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111
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1 /* enough.c -- determine the maximum size of inflate's Huffman code tables over
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2 * all possible valid and complete Huffman codes, subject to a length limit.
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3 * Copyright (C) 2007, 2008, 2012 Mark Adler
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4 * Version 1.4 18 August 2012 Mark Adler
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5 */
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6
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7 /* Version history:
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8 1.0 3 Jan 2007 First version (derived from codecount.c version 1.4)
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9 1.1 4 Jan 2007 Use faster incremental table usage computation
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10 Prune examine() search on previously visited states
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11 1.2 5 Jan 2007 Comments clean up
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12 As inflate does, decrease root for short codes
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13 Refuse cases where inflate would increase root
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14 1.3 17 Feb 2008 Add argument for initial root table size
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15 Fix bug for initial root table size == max - 1
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16 Use a macro to compute the history index
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17 1.4 18 Aug 2012 Avoid shifts more than bits in type (caused endless loop!)
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18 Clean up comparisons of different types
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19 Clean up code indentation
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20 */
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21
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22 /*
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23 Examine all possible Huffman codes for a given number of symbols and a
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24 maximum code length in bits to determine the maximum table size for zilb's
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25 inflate. Only complete Huffman codes are counted.
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26
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27 Two codes are considered distinct if the vectors of the number of codes per
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28 length are not identical. So permutations of the symbol assignments result
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29 in the same code for the counting, as do permutations of the assignments of
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30 the bit values to the codes (i.e. only canonical codes are counted).
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31
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32 We build a code from shorter to longer lengths, determining how many symbols
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33 are coded at each length. At each step, we have how many symbols remain to
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34 be coded, what the last code length used was, and how many bit patterns of
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35 that length remain unused. Then we add one to the code length and double the
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36 number of unused patterns to graduate to the next code length. We then
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37 assign all portions of the remaining symbols to that code length that
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38 preserve the properties of a correct and eventually complete code. Those
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39 properties are: we cannot use more bit patterns than are available; and when
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40 all the symbols are used, there are exactly zero possible bit patterns
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41 remaining.
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42
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43 The inflate Huffman decoding algorithm uses two-level lookup tables for
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44 speed. There is a single first-level table to decode codes up to root bits
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45 in length (root == 9 in the current inflate implementation). The table
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46 has 1 << root entries and is indexed by the next root bits of input. Codes
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47 shorter than root bits have replicated table entries, so that the correct
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48 entry is pointed to regardless of the bits that follow the short code. If
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49 the code is longer than root bits, then the table entry points to a second-
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50 level table. The size of that table is determined by the longest code with
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51 that root-bit prefix. If that longest code has length len, then the table
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52 has size 1 << (len - root), to index the remaining bits in that set of
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53 codes. Each subsequent root-bit prefix then has its own sub-table. The
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54 total number of table entries required by the code is calculated
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55 incrementally as the number of codes at each bit length is populated. When
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56 all of the codes are shorter than root bits, then root is reduced to the
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57 longest code length, resulting in a single, smaller, one-level table.
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58
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59 The inflate algorithm also provides for small values of root (relative to
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60 the log2 of the number of symbols), where the shortest code has more bits
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61 than root. In that case, root is increased to the length of the shortest
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62 code. This program, by design, does not handle that case, so it is verified
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63 that the number of symbols is less than 2^(root + 1).
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64
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65 In order to speed up the examination (by about ten orders of magnitude for
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66 the default arguments), the intermediate states in the build-up of a code
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67 are remembered and previously visited branches are pruned. The memory
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68 required for this will increase rapidly with the total number of symbols and
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69 the maximum code length in bits. However this is a very small price to pay
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70 for the vast speedup.
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71
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72 First, all of the possible Huffman codes are counted, and reachable
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73 intermediate states are noted by a non-zero count in a saved-results array.
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74 Second, the intermediate states that lead to (root + 1) bit or longer codes
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75 are used to look at all sub-codes from those junctures for their inflate
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76 memory usage. (The amount of memory used is not affected by the number of
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77 codes of root bits or less in length.) Third, the visited states in the
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78 construction of those sub-codes and the associated calculation of the table
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79 size is recalled in order to avoid recalculating from the same juncture.
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80 Beginning the code examination at (root + 1) bit codes, which is enabled by
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81 identifying the reachable nodes, accounts for about six of the orders of
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82 magnitude of improvement for the default arguments. About another four
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83 orders of magnitude come from not revisiting previous states. Out of
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84 approximately 2x10^16 possible Huffman codes, only about 2x10^6 sub-codes
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85 need to be examined to cover all of the possible table memory usage cases
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86 for the default arguments of 286 symbols limited to 15-bit codes.
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87
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88 Note that an unsigned long long type is used for counting. It is quite easy
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89 to exceed the capacity of an eight-byte integer with a large number of
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90 symbols and a large maximum code length, so multiple-precision arithmetic
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91 would need to replace the unsigned long long arithmetic in that case. This
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92 program will abort if an overflow occurs. The big_t type identifies where
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93 the counting takes place.
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94
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95 An unsigned long long type is also used for calculating the number of
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96 possible codes remaining at the maximum length. This limits the maximum
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97 code length to the number of bits in a long long minus the number of bits
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98 needed to represent the symbols in a flat code. The code_t type identifies
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99 where the bit pattern counting takes place.
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100 */
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101
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102 #include <stdio.h>
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103 #include <stdlib.h>
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104 #include <string.h>
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105 #include <assert.h>
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106
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107 #define local static
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108
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109 /* special data types */
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110 typedef unsigned long long big_t; /* type for code counting */
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111 typedef unsigned long long code_t; /* type for bit pattern counting */
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112 struct tab { /* type for been here check */
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113 size_t len; /* length of bit vector in char's */
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114 char *vec; /* allocated bit vector */
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115 };
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116
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117 /* The array for saving results, num[], is indexed with this triplet:
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118
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119 syms: number of symbols remaining to code
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120 left: number of available bit patterns at length len
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121 len: number of bits in the codes currently being assigned
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122
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123 Those indices are constrained thusly when saving results:
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124
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125 syms: 3..totsym (totsym == total symbols to code)
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126 left: 2..syms - 1, but only the evens (so syms == 8 -> 2, 4, 6)
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127 len: 1..max - 1 (max == maximum code length in bits)
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128
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129 syms == 2 is not saved since that immediately leads to a single code. left
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130 must be even, since it represents the number of available bit patterns at
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131 the current length, which is double the number at the previous length.
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132 left ends at syms-1 since left == syms immediately results in a single code.
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133 (left > sym is not allowed since that would result in an incomplete code.)
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134 len is less than max, since the code completes immediately when len == max.
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135
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136 The offset into the array is calculated for the three indices with the
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137 first one (syms) being outermost, and the last one (len) being innermost.
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138 We build the array with length max-1 lists for the len index, with syms-3
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139 of those for each symbol. There are totsym-2 of those, with each one
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140 varying in length as a function of sym. See the calculation of index in
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141 count() for the index, and the calculation of size in main() for the size
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142 of the array.
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143
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144 For the deflate example of 286 symbols limited to 15-bit codes, the array
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145 has 284,284 entries, taking up 2.17 MB for an 8-byte big_t. More than
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146 half of the space allocated for saved results is actually used -- not all
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147 possible triplets are reached in the generation of valid Huffman codes.
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148 */
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149
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150 /* The array for tracking visited states, done[], is itself indexed identically
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151 to the num[] array as described above for the (syms, left, len) triplet.
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152 Each element in the array is further indexed by the (mem, rem) doublet,
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153 where mem is the amount of inflate table space used so far, and rem is the
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154 remaining unused entries in the current inflate sub-table. Each indexed
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155 element is simply one bit indicating whether the state has been visited or
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156 not. Since the ranges for mem and rem are not known a priori, each bit
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157 vector is of a variable size, and grows as needed to accommodate the visited
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158 states. mem and rem are used to calculate a single index in a triangular
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159 array. Since the range of mem is expected in the default case to be about
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160 ten times larger than the range of rem, the array is skewed to reduce the
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161 memory usage, with eight times the range for mem than for rem. See the
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162 calculations for offset and bit in beenhere() for the details.
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163
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164 For the deflate example of 286 symbols limited to 15-bit codes, the bit
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165 vectors grow to total approximately 21 MB, in addition to the 4.3 MB done[]
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166 array itself.
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167 */
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168
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169 /* Globals to avoid propagating constants or constant pointers recursively */
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170 local int max; /* maximum allowed bit length for the codes */
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171 local int root; /* size of base code table in bits */
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172 local int large; /* largest code table so far */
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173 local size_t size; /* number of elements in num and done */
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174 local int *code; /* number of symbols assigned to each bit length */
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175 local big_t *num; /* saved results array for code counting */
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176 local struct tab *done; /* states already evaluated array */
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177
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178 /* Index function for num[] and done[] */
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179 #define INDEX(i,j,k) (((size_t)((i-1)>>1)*((i-2)>>1)+(j>>1)-1)*(max-1)+k-1)
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180
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181 /* Free allocated space. Uses globals code, num, and done. */
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182 local void cleanup(void)
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183 {
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184 size_t n;
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185
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186 if (done != NULL) {
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187 for (n = 0; n < size; n++)
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188 if (done[n].len)
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189 free(done[n].vec);
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190 free(done);
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191 }
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192 if (num != NULL)
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193 free(num);
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194 if (code != NULL)
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195 free(code);
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196 }
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197
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198 /* Return the number of possible Huffman codes using bit patterns of lengths
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199 len through max inclusive, coding syms symbols, with left bit patterns of
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200 length len unused -- return -1 if there is an overflow in the counting.
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201 Keep a record of previous results in num to prevent repeating the same
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202 calculation. Uses the globals max and num. */
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203 local big_t count(int syms, int len, int left)
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204 {
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205 big_t sum; /* number of possible codes from this juncture */
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206 big_t got; /* value returned from count() */
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207 int least; /* least number of syms to use at this juncture */
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208 int most; /* most number of syms to use at this juncture */
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209 int use; /* number of bit patterns to use in next call */
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210 size_t index; /* index of this case in *num */
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211
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212 /* see if only one possible code */
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213 if (syms == left)
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214 return 1;
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215
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216 /* note and verify the expected state */
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217 assert(syms > left && left > 0 && len < max);
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218
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219 /* see if we've done this one already */
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220 index = INDEX(syms, left, len);
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221 got = num[index];
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222 if (got)
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223 return got; /* we have -- return the saved result */
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224
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225 /* we need to use at least this many bit patterns so that the code won't be
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226 incomplete at the next length (more bit patterns than symbols) */
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227 least = (left << 1) - syms;
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228 if (least < 0)
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229 least = 0;
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230
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231 /* we can use at most this many bit patterns, lest there not be enough
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232 available for the remaining symbols at the maximum length (if there were
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233 no limit to the code length, this would become: most = left - 1) */
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234 most = (((code_t)left << (max - len)) - syms) /
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235 (((code_t)1 << (max - len)) - 1);
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236
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237 /* count all possible codes from this juncture and add them up */
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238 sum = 0;
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239 for (use = least; use <= most; use++) {
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240 got = count(syms - use, len + 1, (left - use) << 1);
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241 sum += got;
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242 if (got == (big_t)0 - 1 || sum < got) /* overflow */
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243 return (big_t)0 - 1;
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244 }
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245
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246 /* verify that all recursive calls are productive */
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247 assert(sum != 0);
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248
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249 /* save the result and return it */
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250 num[index] = sum;
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251 return sum;
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252 }
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253
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254 /* Return true if we've been here before, set to true if not. Set a bit in a
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255 bit vector to indicate visiting this state. Each (syms,len,left) state
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256 has a variable size bit vector indexed by (mem,rem). The bit vector is
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257 lengthened if needed to allow setting the (mem,rem) bit. */
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258 local int beenhere(int syms, int len, int left, int mem, int rem)
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259 {
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260 size_t index; /* index for this state's bit vector */
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261 size_t offset; /* offset in this state's bit vector */
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262 int bit; /* mask for this state's bit */
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263 size_t length; /* length of the bit vector in bytes */
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264 char *vector; /* new or enlarged bit vector */
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265
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266 /* point to vector for (syms,left,len), bit in vector for (mem,rem) */
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267 index = INDEX(syms, left, len);
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268 mem -= 1 << root;
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269 offset = (mem >> 3) + rem;
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270 offset = ((offset * (offset + 1)) >> 1) + rem;
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271 bit = 1 << (mem & 7);
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272
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273 /* see if we've been here */
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274 length = done[index].len;
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275 if (offset < length && (done[index].vec[offset] & bit) != 0)
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276 return 1; /* done this! */
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277
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278 /* we haven't been here before -- set the bit to show we have now */
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279
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280 /* see if we need to lengthen the vector in order to set the bit */
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281 if (length <= offset) {
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282 /* if we have one already, enlarge it, zero out the appended space */
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283 if (length) {
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284 do {
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285 length <<= 1;
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286 } while (length <= offset);
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287 vector = realloc(done[index].vec, length);
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288 if (vector != NULL)
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289 memset(vector + done[index].len, 0, length - done[index].len);
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290 }
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291
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292 /* otherwise we need to make a new vector and zero it out */
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293 else {
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294 length = 1 << (len - root);
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295 while (length <= offset)
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296 length <<= 1;
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297 vector = calloc(length, sizeof(char));
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298 }
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299
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300 /* in either case, bail if we can't get the memory */
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301 if (vector == NULL) {
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302 fputs("abort: unable to allocate enough memory\n", stderr);
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303 cleanup();
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304 exit(1);
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305 }
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306
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307 /* install the new vector */
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308 done[index].len = length;
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309 done[index].vec = vector;
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310 }
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311
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312 /* set the bit */
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313 done[index].vec[offset] |= bit;
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314 return 0;
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315 }
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316
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317 /* Examine all possible codes from the given node (syms, len, left). Compute
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318 the amount of memory required to build inflate's decoding tables, where the
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319 number of code structures used so far is mem, and the number remaining in
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320 the current sub-table is rem. Uses the globals max, code, root, large, and
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321 done. */
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322 local void examine(int syms, int len, int left, int mem, int rem)
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323 {
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324 int least; /* least number of syms to use at this juncture */
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325 int most; /* most number of syms to use at this juncture */
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326 int use; /* number of bit patterns to use in next call */
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327
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328 /* see if we have a complete code */
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329 if (syms == left) {
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330 /* set the last code entry */
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331 code[len] = left;
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332
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333 /* complete computation of memory used by this code */
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334 while (rem < left) {
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335 left -= rem;
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336 rem = 1 << (len - root);
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337 mem += rem;
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338 }
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339 assert(rem == left);
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340
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341 /* if this is a new maximum, show the entries used and the sub-code */
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342 if (mem > large) {
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343 large = mem;
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344 printf("max %d: ", mem);
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345 for (use = root + 1; use <= max; use++)
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346 if (code[use])
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347 printf("%d[%d] ", code[use], use);
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348 putchar('\n');
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349 fflush(stdout);
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350 }
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351
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352 /* remove entries as we drop back down in the recursion */
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353 code[len] = 0;
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354 return;
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355 }
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356
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357 /* prune the tree if we can */
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358 if (beenhere(syms, len, left, mem, rem))
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359 return;
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360
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361 /* we need to use at least this many bit patterns so that the code won't be
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362 incomplete at the next length (more bit patterns than symbols) */
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363 least = (left << 1) - syms;
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364 if (least < 0)
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365 least = 0;
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366
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367 /* we can use at most this many bit patterns, lest there not be enough
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368 available for the remaining symbols at the maximum length (if there were
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369 no limit to the code length, this would become: most = left - 1) */
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370 most = (((code_t)left << (max - len)) - syms) /
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371 (((code_t)1 << (max - len)) - 1);
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372
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373 /* occupy least table spaces, creating new sub-tables as needed */
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374 use = least;
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375 while (rem < use) {
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376 use -= rem;
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377 rem = 1 << (len - root);
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378 mem += rem;
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379 }
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380 rem -= use;
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381
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382 /* examine codes from here, updating table space as we go */
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383 for (use = least; use <= most; use++) {
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384 code[len] = use;
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385 examine(syms - use, len + 1, (left - use) << 1,
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386 mem + (rem ? 1 << (len - root) : 0), rem << 1);
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387 if (rem == 0) {
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388 rem = 1 << (len - root);
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389 mem += rem;
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390 }
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391 rem--;
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392 }
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393
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394 /* remove entries as we drop back down in the recursion */
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395 code[len] = 0;
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396 }
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397
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398 /* Look at all sub-codes starting with root + 1 bits. Look at only the valid
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399 intermediate code states (syms, left, len). For each completed code,
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400 calculate the amount of memory required by inflate to build the decoding
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401 tables. Find the maximum amount of memory required and show the code that
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402 requires that maximum. Uses the globals max, root, and num. */
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403 local void enough(int syms)
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404 {
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405 int n; /* number of remaing symbols for this node */
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406 int left; /* number of unused bit patterns at this length */
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407 size_t index; /* index of this case in *num */
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408
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409 /* clear code */
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410 for (n = 0; n <= max; n++)
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411 code[n] = 0;
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412
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413 /* look at all (root + 1) bit and longer codes */
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414 large = 1 << root; /* base table */
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415 if (root < max) /* otherwise, there's only a base table */
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416 for (n = 3; n <= syms; n++)
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417 for (left = 2; left < n; left += 2)
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418 {
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419 /* look at all reachable (root + 1) bit nodes, and the
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420 resulting codes (complete at root + 2 or more) */
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421 index = INDEX(n, left, root + 1);
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422 if (root + 1 < max && num[index]) /* reachable node */
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423 examine(n, root + 1, left, 1 << root, 0);
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424
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425 /* also look at root bit codes with completions at root + 1
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426 bits (not saved in num, since complete), just in case */
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427 if (num[index - 1] && n <= left << 1)
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428 examine((n - left) << 1, root + 1, (n - left) << 1,
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429 1 << root, 0);
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430 }
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431
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432 /* done */
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433 printf("done: maximum of %d table entries\n", large);
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434 }
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435
|
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436 /*
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437 Examine and show the total number of possible Huffman codes for a given
|
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438 maximum number of symbols, initial root table size, and maximum code length
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439 in bits -- those are the command arguments in that order. The default
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440 values are 286, 9, and 15 respectively, for the deflate literal/length code.
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441 The possible codes are counted for each number of coded symbols from two to
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442 the maximum. The counts for each of those and the total number of codes are
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443 shown. The maximum number of inflate table entires is then calculated
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444 across all possible codes. Each new maximum number of table entries and the
|
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445 associated sub-code (starting at root + 1 == 10 bits) is shown.
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446
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447 To count and examine Huffman codes that are not length-limited, provide a
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448 maximum length equal to the number of symbols minus one.
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449
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|
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450 For the deflate literal/length code, use "enough". For the deflate distance
|
|
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451 code, use "enough 30 6".
|
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452
|
|
|
453 This uses the %llu printf format to print big_t numbers, which assumes that
|
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|
454 big_t is an unsigned long long. If the big_t type is changed (for example
|
|
|
455 to a multiple precision type), the method of printing will also need to be
|
|
|
456 updated.
|
|
|
457 */
|
|
|
458 int main(int argc, char **argv)
|
|
|
459 {
|
|
|
460 int syms; /* total number of symbols to code */
|
|
|
461 int n; /* number of symbols to code for this run */
|
|
|
462 big_t got; /* return value of count() */
|
|
|
463 big_t sum; /* accumulated number of codes over n */
|
|
|
464 code_t word; /* for counting bits in code_t */
|
|
|
465
|
|
|
466 /* set up globals for cleanup() */
|
|
|
467 code = NULL;
|
|
|
468 num = NULL;
|
|
|
469 done = NULL;
|
|
|
470
|
|
|
471 /* get arguments -- default to the deflate literal/length code */
|
|
|
472 syms = 286;
|
|
|
473 root = 9;
|
|
|
474 max = 15;
|
|
|
475 if (argc > 1) {
|
|
|
476 syms = atoi(argv[1]);
|
|
|
477 if (argc > 2) {
|
|
|
478 root = atoi(argv[2]);
|
|
|
479 if (argc > 3)
|
|
|
480 max = atoi(argv[3]);
|
|
|
481 }
|
|
|
482 }
|
|
|
483 if (argc > 4 || syms < 2 || root < 1 || max < 1) {
|
|
|
484 fputs("invalid arguments, need: [sym >= 2 [root >= 1 [max >= 1]]]\n",
|
|
|
485 stderr);
|
|
|
486 return 1;
|
|
|
487 }
|
|
|
488
|
|
|
489 /* if not restricting the code length, the longest is syms - 1 */
|
|
|
490 if (max > syms - 1)
|
|
|
491 max = syms - 1;
|
|
|
492
|
|
|
493 /* determine the number of bits in a code_t */
|
|
|
494 for (n = 0, word = 1; word; n++, word <<= 1)
|
|
|
495 ;
|
|
|
496
|
|
|
497 /* make sure that the calculation of most will not overflow */
|
|
|
498 if (max > n || (code_t)(syms - 2) >= (((code_t)0 - 1) >> (max - 1))) {
|
|
|
499 fputs("abort: code length too long for internal types\n", stderr);
|
|
|
500 return 1;
|
|
|
501 }
|
|
|
502
|
|
|
503 /* reject impossible code requests */
|
|
|
504 if ((code_t)(syms - 1) > ((code_t)1 << max) - 1) {
|
|
|
505 fprintf(stderr, "%d symbols cannot be coded in %d bits\n",
|
|
|
506 syms, max);
|
|
|
507 return 1;
|
|
|
508 }
|
|
|
509
|
|
|
510 /* allocate code vector */
|
|
|
511 code = calloc(max + 1, sizeof(int));
|
|
|
512 if (code == NULL) {
|
|
|
513 fputs("abort: unable to allocate enough memory\n", stderr);
|
|
|
514 return 1;
|
|
|
515 }
|
|
|
516
|
|
|
517 /* determine size of saved results array, checking for overflows,
|
|
|
518 allocate and clear the array (set all to zero with calloc()) */
|
|
|
519 if (syms == 2) /* iff max == 1 */
|
|
|
520 num = NULL; /* won't be saving any results */
|
|
|
521 else {
|
|
|
522 size = syms >> 1;
|
|
|
523 if (size > ((size_t)0 - 1) / (n = (syms - 1) >> 1) ||
|
|
|
524 (size *= n, size > ((size_t)0 - 1) / (n = max - 1)) ||
|
|
|
525 (size *= n, size > ((size_t)0 - 1) / sizeof(big_t)) ||
|
|
|
526 (num = calloc(size, sizeof(big_t))) == NULL) {
|
|
|
527 fputs("abort: unable to allocate enough memory\n", stderr);
|
|
|
528 cleanup();
|
|
|
529 return 1;
|
|
|
530 }
|
|
|
531 }
|
|
|
532
|
|
|
533 /* count possible codes for all numbers of symbols, add up counts */
|
|
|
534 sum = 0;
|
|
|
535 for (n = 2; n <= syms; n++) {
|
|
|
536 got = count(n, 1, 2);
|
|
|
537 sum += got;
|
|
|
538 if (got == (big_t)0 - 1 || sum < got) { /* overflow */
|
|
|
539 fputs("abort: can't count that high!\n", stderr);
|
|
|
540 cleanup();
|
|
|
541 return 1;
|
|
|
542 }
|
|
|
543 printf("%llu %d-codes\n", got, n);
|
|
|
544 }
|
|
|
545 printf("%llu total codes for 2 to %d symbols", sum, syms);
|
|
|
546 if (max < syms - 1)
|
|
|
547 printf(" (%d-bit length limit)\n", max);
|
|
|
548 else
|
|
|
549 puts(" (no length limit)");
|
|
|
550
|
|
|
551 /* allocate and clear done array for beenhere() */
|
|
|
552 if (syms == 2)
|
|
|
553 done = NULL;
|
|
|
554 else if (size > ((size_t)0 - 1) / sizeof(struct tab) ||
|
|
|
555 (done = calloc(size, sizeof(struct tab))) == NULL) {
|
|
|
556 fputs("abort: unable to allocate enough memory\n", stderr);
|
|
|
557 cleanup();
|
|
|
558 return 1;
|
|
|
559 }
|
|
|
560
|
|
|
561 /* find and show maximum inflate table usage */
|
|
|
562 if (root > max) /* reduce root to max length */
|
|
|
563 root = max;
|
|
|
564 if ((code_t)syms < ((code_t)1 << (root + 1)))
|
|
|
565 enough(syms);
|
|
|
566 else
|
|
|
567 puts("cannot handle minimum code lengths > root");
|
|
|
568
|
|
|
569 /* done */
|
|
|
570 cleanup();
|
|
|
571 return 0;
|
|
|
572 }
|
