Use MIT license instead of CC0

[wimlib] / src / compress_common.c

/*
 * compress_common.c
 *
 * Code for compression shared among multiple compression formats.
 *
 * Copyright 2022 Eric Biggers
 *
 * Permission is hereby granted, free of charge, to any person
 * obtaining a copy of this software and associated documentation
 * files (the "Software"), to deal in the Software without
 * restriction, including without limitation the rights to use,
 * copy, modify, merge, publish, distribute, sublicense, and/or sell
 * copies of the Software, and to permit persons to whom the
 * Software is furnished to do so, subject to the following
 * conditions:
 *
 * The above copyright notice and this permission notice shall be
 * included in all copies or substantial portions of the Software.
 *
 * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
 * EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
 * OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
 * NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
 * HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
 * WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
 * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
 * OTHER DEALINGS IN THE SOFTWARE.
 */

#ifdef HAVE_CONFIG_H
#  include "config.h"
#endif

#include <string.h>

#include "wimlib/compress_common.h"
#include "wimlib/util.h"

/* Given the binary tree node A[subtree_idx] whose children already
 * satisfy the maxheap property, swap the node with its greater child
 * until it is greater than both its children, so that the maxheap
 * property is satisfied in the subtree rooted at A[subtree_idx].  */
static void
heapify_subtree(u32 A[], unsigned length, unsigned subtree_idx)
{
        unsigned parent_idx;
        unsigned child_idx;
        u32 v;

        v = A[subtree_idx];
        parent_idx = subtree_idx;
        while ((child_idx = parent_idx * 2) <= length) {
                if (child_idx < length && A[child_idx + 1] > A[child_idx])
                        child_idx++;
                if (v >= A[child_idx])
                        break;
                A[parent_idx] = A[child_idx];
                parent_idx = child_idx;
        }
        A[parent_idx] = v;
}

/* Rearrange the array 'A' so that it satisfies the maxheap property.
 * 'A' uses 1-based indices, so the children of A[i] are A[i*2] and A[i*2 + 1].
 */
static void
heapify_array(u32 A[], unsigned length)
{
        for (unsigned subtree_idx = length / 2; subtree_idx >= 1; subtree_idx--)
                heapify_subtree(A, length, subtree_idx);
}

/* Sort the array 'A', which contains 'length' unsigned 32-bit integers.  */
static void
heapsort(u32 A[], unsigned length)
{
        A--; /* Use 1-based indices  */

        heapify_array(A, length);

        while (length >= 2) {
                swap(A[1], A[length]);
                length--;
                heapify_subtree(A, length, 1);
        }
}

#define NUM_SYMBOL_BITS 10
#define SYMBOL_MASK ((1 << NUM_SYMBOL_BITS) - 1)

/*
 * Sort the symbols primarily by frequency and secondarily by symbol
 * value.  Discard symbols with zero frequency and fill in an array with
 * the remaining symbols, along with their frequencies.  The low
 * NUM_SYMBOL_BITS bits of each array entry will contain the symbol
 * value, and the remaining bits will contain the frequency.
 *
 * @num_syms
 *      Number of symbols in the alphabet.
 *      Can't be greater than (1 << NUM_SYMBOL_BITS).
 *
 * @freqs[num_syms]
 *      The frequency of each symbol.
 *
 * @lens[num_syms]
 *      An array that eventually will hold the length of each codeword.
 *      This function only fills in the codeword lengths for symbols that
 *      have zero frequency, which are not well defined per se but will
 *      be set to 0.
 *
 * @symout[num_syms]
 *      The output array, described above.
 *
 * Returns the number of entries in 'symout' that were filled.  This is
 * the number of symbols that have nonzero frequency.
 */
static unsigned
sort_symbols(unsigned num_syms, const u32 freqs[restrict],
             u8 lens[restrict], u32 symout[restrict])
{
        unsigned num_used_syms;
        unsigned num_counters;

        /* We rely on heapsort, but with an added optimization.  Since
         * it's common for most symbol frequencies to be low, we first do
         * a count sort using a limited number of counters.  High
         * frequencies will be counted in the last counter, and only they
         * will be sorted with heapsort.
         *
         * Note: with more symbols, it is generally beneficial to have more
         * counters.  About 1 counter per 4 symbols seems fast.
         *
         * Note: I also tested radix sort, but even for large symbol
         * counts (> 255) and frequencies bounded at 16 bits (enabling
         * radix sort by just two base-256 digits), it didn't seem any
         * faster than the method implemented here.
         *
         * Note: I tested the optimized quicksort implementation from
         * glibc (with indirection overhead removed), but it was only
         * marginally faster than the simple heapsort implemented here.
         *
         * Tests were done with building the codes for LZX.  Results may
         * vary for different compression algorithms...!  */

        num_counters = ALIGN(DIV_ROUND_UP(num_syms, 4), 4);

        unsigned counters[num_counters];

        memset(counters, 0, sizeof(counters));

        /* Count the frequencies.  */
        for (unsigned sym = 0; sym < num_syms; sym++)
                counters[min(freqs[sym], num_counters - 1)]++;

        /* Make the counters cumulative, ignoring the zero-th, which
         * counted symbols with zero frequency.  As a side effect, this
         * calculates the number of symbols with nonzero frequency.  */
        num_used_syms = 0;
        for (unsigned i = 1; i < num_counters; i++) {
                unsigned count = counters[i];
                counters[i] = num_used_syms;
                num_used_syms += count;
        }

        /* Sort nonzero-frequency symbols using the counters.  At the
         * same time, set the codeword lengths of zero-frequency symbols
         * to 0.  */
        for (unsigned sym = 0; sym < num_syms; sym++) {
                u32 freq = freqs[sym];
                if (freq != 0) {
                        symout[counters[min(freq, num_counters - 1)]++] =
                                sym | (freq << NUM_SYMBOL_BITS);
                } else {
                        lens[sym] = 0;
                }
        }

        /* Sort the symbols counted in the last counter.  */
        heapsort(symout + counters[num_counters - 2],
                 counters[num_counters - 1] - counters[num_counters - 2]);

        return num_used_syms;
}

/*
 * Build the Huffman tree.
 *
 * This is an optimized implementation that
 *      (a) takes advantage of the frequencies being already sorted;
 *      (b) only generates non-leaf nodes, since the non-leaf nodes of a
 *          Huffman tree are sufficient to generate a canonical code;
 *      (c) Only stores parent pointers, not child pointers;
 *      (d) Produces the nodes in the same memory used for input
 *          frequency information.
 *
 * Array 'A', which contains 'sym_count' entries, is used for both input
 * and output.  For this function, 'sym_count' must be at least 2.
 *
 * For input, the array must contain the frequencies of the symbols,
 * sorted in increasing order.  Specifically, each entry must contain a
 * frequency left shifted by NUM_SYMBOL_BITS bits.  Any data in the low
 * NUM_SYMBOL_BITS bits of the entries will be ignored by this function.
 * Although these bits will, in fact, contain the symbols that correspond
 * to the frequencies, this function is concerned with frequencies only
 * and keeps the symbols as-is.
 *
 * For output, this function will produce the non-leaf nodes of the
 * Huffman tree.  These nodes will be stored in the first (sym_count - 1)
 * entries of the array.  Entry A[sym_count - 2] will represent the root
 * node.  Each other node will contain the zero-based index of its parent
 * node in 'A', left shifted by NUM_SYMBOL_BITS bits.  The low
 * NUM_SYMBOL_BITS bits of each entry in A will be kept as-is.  Again,
 * note that although these low bits will, in fact, contain a symbol
 * value, this symbol will have *no relationship* with the Huffman tree
 * node that happens to occupy the same slot.  This is because this
 * implementation only generates the non-leaf nodes of the tree.
 */
static void
build_tree(u32 A[], unsigned sym_count)
{
        /* Index, in 'A', of next lowest frequency symbol that has not
         * yet been processed.  */
        unsigned i = 0;

        /* Index, in 'A', of next lowest frequency parentless non-leaf
         * node; or, if equal to 'e', then no such node exists yet.  */
        unsigned b = 0;

        /* Index, in 'A', of next node to allocate as a non-leaf.  */
        unsigned e = 0;

        do {
                unsigned m, n;
                u32 freq_shifted;

                /* Choose the two next lowest frequency entries.  */

                if (i != sym_count &&
                    (b == e || (A[i] >> NUM_SYMBOL_BITS) <= (A[b] >> NUM_SYMBOL_BITS)))
                        m = i++;
                else
                        m = b++;

                if (i != sym_count &&
                    (b == e || (A[i] >> NUM_SYMBOL_BITS) <= (A[b] >> NUM_SYMBOL_BITS)))
                        n = i++;
                else
                        n = b++;

                /* Allocate a non-leaf node and link the entries to it.
                 *
                 * If we link an entry that we're visiting for the first
                 * time (via index 'i'), then we're actually linking a
                 * leaf node and it will have no effect, since the leaf
                 * will be overwritten with a non-leaf when index 'e'
                 * catches up to it.  But it's not any slower to
                 * unconditionally set the parent index.
                 *
                 * We also compute the frequency of the non-leaf node as
                 * the sum of its two children's frequencies.  */

                freq_shifted = (A[m] & ~SYMBOL_MASK) + (A[n] & ~SYMBOL_MASK);

                A[m] = (A[m] & SYMBOL_MASK) | (e << NUM_SYMBOL_BITS);
                A[n] = (A[n] & SYMBOL_MASK) | (e << NUM_SYMBOL_BITS);
                A[e] = (A[e] & SYMBOL_MASK) | freq_shifted;
                e++;
        } while (sym_count - e > 1);
                /* When just one entry remains, it is a "leaf" that was
                 * linked to some other node.  We ignore it, since the
                 * rest of the array contains the non-leaves which we
                 * need.  (Note that we're assuming the cases with 0 or 1
                 * symbols were handled separately.) */
}

/*
 * Given the stripped-down Huffman tree constructed by build_tree(),
 * determine the number of codewords that should be assigned each
 * possible length, taking into account the length-limited constraint.
 *
 * @A
 *      The array produced by build_tree(), containing parent index
 *      information for the non-leaf nodes of the Huffman tree.  Each
 *      entry in this array is a node; a node's parent always has a
 *      greater index than that node itself.  This function will
 *      overwrite the parent index information in this array, so
 *      essentially it will destroy the tree.  However, the data in the
 *      low NUM_SYMBOL_BITS of each entry will be preserved.
 *
 * @root_idx
 *      The 0-based index of the root node in 'A', and consequently one
 *      less than the number of tree node entries in 'A'.  (Or, really 2
 *      less than the actual length of 'A'.)
 *
 * @len_counts
 *      An array of length ('max_codeword_len' + 1) in which the number of
 *      codewords having each length <= max_codeword_len will be
 *      returned.
 *
 * @max_codeword_len
 *      The maximum permissible codeword length.
 */
static void
compute_length_counts(u32 A[restrict], unsigned root_idx,
                      unsigned len_counts[restrict], unsigned max_codeword_len)
{
        /* The key observations are:
         *
         * (1) We can traverse the non-leaf nodes of the tree, always
         * visiting a parent before its children, by simply iterating
         * through the array in reverse order.  Consequently, we can
         * compute the depth of each node in one pass, overwriting the
         * parent indices with depths.
         *
         * (2) We can initially assume that in the real Huffman tree,
         * both children of the root are leaves.  This corresponds to two
         * codewords of length 1.  Then, whenever we visit a (non-leaf)
         * node during the traversal, we modify this assumption to
         * account for the current node *not* being a leaf, but rather
         * its two children being leaves.  This causes the loss of one
         * codeword for the current depth and the addition of two
         * codewords for the current depth plus one.
         *
         * (3) We can handle the length-limited constraint fairly easily
         * by simply using the largest length available when a depth
         * exceeds max_codeword_len.
         */

        for (unsigned len = 0; len <= max_codeword_len; len++)
                len_counts[len] = 0;
        len_counts[1] = 2;

        /* Set the root node's depth to 0.  */
        A[root_idx] &= SYMBOL_MASK;

        for (int node = root_idx - 1; node >= 0; node--) {

                /* Calculate the depth of this node.  */

                unsigned parent = A[node] >> NUM_SYMBOL_BITS;
                unsigned parent_depth = A[parent] >> NUM_SYMBOL_BITS;
                unsigned depth = parent_depth + 1;
                unsigned len = depth;

                /* Set the depth of this node so that it is available
                 * when its children (if any) are processed.  */

                A[node] = (A[node] & SYMBOL_MASK) | (depth << NUM_SYMBOL_BITS);

                /* If needed, decrease the length to meet the
                 * length-limited constraint.  This is not the optimal
                 * method for generating length-limited Huffman codes!
                 * But it should be good enough.  */
                if (len >= max_codeword_len) {
                        len = max_codeword_len;
                        do {
                                len--;
                        } while (len_counts[len] == 0);
                }

                /* Account for the fact that we have a non-leaf node at
                 * the current depth.  */
                len_counts[len]--;
                len_counts[len + 1] += 2;
        }
}

/*
 * Generate the codewords for a canonical Huffman code.
 *
 * @A
 *      The output array for codewords.  In addition, initially this
 *      array must contain the symbols, sorted primarily by frequency and
 *      secondarily by symbol value, in the low NUM_SYMBOL_BITS bits of
 *      each entry.
 *
 * @len
 *      Output array for codeword lengths.
 *
 * @len_counts
 *      An array that provides the number of codewords that will have
 *      each possible length <= max_codeword_len.
 *
 * @max_codeword_len
 *      Maximum length, in bits, of each codeword.
 *
 * @num_syms
 *      Number of symbols in the alphabet, including symbols with zero
 *      frequency.  This is the length of the 'A' and 'len' arrays.
 */
static void
gen_codewords(u32 A[restrict], u8 lens[restrict],
              const unsigned len_counts[restrict],
              unsigned max_codeword_len, unsigned num_syms)
{
        u32 next_codewords[max_codeword_len + 1];

        /* Given the number of codewords that will have each length,
         * assign codeword lengths to symbols.  We do this by assigning
         * the lengths in decreasing order to the symbols sorted
         * primarily by increasing frequency and secondarily by
         * increasing symbol value.  */
        for (unsigned i = 0, len = max_codeword_len; len >= 1; len--) {
                unsigned count = len_counts[len];
                while (count--)
                        lens[A[i++] & SYMBOL_MASK] = len;
        }

        /* Generate the codewords themselves.  We initialize the
         * 'next_codewords' array to provide the lexicographically first
         * codeword of each length, then assign codewords in symbol
         * order.  This produces a canonical code.  */
        next_codewords[0] = 0;
        next_codewords[1] = 0;
        for (unsigned len = 2; len <= max_codeword_len; len++)
                next_codewords[len] =
                        (next_codewords[len - 1] + len_counts[len - 1]) << 1;

        for (unsigned sym = 0; sym < num_syms; sym++)
                A[sym] = next_codewords[lens[sym]]++;
}

/*
 * ---------------------------------------------------------------------
 *                      make_canonical_huffman_code()
 * ---------------------------------------------------------------------
 *
 * Given an alphabet and the frequency of each symbol in it, construct a
 * length-limited canonical Huffman code.
 *
 * @num_syms
 *      The number of symbols in the alphabet.  The symbols are the
 *      integers in the range [0, num_syms - 1].  This parameter must be
 *      at least 2 and can't be greater than (1 << NUM_SYMBOL_BITS).
 *
 * @max_codeword_len
 *      The maximum permissible codeword length.
 *
 * @freqs
 *      An array of @num_syms entries, each of which specifies the
 *      frequency of the corresponding symbol.  It is valid for some,
 *      none, or all of the frequencies to be 0.
 *
 * @lens
 *      An array of @num_syms entries in which this function will return
 *      the length, in bits, of the codeword assigned to each symbol.
 *      Symbols with 0 frequency will not have codewords per se, but
 *      their entries in this array will be set to 0.  No lengths greater
 *      than @max_codeword_len will be assigned.
 *
 * @codewords
 *      An array of @num_syms entries in which this function will return
 *      the codeword for each symbol, right-justified and padded on the
 *      left with zeroes.  Codewords for symbols with 0 frequency will be
 *      undefined.
 *
 * ---------------------------------------------------------------------
 *
 * This function builds a length-limited canonical Huffman code.
 *
 * A length-limited Huffman code contains no codewords longer than some
 * specified length, and has exactly (with some algorithms) or
 * approximately (with the algorithm used here) the minimum weighted path
 * length from the root, given this constraint.
 *
 * A canonical Huffman code satisfies the properties that a longer
 * codeword never lexicographically precedes a shorter codeword, and the
 * lexicographic ordering of codewords of the same length is the same as
 * the lexicographic ordering of the corresponding symbols.  A canonical
 * Huffman code, or more generally a canonical prefix code, can be
 * reconstructed from only a list containing the codeword length of each
 * symbol.
 *
 * The classic algorithm to generate a Huffman code creates a node for
 * each symbol, then inserts these nodes into a min-heap keyed by symbol
 * frequency.  Then, repeatedly, the two lowest-frequency nodes are
 * removed from the min-heap and added as the children of a new node
 * having frequency equal to the sum of its two children, which is then
 * inserted into the min-heap.  When only a single node remains in the
 * min-heap, it is the root of the Huffman tree.  The codeword for each
 * symbol is determined by the path needed to reach the corresponding
 * node from the root.  Descending to the left child appends a 0 bit,
 * whereas descending to the right child appends a 1 bit.
 *
 * The classic algorithm is relatively easy to understand, but it is
 * subject to a number of inefficiencies.  In practice, it is fastest to
 * first sort the symbols by frequency.  (This itself can be subject to
 * an optimization based on the fact that most frequencies tend to be
 * low.)  At the same time, we sort secondarily by symbol value, which
 * aids the process of generating a canonical code.  Then, during tree
 * construction, no heap is necessary because both the leaf nodes and the
 * unparented non-leaf nodes can be easily maintained in sorted order.
 * Consequently, there can never be more than two possibilities for the
 * next-lowest-frequency node.
 *
 * In addition, because we're generating a canonical code, we actually
 * don't need the leaf nodes of the tree at all, only the non-leaf nodes.
 * This is because for canonical code generation we don't need to know
 * where the symbols are in the tree.  Rather, we only need to know how
 * many leaf nodes have each depth (codeword length).  And this
 * information can, in fact, be quickly generated from the tree of
 * non-leaves only.
 *
 * Furthermore, we can build this stripped-down Huffman tree directly in
 * the array in which the codewords are to be generated, provided that
 * these array slots are large enough to hold a symbol and frequency
 * value.
 *
 * Still furthermore, we don't even need to maintain explicit child
 * pointers.  We only need the parent pointers, and even those can be
 * overwritten in-place with depth information as part of the process of
 * extracting codeword lengths from the tree.  So in summary, we do NOT
 * need a big structure like:
 *
 *      struct huffman_tree_node {
 *              unsigned int symbol;
 *              unsigned int frequency;
 *              unsigned int depth;
 *              struct huffman_tree_node *left_child;
 *              struct huffman_tree_node *right_child;
 *      };
 *
 *
 *   ... which often gets used in "naive" implementations of Huffman code
 *   generation.
 *
 * Most of these optimizations are based on the implementation in 7-Zip
 * (source file: C/HuffEnc.c), which has been placed in the public domain
 * by Igor Pavlov.  But I've rewritten the code with extensive comments,
 * as it took me a while to figure out what it was doing...!
 *
 * ---------------------------------------------------------------------
 *
 * NOTE: in general, the same frequencies can be used to generate
 * different length-limited canonical Huffman codes.  One choice we have
 * is during tree construction, when we must decide whether to prefer a
 * leaf or non-leaf when there is a tie in frequency.  Another choice we
 * have is how to deal with codewords that would exceed @max_codeword_len
 * bits in length.  Both of these choices affect the resulting codeword
 * lengths, which otherwise can be mapped uniquely onto the resulting
 * canonical Huffman code.
 *
 * Normally, there is no problem with choosing one valid code over
 * another, provided that they produce similar compression ratios.
 * However, the LZMS compression format uses adaptive Huffman coding.  It
 * requires that both the decompressor and compressor build a canonical
 * code equivalent to that which can be generated by using the classic
 * Huffman tree construction algorithm and always processing leaves
 * before non-leaves when there is a frequency tie.  Therefore, we make
 * sure to do this.  This method also has the advantage of sometimes
 * shortening the longest codeword that is generated.
 *
 * There also is the issue of how codewords longer than @max_codeword_len
 * are dealt with.  Fortunately, for LZMS this is irrelevant because for
 * the LZMS alphabets no codeword can ever exceed LZMS_MAX_CODEWORD_LEN
 * (= 15).  Since the LZMS algorithm regularly halves all frequencies,
 * the frequencies cannot become high enough for a length 16 codeword to
 * be generated.  Specifically, I think that if ties are broken in favor
 * of non-leaves (as we do), the lowest total frequency that would give a
 * length-16 codeword would be the sum of the frequencies 1 1 1 3 4 7 11
 * 18 29 47 76 123 199 322 521 843 1364, which is 3570.  And in LZMS we
 * can't get a frequency that high based on the alphabet sizes, rebuild
 * frequencies, and scaling factors.  This worst-case scenario is based
 * on the following degenerate case (only the bottom of the tree shown):
 *
 *                          ...
 *                        17
 *                       /  \
 *                      10   7
 *                     / \
 *                    6   4
 *                   / \
 *                  3   3
 *                 / \
 *                2   1
 *               / \
 *              1   1
 *
 * Excluding the first leaves (those with value 1), each leaf value must
 * be greater than the non-leaf up 1 and down 2 from it; otherwise that
 * leaf would have taken precedence over that non-leaf and been combined
 * with the leaf below, thereby decreasing the height compared to that
 * shown.
 *
 * Interesting fact: if we were to instead prioritize non-leaves over
 * leaves, then the worst case frequencies would be the Fibonacci
 * sequence, plus an extra frequency of 1.  In this hypothetical
 * scenario, it would be slightly easier for longer codewords to be
 * generated.
 */
void
make_canonical_huffman_code(unsigned num_syms, unsigned max_codeword_len,
                            const u32 freqs[restrict],
                            u8 lens[restrict], u32 codewords[restrict])
{
        u32 *A = codewords;
        unsigned num_used_syms;

        /* We begin by sorting the symbols primarily by frequency and
         * secondarily by symbol value.  As an optimization, the array
         * used for this purpose ('A') shares storage with the space in
         * which we will eventually return the codewords.  */

        num_used_syms = sort_symbols(num_syms, freqs, lens, A);

        /* 'num_used_syms' is the number of symbols with nonzero
         * frequency.  This may be less than @num_syms.  'num_used_syms'
         * is also the number of entries in 'A' that are valid.  Each
         * entry consists of a distinct symbol and a nonzero frequency
         * packed into a 32-bit integer.  */

        /* Handle special cases where only 0 or 1 symbols were used (had
         * nonzero frequency).  */

        if (unlikely(num_used_syms == 0)) {
                /* Code is empty.  sort_symbols() already set all lengths
                 * to 0, so there is nothing more to do.  */
                return;
        }

        if (unlikely(num_used_syms == 1)) {
                /* Only one symbol was used, so we only need one
                 * codeword.  But two codewords are needed to form the
                 * smallest complete Huffman code, which uses codewords 0
                 * and 1.  Therefore, we choose another symbol to which
                 * to assign a codeword.  We use 0 (if the used symbol is
                 * not 0) or 1 (if the used symbol is 0).  In either
                 * case, the lesser-valued symbol must be assigned
                 * codeword 0 so that the resulting code is canonical.  */

                unsigned sym = A[0] & SYMBOL_MASK;
                unsigned nonzero_idx = sym ? sym : 1;

                codewords[0] = 0;
                lens[0] = 1;
                codewords[nonzero_idx] = 1;
                lens[nonzero_idx] = 1;
                return;
        }

        /* Build a stripped-down version of the Huffman tree, sharing the
         * array 'A' with the symbol values.  Then extract length counts
         * from the tree and use them to generate the final codewords.  */

        build_tree(A, num_used_syms);

        {
                unsigned len_counts[max_codeword_len + 1];

                compute_length_counts(A, num_used_syms - 2,
                                      len_counts, max_codeword_len);

                gen_codewords(A, lens, len_counts, max_codeword_len, num_syms);
        }
}