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