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