/* enough.c -- determine the maximum size of inflate's Huffman code tables over
* all possible valid and complete prefix codes, subject to a length limit.
* Copyright (C) 2007, 2008, 2012, 2018 Mark Adler
* Version 1.5 5 August 2018 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
1.4 18 Aug 2012 Avoid shifts more than bits in type (caused endless loop!)
Clean up comparisons of different types
Clean up code indentation
1.5 5 Aug 2018 Clean up code style, formatting, and comments
Show all the codes for the maximum, and only the maximum
*/
/*
Examine all possible prefix codes for a given number of symbols and a
maximum code length in bits to determine the maximum table size for zlib's
inflate. Only complete prefix 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 left
unused.
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 for literal/length codes and root == 6 for distance
codes, in the current inflate implementation). The base 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 1 << (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 prefix 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 prefix 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 the uintmax_t 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 integer arithmetic in that case. This program will abort if
an overflow occurs. The big_t type identifies where the counting takes
place.
The uintmax_t 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 <stdarg.h>
#include <stdint.h>
#include <assert.h>
#define local static
// Special data types.
typedef uintmax_t big_t; // type for code counting
#define PRIbig "ju" // printf format for big_t
typedef uintmax_t code_t; // type for bit pattern counting
struct tab { // type for been-here check
size_t len; // allocated length of bit vector in octets
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 map() 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 prefix 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 been_here() for the details.
For the deflate example of 286 symbols limited to 15-bit codes, the bit
vectors grow to total 5.5 MB, in addition to the 4.3 MB done array itself.
*/
// Type for a variable-length, allocated string.
typedef struct {
char *str; // pointer to allocated string
size_t size; // size of allocation
size_t len; // length of string, not including terminating zero
} string_t;
// Clear a string_t.
local void string_clear(string_t *s) {
s->str[0] = 0;
s->len = 0;
}
// Initialize a string_t.
local void string_init(string_t *s) {
s->size = 16;
s->str = malloc(s->size);
assert(s->str != NULL && "out of memory");
string_clear(s);
}
// Release the allocation of a string_t.
local void string_free(string_t *s) {
free(s->str);
s->str = NULL;
s->size = 0;
s->len = 0;
}
// Save the results of printf with fmt and the subsequent argument list to s.
// Each call appends to s. The allocated space for s is increased as needed.
local void string_printf(string_t *s, char *fmt, ...) {
va_list ap;
va_start(ap, fmt);
size_t len = s->len;
int ret = vsnprintf(s->str + len, s->size - len, fmt, ap);
assert(ret >= 0 && "out of memory");
s->len += ret;
if (s->size < s->len + 1) {
do {
s->size <<= 1;
assert(s->size != 0 && "overflow");
} while (s->size < s->len + 1);
s->str = realloc(s->str, s->size);
assert(s->str != NULL && "out of memory");
vsnprintf(s->str + len, s->size - len, fmt, ap);
}
va_end(ap);
}
// Globals to avoid propagating constants or constant pointers recursively.
struct {
int max; // maximum allowed bit length for the codes
int root; // size of base code table in bits
int large; // largest code table so far
size_t size; // number of elements in num and done
big_t tot; // total number of codes with maximum tables size
string_t out; // display of subcodes for maximum tables size
int *code; // number of symbols assigned to each bit length
big_t *num; // saved results array for code counting
struct tab *done; // states already evaluated array
} g;
// Index function for num[] and done[].
local inline size_t map(int syms, int left, int len) {
return ((size_t)((syms - 1) >> 1) * ((syms - 2) >> 1) +
(left >> 1) - 1) * (g.max - 1) +
len - 1;
}
// Free allocated space in globals.
local void cleanup(void) {
if (g.done != NULL) {
for (size_t n = 0; n < g.size; n++)
if (g.done[n].len)
free(g.done[n].vec);
g.size = 0;
free(g.done); g.done = NULL;
}
free(g.num); g.num = NULL;
free(g.code); g.code = NULL;
string_free(&g.out);
}
// Return the number of possible prefix 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.
local big_t count(int syms, int left, int len) {
// see if only one possible code
if (syms == left)
return 1;
// note and verify the expected state
assert(syms > left && left > 0 && len < g.max);
// see if we've done this one already
size_t index = map(syms, left, len);
big_t got = g.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)
int 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)
int most = (((code_t)left << (g.max - len)) - syms) /
(((code_t)1 << (g.max - len)) - 1);
// count all possible codes from this juncture and add them up
big_t sum = 0;
for (int use = least; use <= most; use++) {
got = count(syms - use, (left - use) << 1, len + 1);
sum += got;
if (got == (big_t)-1 || sum < got) // overflow
return (big_t)-1;
}
// verify that all recursive calls are productive
assert(sum != 0);
// save the result and return it
g.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
// as needed to allow setting the (mem,rem) bit.
local int been_here(int syms, int left, int len, int mem, int rem) {
// point to vector for (syms,left,len), bit in vector for (mem,rem)
size_t index = map(syms, left, len);
mem -= 1 << g.root; // mem always includes the root table
mem >>= 1; // mem and rem are always even
rem >>= 1;
size_t offset = (mem >> 3) + rem;
offset = ((offset * (offset + 1)) >> 1) + rem;
int bit = 1 << (mem & 7);
// see if we've been here
size_t length = g.done[index].len;
if (offset < length && (g.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
char *vector;
if (length) {
do {
length <<= 1;
} while (length <= offset);
vector = realloc(g.done[index].vec, length);
assert(vector != NULL && "out of memory");
memset(vector + g.done[index].len, 0, length - g.done[index].len);
}
// otherwise we need to make a new vector and zero it out
else {
length = 16;
while (length <= offset)
length <<= 1;
vector = calloc(length, 1);
assert(vector != NULL && "out of memory");
}
// install the new vector
g.done[index].len = length;
g.done[index].vec = vector;
}
// set the bit
g.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.
local void examine(int syms, int left, int len, int mem, int rem) {
// see if we have a complete code
if (syms == left) {
// set the last code entry
g.code[len] = left;
// complete computation of memory used by this code
while (rem < left) {
left -= rem;
rem = 1 << (len - g.root);
mem += rem;
}
assert(rem == left);
// if this is at the maximum, show the sub-code
if (mem >= g.large) {
// if this is a new maximum, update the maximum and clear out the
// printed sub-codes from the previous maximum
if (mem > g.large) {
g.large = mem;
string_clear(&g.out);
}
// compute the starting state for this sub-code
syms = 0;
left = 1 << g.max;
for (int bits = g.max; bits > g.root; bits--) {
syms += g.code[bits];
left -= g.code[bits];
assert((left & 1) == 0);
left >>= 1;
}
// print the starting state and the resulting sub-code to g.out
string_printf(&g.out, "<%u, %u, %u>:",
syms, g.root + 1, ((1 << g.root) - left) << 1);
for (int bits = g.root + 1; bits <= g.max; bits++)
if (g.code[bits])
string_printf(&g.out, " %d[%d]", g.code[bits], bits);
string_printf(&g.out, "\n");
}
// remove entries as we drop back down in the recursion
g.code[len] = 0;
return;
}
// prune the tree if we can
if (been_here(syms, left, len, 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)
int 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)
int most = (((code_t)left << (g.max - len)) - syms) /
(((code_t)1 << (g.max - len)) - 1);
// occupy least table spaces, creating new sub-tables as needed
int use = least;
while (rem < use) {
use -= rem;
rem = 1 << (len - g.root);
mem += rem;
}
rem -= use;
// examine codes from here, updating table space as we go
for (use = least; use <= most; use++) {
g.code[len] = use;
examine(syms - use, (left - use) << 1, len + 1,
mem + (rem ? 1 << (len - g.root) : 0), rem << 1);
if (rem == 0) {
rem = 1 << (len - g.root);
mem += rem;
}
rem--;
}
// remove entries as we drop back down in the recursion
g.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 codes that
// require that maximum.
local void enough(int syms) {
// clear code
for (int n = 0; n <= g.max; n++)
g.code[n] = 0;
// look at all (root + 1) bit and longer codes
string_clear(&g.out); // empty saved results
g.large = 1 << g.root; // base table
if (g.root < g.max) // otherwise, there's only a base table
for (int n = 3; n <= syms; n++)
for (int left = 2; left < n; left += 2) {
// look at all reachable (root + 1) bit nodes, and the
// resulting codes (complete at root + 2 or more)
size_t index = map(n, left, g.root + 1);
if (g.root + 1 < g.max && g.num[index]) // reachable node
examine(n, left, g.root + 1, 1 << g.root, 0);
// also look at root bit codes with completions at root + 1
// bits (not saved in num, since complete), just in case
if (g.num[index - 1] && n <= left << 1)
examine((n - left) << 1, (n - left) << 1, g.root + 1,
1 << g.root, 0);
}
// done
printf("maximum of %d table entries for root = %d\n", g.large, g.root);
fputs(g.out.str, stdout);
}
// Examine and show the total number of possible prefix 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 prefix 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".
int main(int argc, char **argv) {
// set up globals for cleanup()
g.code = NULL;
g.num = NULL;
g.done = NULL;
string_init(&g.out);
// get arguments -- default to the deflate literal/length code
int syms = 286;
g.root = 9;
g.max = 15;
if (argc > 1) {
syms = atoi(argv[1]);
if (argc > 2) {
g.root = atoi(argv[2]);
if (argc > 3)
g.max = atoi(argv[3]);
}
}
if (argc > 4 || syms < 2 || g.root < 1 || g.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 (g.max > syms - 1)
g.max = syms - 1;
// determine the number of bits in a code_t
int bits = 0;
for (code_t word = 1; word; word <<= 1)
bits++;
// make sure that the calculation of most will not overflow
if (g.max > bits || (code_t)(syms - 2) >= ((code_t)-1 >> (g.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 << g.max) - 1) {
fprintf(stderr, "%d symbols cannot be coded in %d bits\n",
syms, g.max);
return 1;
}
// allocate code vector
g.code = calloc(g.max + 1, sizeof(int));
assert(g.code != NULL && "out of memory");
// 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
g.num = NULL; // won't be saving any results
else {
g.size = syms >> 1;
int n = (syms - 1) >> 1;
assert(g.size <= (size_t)-1 / n && "overflow");
g.size *= n;
n = g.max - 1;
assert(g.size <= (size_t)-1 / n && "overflow");
g.size *= n;
g.num = calloc(g.size, sizeof(big_t));
assert(g.num != NULL && "out of memory");
}
// count possible codes for all numbers of symbols, add up counts
big_t sum = 0;
for (int n = 2; n <= syms; n++) {
big_t got = count(n, 2, 1);
sum += got;
assert(got != (big_t)-1 && sum >= got && "overflow");
}
printf("%"PRIbig" total codes for 2 to %d symbols", sum, syms);
if (g.max < syms - 1)
printf(" (%d-bit length limit)\n", g.max);
else
puts(" (no length limit)");
// allocate and clear done array for been_here()
if (syms == 2)
g.done = NULL;
else {
g.done = calloc(g.size, sizeof(struct tab));
assert(g.done != NULL && "out of memory");
}
// find and show maximum inflate table usage
if (g.root > g.max) // reduce root to max length
g.root = g.max;
if ((code_t)syms < ((code_t)1 << (g.root + 1)))
enough(syms);
else
fputs("cannot handle minimum code lengths > root", stderr);
// done
cleanup();
return 0;
}
