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+/* 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 entries 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;
+}