. Notable among these is |
---|
1094 | that the architecture provides a framework for systematic |
---|
1095 | support of horizontal SIMD operations. |
---|
1096 | |
---|
1097 | |
---|
1098 | |
---|
1099 | %\appendix |
---|
1100 | %\section{Appendix Title} |
---|
1101 | % |
---|
1102 | %This is the text of the appendix, if you need one. |
---|
1103 | |
---|
1104 | \acks |
---|
1105 | |
---|
1106 | This research was supported by a Discovery Grant from the |
---|
1107 | Natural Sciences and Engineering Research Council of Canada. |
---|
1108 | |
---|
1109 | %\bibliographystyle{plainnat} |
---|
1110 | \bibliographystyle{plain} |
---|
1111 | \bibliography{asplos094-cameron} |
---|
1112 | %\begin{thebibliography}{} |
---|
1113 | |
---|
1114 | %\bibitem{smith02} |
---|
1115 | %Smith, P. Q. reference text |
---|
1116 | |
---|
1117 | %\end{thebibliography} |
---|
1118 | |
---|
1119 | |
---|
1120 | \end{document} |
---|