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573 lines
24 KiB
C
573 lines
24 KiB
C
/* enough.c -- determine the maximum size of inflate's Huffman code tables over
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* all possible valid and complete Huffman codes, subject to a length limit.
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* Copyright (C) 2007, 2008, 2012 Mark Adler
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* Version 1.4 18 August 2012 Mark Adler
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*/
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/* Version history:
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1.0 3 Jan 2007 First version (derived from codecount.c version 1.4)
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1.1 4 Jan 2007 Use faster incremental table usage computation
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Prune examine() search on previously visited states
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1.2 5 Jan 2007 Comments clean up
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As inflate does, decrease root for short codes
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Refuse cases where inflate would increase root
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1.3 17 Feb 2008 Add argument for initial root table size
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Fix bug for initial root table size == max - 1
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Use a macro to compute the history index
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1.4 18 Aug 2012 Avoid shifts more than bits in type (caused endless loop!)
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Clean up comparisons of different types
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Clean up code indentation
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*/
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/*
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Examine all possible Huffman codes for a given number of symbols and a
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maximum code length in bits to determine the maximum table size for zilb's
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inflate. Only complete Huffman codes are counted.
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Two codes are considered distinct if the vectors of the number of codes per
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length are not identical. So permutations of the symbol assignments result
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in the same code for the counting, as do permutations of the assignments of
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the bit values to the codes (i.e. only canonical codes are counted).
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We build a code from shorter to longer lengths, determining how many symbols
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are coded at each length. At each step, we have how many symbols remain to
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be coded, what the last code length used was, and how many bit patterns of
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that length remain unused. Then we add one to the code length and double the
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number of unused patterns to graduate to the next code length. We then
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assign all portions of the remaining symbols to that code length that
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preserve the properties of a correct and eventually complete code. Those
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properties are: we cannot use more bit patterns than are available; and when
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all the symbols are used, there are exactly zero possible bit patterns
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remaining.
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The inflate Huffman decoding algorithm uses two-level lookup tables for
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speed. There is a single first-level table to decode codes up to root bits
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in length (root == 9 in the current inflate implementation). The table
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has 1 << root entries and is indexed by the next root bits of input. Codes
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shorter than root bits have replicated table entries, so that the correct
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entry is pointed to regardless of the bits that follow the short code. If
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the code is longer than root bits, then the table entry points to a second-
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level table. The size of that table is determined by the longest code with
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that root-bit prefix. If that longest code has length len, then the table
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has size 1 << (len - root), to index the remaining bits in that set of
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codes. Each subsequent root-bit prefix then has its own sub-table. The
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total number of table entries required by the code is calculated
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incrementally as the number of codes at each bit length is populated. When
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all of the codes are shorter than root bits, then root is reduced to the
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longest code length, resulting in a single, smaller, one-level table.
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The inflate algorithm also provides for small values of root (relative to
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the log2 of the number of symbols), where the shortest code has more bits
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than root. In that case, root is increased to the length of the shortest
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code. This program, by design, does not handle that case, so it is verified
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that the number of symbols is less than 2^(root + 1).
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In order to speed up the examination (by about ten orders of magnitude for
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the default arguments), the intermediate states in the build-up of a code
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are remembered and previously visited branches are pruned. The memory
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required for this will increase rapidly with the total number of symbols and
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the maximum code length in bits. However this is a very small price to pay
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for the vast speedup.
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First, all of the possible Huffman codes are counted, and reachable
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intermediate states are noted by a non-zero count in a saved-results array.
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Second, the intermediate states that lead to (root + 1) bit or longer codes
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are used to look at all sub-codes from those junctures for their inflate
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memory usage. (The amount of memory used is not affected by the number of
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codes of root bits or less in length.) Third, the visited states in the
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construction of those sub-codes and the associated calculation of the table
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size is recalled in order to avoid recalculating from the same juncture.
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Beginning the code examination at (root + 1) bit codes, which is enabled by
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identifying the reachable nodes, accounts for about six of the orders of
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magnitude of improvement for the default arguments. About another four
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orders of magnitude come from not revisiting previous states. Out of
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approximately 2x10^16 possible Huffman codes, only about 2x10^6 sub-codes
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need to be examined to cover all of the possible table memory usage cases
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for the default arguments of 286 symbols limited to 15-bit codes.
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Note that an unsigned long long type is used for counting. It is quite easy
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to exceed the capacity of an eight-byte integer with a large number of
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symbols and a large maximum code length, so multiple-precision arithmetic
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would need to replace the unsigned long long arithmetic in that case. This
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program will abort if an overflow occurs. The big_t type identifies where
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the counting takes place.
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An unsigned long long type is also used for calculating the number of
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possible codes remaining at the maximum length. This limits the maximum
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code length to the number of bits in a long long minus the number of bits
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needed to represent the symbols in a flat code. The code_t type identifies
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where the bit pattern counting takes place.
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*/
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#include <stdio.h>
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#include <stdlib.h>
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#include <string.h>
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#include <assert.h>
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#define local static
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/* special data types */
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typedef unsigned long long big_t; /* type for code counting */
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typedef unsigned long long code_t; /* type for bit pattern counting */
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struct tab { /* type for been here check */
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size_t len; /* length of bit vector in char's */
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char *vec; /* allocated bit vector */
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};
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/* The array for saving results, num[], is indexed with this triplet:
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syms: number of symbols remaining to code
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left: number of available bit patterns at length len
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len: number of bits in the codes currently being assigned
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Those indices are constrained thusly when saving results:
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syms: 3..totsym (totsym == total symbols to code)
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left: 2..syms - 1, but only the evens (so syms == 8 -> 2, 4, 6)
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len: 1..max - 1 (max == maximum code length in bits)
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syms == 2 is not saved since that immediately leads to a single code. left
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must be even, since it represents the number of available bit patterns at
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the current length, which is double the number at the previous length.
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left ends at syms-1 since left == syms immediately results in a single code.
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(left > sym is not allowed since that would result in an incomplete code.)
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len is less than max, since the code completes immediately when len == max.
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The offset into the array is calculated for the three indices with the
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first one (syms) being outermost, and the last one (len) being innermost.
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We build the array with length max-1 lists for the len index, with syms-3
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of those for each symbol. There are totsym-2 of those, with each one
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varying in length as a function of sym. See the calculation of index in
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count() for the index, and the calculation of size in main() for the size
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of the array.
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For the deflate example of 286 symbols limited to 15-bit codes, the array
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has 284,284 entries, taking up 2.17 MB for an 8-byte big_t. More than
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half of the space allocated for saved results is actually used -- not all
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possible triplets are reached in the generation of valid Huffman codes.
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*/
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/* The array for tracking visited states, done[], is itself indexed identically
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to the num[] array as described above for the (syms, left, len) triplet.
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Each element in the array is further indexed by the (mem, rem) doublet,
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where mem is the amount of inflate table space used so far, and rem is the
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remaining unused entries in the current inflate sub-table. Each indexed
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element is simply one bit indicating whether the state has been visited or
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not. Since the ranges for mem and rem are not known a priori, each bit
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vector is of a variable size, and grows as needed to accommodate the visited
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states. mem and rem are used to calculate a single index in a triangular
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array. Since the range of mem is expected in the default case to be about
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ten times larger than the range of rem, the array is skewed to reduce the
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memory usage, with eight times the range for mem than for rem. See the
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calculations for offset and bit in beenhere() for the details.
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For the deflate example of 286 symbols limited to 15-bit codes, the bit
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vectors grow to total approximately 21 MB, in addition to the 4.3 MB done[]
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array itself.
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*/
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/* Globals to avoid propagating constants or constant pointers recursively */
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local int max; /* maximum allowed bit length for the codes */
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local int root; /* size of base code table in bits */
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local int large; /* largest code table so far */
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local size_t size; /* number of elements in num and done */
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local int *code; /* number of symbols assigned to each bit length */
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local big_t *num; /* saved results array for code counting */
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local struct tab *done; /* states already evaluated array */
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/* Index function for num[] and done[] */
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#define INDEX(i,j,k) (((size_t)((i-1)>>1)*((i-2)>>1)+(j>>1)-1)*(max-1)+k-1)
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/* Free allocated space. Uses globals code, num, and done. */
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local void cleanup(void)
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{
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size_t n;
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if (done != NULL) {
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for (n = 0; n < size; n++)
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if (done[n].len)
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free(done[n].vec);
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free(done);
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}
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if (num != NULL)
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free(num);
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if (code != NULL)
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free(code);
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}
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/* Return the number of possible Huffman codes using bit patterns of lengths
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len through max inclusive, coding syms symbols, with left bit patterns of
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length len unused -- return -1 if there is an overflow in the counting.
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Keep a record of previous results in num to prevent repeating the same
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calculation. Uses the globals max and num. */
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local big_t count(int syms, int len, int left)
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{
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big_t sum; /* number of possible codes from this juncture */
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big_t got; /* value returned from count() */
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int least; /* least number of syms to use at this juncture */
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int most; /* most number of syms to use at this juncture */
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int use; /* number of bit patterns to use in next call */
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size_t index; /* index of this case in *num */
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/* see if only one possible code */
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if (syms == left)
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return 1;
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/* note and verify the expected state */
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assert(syms > left && left > 0 && len < max);
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/* see if we've done this one already */
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index = INDEX(syms, left, len);
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got = num[index];
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if (got)
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return got; /* we have -- return the saved result */
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/* we need to use at least this many bit patterns so that the code won't be
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incomplete at the next length (more bit patterns than symbols) */
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least = (left << 1) - syms;
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if (least < 0)
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least = 0;
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/* we can use at most this many bit patterns, lest there not be enough
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available for the remaining symbols at the maximum length (if there were
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no limit to the code length, this would become: most = left - 1) */
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most = (((code_t)left << (max - len)) - syms) /
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(((code_t)1 << (max - len)) - 1);
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/* count all possible codes from this juncture and add them up */
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sum = 0;
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for (use = least; use <= most; use++) {
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got = count(syms - use, len + 1, (left - use) << 1);
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sum += got;
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if (got == (big_t)0 - 1 || sum < got) /* overflow */
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return (big_t)0 - 1;
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}
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/* verify that all recursive calls are productive */
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assert(sum != 0);
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/* save the result and return it */
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num[index] = sum;
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return sum;
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}
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/* Return true if we've been here before, set to true if not. Set a bit in a
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bit vector to indicate visiting this state. Each (syms,len,left) state
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has a variable size bit vector indexed by (mem,rem). The bit vector is
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lengthened if needed to allow setting the (mem,rem) bit. */
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local int beenhere(int syms, int len, int left, int mem, int rem)
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{
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size_t index; /* index for this state's bit vector */
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size_t offset; /* offset in this state's bit vector */
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int bit; /* mask for this state's bit */
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size_t length; /* length of the bit vector in bytes */
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char *vector; /* new or enlarged bit vector */
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/* point to vector for (syms,left,len), bit in vector for (mem,rem) */
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index = INDEX(syms, left, len);
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mem -= 1 << root;
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offset = (mem >> 3) + rem;
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offset = ((offset * (offset + 1)) >> 1) + rem;
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bit = 1 << (mem & 7);
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/* see if we've been here */
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length = done[index].len;
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if (offset < length && (done[index].vec[offset] & bit) != 0)
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return 1; /* done this! */
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/* we haven't been here before -- set the bit to show we have now */
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/* see if we need to lengthen the vector in order to set the bit */
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if (length <= offset) {
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/* if we have one already, enlarge it, zero out the appended space */
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if (length) {
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do {
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length <<= 1;
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} while (length <= offset);
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vector = realloc(done[index].vec, length);
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if (vector != NULL)
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memset(vector + done[index].len, 0, length - done[index].len);
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}
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/* otherwise we need to make a new vector and zero it out */
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else {
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length = 1 << (len - root);
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while (length <= offset)
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length <<= 1;
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vector = calloc(length, sizeof(char));
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}
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/* in either case, bail if we can't get the memory */
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if (vector == NULL) {
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fputs("abort: unable to allocate enough memory\n", stderr);
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cleanup();
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exit(1);
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}
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/* install the new vector */
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done[index].len = length;
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done[index].vec = vector;
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}
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/* set the bit */
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done[index].vec[offset] |= bit;
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return 0;
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}
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/* Examine all possible codes from the given node (syms, len, left). Compute
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the amount of memory required to build inflate's decoding tables, where the
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number of code structures used so far is mem, and the number remaining in
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the current sub-table is rem. Uses the globals max, code, root, large, and
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done. */
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local void examine(int syms, int len, int left, int mem, int rem)
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{
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int least; /* least number of syms to use at this juncture */
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int most; /* most number of syms to use at this juncture */
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int use; /* number of bit patterns to use in next call */
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/* see if we have a complete code */
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if (syms == left) {
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/* set the last code entry */
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code[len] = left;
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/* complete computation of memory used by this code */
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while (rem < left) {
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left -= rem;
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rem = 1 << (len - root);
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mem += rem;
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}
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assert(rem == left);
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/* if this is a new maximum, show the entries used and the sub-code */
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if (mem > large) {
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large = mem;
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printf("max %d: ", mem);
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for (use = root + 1; use <= max; use++)
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if (code[use])
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printf("%d[%d] ", code[use], use);
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putchar('\n');
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fflush(stdout);
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}
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/* remove entries as we drop back down in the recursion */
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code[len] = 0;
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return;
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}
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/* prune the tree if we can */
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if (beenhere(syms, len, left, mem, rem))
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return;
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/* we need to use at least this many bit patterns so that the code won't be
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incomplete at the next length (more bit patterns than symbols) */
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least = (left << 1) - syms;
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if (least < 0)
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least = 0;
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/* we can use at most this many bit patterns, lest there not be enough
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available for the remaining symbols at the maximum length (if there were
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no limit to the code length, this would become: most = left - 1) */
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most = (((code_t)left << (max - len)) - syms) /
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(((code_t)1 << (max - len)) - 1);
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/* occupy least table spaces, creating new sub-tables as needed */
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use = least;
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while (rem < use) {
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use -= rem;
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rem = 1 << (len - root);
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mem += rem;
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}
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rem -= use;
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/* examine codes from here, updating table space as we go */
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for (use = least; use <= most; use++) {
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code[len] = use;
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examine(syms - use, len + 1, (left - use) << 1,
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mem + (rem ? 1 << (len - root) : 0), rem << 1);
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if (rem == 0) {
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rem = 1 << (len - root);
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mem += rem;
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}
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rem--;
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}
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/* remove entries as we drop back down in the recursion */
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code[len] = 0;
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}
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/* Look at all sub-codes starting with root + 1 bits. Look at only the valid
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intermediate code states (syms, left, len). For each completed code,
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calculate the amount of memory required by inflate to build the decoding
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tables. Find the maximum amount of memory required and show the code that
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requires that maximum. Uses the globals max, root, and num. */
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local void enough(int syms)
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{
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int n; /* number of remaing symbols for this node */
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int left; /* number of unused bit patterns at this length */
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size_t index; /* index of this case in *num */
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/* clear code */
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for (n = 0; n <= max; n++)
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code[n] = 0;
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/* look at all (root + 1) bit and longer codes */
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large = 1 << root; /* base table */
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if (root < max) /* otherwise, there's only a base table */
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for (n = 3; n <= syms; n++)
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for (left = 2; left < n; left += 2)
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{
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/* look at all reachable (root + 1) bit nodes, and the
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resulting codes (complete at root + 2 or more) */
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index = INDEX(n, left, root + 1);
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if (root + 1 < max && num[index]) /* reachable node */
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examine(n, root + 1, left, 1 << root, 0);
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/* also look at root bit codes with completions at root + 1
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bits (not saved in num, since complete), just in case */
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if (num[index - 1] && n <= left << 1)
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examine((n - left) << 1, root + 1, (n - left) << 1,
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1 << root, 0);
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}
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/* done */
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printf("done: maximum of %d table entries\n", large);
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}
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/*
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Examine and show the total number of possible Huffman codes for a given
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maximum number of symbols, initial root table size, and maximum code length
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in bits -- those are the command arguments in that order. The default
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values are 286, 9, and 15 respectively, for the deflate literal/length code.
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The possible codes are counted for each number of coded symbols from two to
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the maximum. The counts for each of those and the total number of codes are
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shown. The maximum number of inflate table entires is then calculated
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across all possible codes. Each new maximum number of table entries and the
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associated sub-code (starting at root + 1 == 10 bits) is shown.
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To count and examine Huffman codes that are not length-limited, provide a
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maximum length equal to the number of symbols minus one.
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For the deflate literal/length code, use "enough". For the deflate distance
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code, use "enough 30 6".
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This uses the %llu printf format to print big_t numbers, which assumes that
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big_t is an unsigned long long. If the big_t type is changed (for example
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to a multiple precision type), the method of printing will also need to be
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updated.
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*/
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int main(int argc, char **argv)
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{
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int syms; /* total number of symbols to code */
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int n; /* number of symbols to code for this run */
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big_t got; /* return value of count() */
|
|
big_t sum; /* accumulated number of codes over n */
|
|
code_t word; /* for counting bits in code_t */
|
|
|
|
/* set up globals for cleanup() */
|
|
code = NULL;
|
|
num = NULL;
|
|
done = NULL;
|
|
|
|
/* get arguments -- default to the deflate literal/length code */
|
|
syms = 286;
|
|
root = 9;
|
|
max = 15;
|
|
if (argc > 1) {
|
|
syms = atoi(argv[1]);
|
|
if (argc > 2) {
|
|
root = atoi(argv[2]);
|
|
if (argc > 3)
|
|
max = atoi(argv[3]);
|
|
}
|
|
}
|
|
if (argc > 4 || syms < 2 || root < 1 || max < 1) {
|
|
fputs("invalid arguments, need: [sym >= 2 [root >= 1 [max >= 1]]]\n",
|
|
stderr);
|
|
return 1;
|
|
}
|
|
|
|
/* if not restricting the code length, the longest is syms - 1 */
|
|
if (max > syms - 1)
|
|
max = syms - 1;
|
|
|
|
/* determine the number of bits in a code_t */
|
|
for (n = 0, word = 1; word; n++, word <<= 1)
|
|
;
|
|
|
|
/* make sure that the calculation of most will not overflow */
|
|
if (max > n || (code_t)(syms - 2) >= (((code_t)0 - 1) >> (max - 1))) {
|
|
fputs("abort: code length too long for internal types\n", stderr);
|
|
return 1;
|
|
}
|
|
|
|
/* reject impossible code requests */
|
|
if ((code_t)(syms - 1) > ((code_t)1 << max) - 1) {
|
|
fprintf(stderr, "%d symbols cannot be coded in %d bits\n",
|
|
syms, max);
|
|
return 1;
|
|
}
|
|
|
|
/* allocate code vector */
|
|
code = calloc(max + 1, sizeof(int));
|
|
if (code == NULL) {
|
|
fputs("abort: unable to allocate enough memory\n", stderr);
|
|
return 1;
|
|
}
|
|
|
|
/* determine size of saved results array, checking for overflows,
|
|
allocate and clear the array (set all to zero with calloc()) */
|
|
if (syms == 2) /* iff max == 1 */
|
|
num = NULL; /* won't be saving any results */
|
|
else {
|
|
size = syms >> 1;
|
|
if (size > ((size_t)0 - 1) / (n = (syms - 1) >> 1) ||
|
|
(size *= n, size > ((size_t)0 - 1) / (n = max - 1)) ||
|
|
(size *= n, size > ((size_t)0 - 1) / sizeof(big_t)) ||
|
|
(num = calloc(size, sizeof(big_t))) == NULL) {
|
|
fputs("abort: unable to allocate enough memory\n", stderr);
|
|
cleanup();
|
|
return 1;
|
|
}
|
|
}
|
|
|
|
/* count possible codes for all numbers of symbols, add up counts */
|
|
sum = 0;
|
|
for (n = 2; n <= syms; n++) {
|
|
got = count(n, 1, 2);
|
|
sum += got;
|
|
if (got == (big_t)0 - 1 || sum < got) { /* overflow */
|
|
fputs("abort: can't count that high!\n", stderr);
|
|
cleanup();
|
|
return 1;
|
|
}
|
|
printf("%llu %d-codes\n", got, n);
|
|
}
|
|
printf("%llu total codes for 2 to %d symbols", sum, syms);
|
|
if (max < syms - 1)
|
|
printf(" (%d-bit length limit)\n", max);
|
|
else
|
|
puts(" (no length limit)");
|
|
|
|
/* allocate and clear done array for beenhere() */
|
|
if (syms == 2)
|
|
done = NULL;
|
|
else if (size > ((size_t)0 - 1) / sizeof(struct tab) ||
|
|
(done = calloc(size, sizeof(struct tab))) == NULL) {
|
|
fputs("abort: unable to allocate enough memory\n", stderr);
|
|
cleanup();
|
|
return 1;
|
|
}
|
|
|
|
/* find and show maximum inflate table usage */
|
|
if (root > max) /* reduce root to max length */
|
|
root = max;
|
|
if ((code_t)syms < ((code_t)1 << (root + 1)))
|
|
enough(syms);
|
|
else
|
|
puts("cannot handle minimum code lengths > root");
|
|
|
|
/* done */
|
|
cleanup();
|
|
return 0;
|
|
}
|