redict/src/hyperloglog.c

/* hyperloglog.c - Redis HyperLogLog probabilistic cardinality approximation.
 * This file implements the algorithm and the exported Redis commands.
 *
 * Copyright (c) 2014, Salvatore Sanfilippo <antirez at gmail dot com>
 * All rights reserved.
 *
 * Redistribution and use in source and binary forms, with or without
 * modification, are permitted provided that the following conditions are met:
 *
 *   * Redistributions of source code must retain the above copyright notice,
 *     this list of conditions and the following disclaimer.
 *   * Redistributions in binary form must reproduce the above copyright
 *     notice, this list of conditions and the following disclaimer in the
 *     documentation and/or other materials provided with the distribution.
 *   * Neither the name of Redis nor the names of its contributors may be used
 *     to endorse or promote products derived from this software without
 *     specific prior written permission.
 *
 * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
 * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
 * ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
 * LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
 * CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
 * SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
 * INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
 * CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
 * POSSIBILITY OF SUCH DAMAGE.
 */

#include <stdint.h>
#include <math.h>
#include "redis.h"

/* The Redis HyperLogLog implementation is based on the following ideas:
 *
 * * The use of a 64 bit hash function as proposed in [1], in order to don't
 *   limited to cardinalities up to 10^9, at the cost of just 1 additional
 *   bit per register.
 * * The use of 16384 6-bit registers for a great level of accuracy, using
 *   a total of 12k per key.
 * * The use of the Redis string data type. No new type is introduced.
 * * No attempt is made to compress the data structure as in [1]. Also the
 *   algorithm used is the original HyperLogLog Algorithm as in [2], with
 *   the only difference that a 64 bit hash function is used, so no correction
 *   is performed for values near 2^32 as in [1].
 *
 * [1] Heule, Nunkesser, Hall: HyperLogLog in Practice: Algorithmic
 *     Engineering of a State of The Art Cardinality Estimation Algorithm.
 *
 * [2] P. Flajolet, Éric Fusy, O. Gandouet, and F. Meunier. Hyperloglog: The
 *     analysis of a near-optimal cardinality estimation algorithm.
 *
 * The representation used by Redis is the following:
 *
 * +--------+--------+--------+------//      //--+---------------------+
 * |11000000|22221111|33333322|55444444 ....     | 64 bit cardinality  |
 * +--------+--------+--------+------//      //--+---------------------+
 *
 * The 6 bits counters are encoded one after the other starting from the
 * LSB to the MSB, and using the next bytes as needed.
 *
 * At the end of the 16k counters, there is an additional 64 bit integer
 * stored in little endian format with the latest cardinality computed that
 * can be reused if the data structure was not modified since the last
 * computation (this is useful because there are high probabilities that
 * HLLADD operations don't modify the actual data structure and hence the
 * approximated cardinality).
 *
 * When the most significant bit in the most significant byte of the cached
 * cardinality is set, it means that the data structure was modified and
 * we can't reuse the cached value that must be recomputed. */

#define REDIS_HLL_P 14 /* The greater is P, the smaller the error. */
#define REDIS_HLL_REGISTERS (1<<REDIS_HLL_P) /* With P=14, 16384 registers. */
#define REDIS_HLL_P_MASK (REDIS_HLL_REGISTERS-1) /* Mask to index register. */
#define REDIS_HLL_BITS 6 /* Enough to count up to 63 leading zeroes. */
#define REDIS_HLL_REGISTER_MAX ((1<<REDIS_HLL_BITS)-1)
/* Note: REDIS_HLL_SIZE define has a final "+8" since we store a 64 bit
 * integer at the end of the HyperLogLog structure to cache the cardinality. */
#define REDIS_HLL_SIZE ((REDIS_HLL_REGISTERS*REDIS_HLL_BITS+7)/8)+8

/* =========================== Low level bit macros ========================= */

/* We need to get and set 6 bit counters in an array of 8 bit bytes.
 * We use macros to make sure the code is inlined since speed is critical
 * especially in order to compute the approximated cardinality in
 * HLLCOUNT where we need to access all the registers at once.
 * For the same reason we also want to avoid conditionals in this code path.
 *
 * +--------+--------+--------+------//
 * |11000000|22221111|33333322|55444444
 * +--------+--------+--------+------//
 *
 * Note: in the above representation the most significant bit (MSB)
 * of every byte is on the left. We start using bits from the LSB to MSB,
 * and so forth passing to the next byte.
 *
 * Example, we want to access to counter at pos = 1 ("111111" in the
 * illustration above).
 *
 * The index of the first byte b0 containing our data is:
 *
 *  b0 = 6 * pos / 8 = 0
 *
 *   +--------+
 *   |11000000|  <- Our byte at b0
 *   +--------+
 *
 * The position of the first bit (counting from the LSB = 0) in the byte
 * is given by:
 *
 *  fb = 6 * pos % 8 -> 6
 *
 * Right shift b0 of 'fb' bits.
 *
 *   +--------+
 *   |11000000|  <- Initial value of b0
 *   |00000011|  <- After right shift of 6 pos.
 *   +--------+
 *
 * Left shift b1 of bits 8-fb bits (2 bits)
 *
 *   +--------+
 *   |22221111|  <- Initial value of b1
 *   |22111100|  <- After left shift of 2 bits.
 *   +--------+
 *
 * OR the two bits, and finally AND with 111111 (63 in decimal) to
 * clean the higher order bits we are not interested in:
 *
 *   +--------+
 *   |00000011|  <- b0 right shifted
 *   |22111100|  <- b1 left shifted
 *   |22111111|  <- b0 OR b1
 *   |  111111|  <- (b0 OR b1) AND 63, our value.
 *   +--------+
 *
 * We can try with a different example, like pos = 0. In this case
 * the 6-bit counter is actually contained in a single byte.
 *
 *  b0 = 6 * pos / 8 = 0
 *
 *   +--------+
 *   |11000000|  <- Our byte at b0
 *   +--------+
 *
 *  fb = 6 * pos % 8 = 0
 *
 *  So we right shift of 0 bits (no shift in practice) and
 *  left shift the next byte of 8 bits, even if we don't use it,
 *  but this has the effect of clearing the bits so the result
 *  will not be affacted after the OR.
 *
 * -------------------------------------------------------------------------
 *
 * Setting the register is a bit more complex, let's assume that 'val'
 * is the value we want to set, already in the right range.
 *
 * We need two steps, in one we need to clear the bits, and in the other
 * we need to bitwise-OR the new bits.
 *
 * Let's try with 'pos' = 1, so our first byte at 'b' is 0,
 *
 * "fb" is 6 in this case.
 *
 *   +--------+
 *   |11000000|  <- Our byte at b0
 *   +--------+
 *
 * To create a AND-mask to clear the bits about this position, we just
 * initialize the mask with the value 63, left shift it of "fs" bits,
 * and finally invert the result.
 *
 *   +--------+
 *   |00111111|  <- "mask" starts at 63
 *   |11000000|  <- "mask" after left shift of "ls" bits.
 *   |00111111|  <- "mask" after invert.
 *   +--------+
 *
 * Now we can bitwise-AND the byte at "b" with the mask, and bitwise-OR
 * it with "val" left-shifted of "ls" bits to set the new bits.
 *
 * Now let's focus on the next byte b1:
 *
 *   +--------+
 *   |22221111|  <- Initial value of b1
 *   +--------+
 *
 * To build the AND mask we start again with the 63 value, right shift
 * it by 8-fb bits, and invert it.
 *
 *   +--------+
 *   |00111111|  <- "mask" set at 2&6-1
 *   |00001111|  <- "mask" after the right shift by 8-fb = 2 bits
 *   |11110000|  <- "mask" after bitwise not.
 *   +--------+
 *
 * Now we can mask it with b+1 to clear the old bits, and bitwise-OR
 * with "val" left-shifted by "rs" bits to set the new value.
 */

/* Note: if we access the last counter, we will also access the b+1 byte
 * that is out of the array, but sds strings always have an implicit null
 * term, so the byte exists, and we can skip the conditional (or the need
 * to allocate 1 byte more explicitly). */

/* Store the value of the register at position 'regnum' into variable 'target'.
 * 'p' is an array of unsigned bytes. */
#define HLL_GET_REGISTER(target,p,regnum) do { \
    uint8_t *_p = (uint8_t*) p; \
    unsigned long _byte = regnum*REDIS_HLL_BITS/8; \
    unsigned long _fb = regnum*REDIS_HLL_BITS&7; \
    unsigned long _fb8 = 8 - _fb; \
    unsigned long b0 = _p[_byte]; \
    unsigned long b1 = _p[_byte+1]; \
    target = ((b0 >> _fb) | (b1 << _fb8)) & REDIS_HLL_REGISTER_MAX; \
} while(0)

/* Set the value of the register at position 'regnum' to 'val'.
 * 'p' is an array of unsigned bytes. */
#define HLL_SET_REGISTER(p,regnum,val) do { \
    uint8_t *_p = (uint8_t*) p; \
    unsigned long _byte = regnum*REDIS_HLL_BITS/8; \
    unsigned long _fb = regnum*REDIS_HLL_BITS&7; \
    unsigned long _fb8 = 8 - _fb; \
    unsigned long _v = val; \
    _p[_byte] &= ~(REDIS_HLL_REGISTER_MAX << _fb); \
    _p[_byte] |= _v << _fb; \
    _p[_byte+1] &= ~(REDIS_HLL_REGISTER_MAX >> _fb8); \
    _p[_byte+1] |= _v >> _fb8; \
} while(0)

/* ========================= HyperLogLog algorithm  ========================= */

/* Our hash function is MurmurHash2, 64 bit version.
 * It was modified for Redis in order to provide the same result in
 * big and little endian archs (endian neutral). */
uint64_t MurmurHash64A (const void * key, int len, unsigned int seed) {
    const uint64_t m = 0xc6a4a7935bd1e995;
    const int r = 47;
    uint64_t h = seed ^ (len * m);
    const uint8_t *data = (const uint8_t *)key;
    const uint8_t *end = data + (len-(len&7));

    while(data != end) {
        uint64_t k;

#if (BYTE_ORDER == LITTLE_ENDIAN)
        k = *((uint64_t*)data);
#else
        k = (uint64_t) data[0];
        k |= (uint64_t) data[1] << 8;
        k |= (uint64_t) data[2] << 16;
        k |= (uint64_t) data[3] << 24;
        k |= (uint64_t) data[4] << 32;
        k |= (uint64_t) data[5] << 40;
        k |= (uint64_t) data[6] << 48;
        k |= (uint64_t) data[7] << 56;
#endif

        k *= m;
        k ^= k >> r;
        k *= m;
        h ^= k;
        h *= m;
        data += 8;
    }

    switch(len & 7) {
    case 7: h ^= (uint64_t)data[6] << 48;
    case 6: h ^= (uint64_t)data[5] << 40;
    case 5: h ^= (uint64_t)data[4] << 32;
    case 4: h ^= (uint64_t)data[3] << 24;
    case 3: h ^= (uint64_t)data[2] << 16;
    case 2: h ^= (uint64_t)data[1] << 8;
    case 1: h ^= (uint64_t)data[0];
            h *= m;
    };

    h ^= h >> r;
    h *= m;
    h ^= h >> r;
    return h;
}

/* "Add" the element in the hyperloglog data structure.
 * Actually nothing is added, but the max 0 pattern counter of the subset
 * the element belongs to is incremented if needed.
 *
 * 'registers' is expected to have room for REDIS_HLL_REGISTERS plus an
 * additional byte on the right. This requirement is met by sds strings
 * automatically since they are implicitly null terminated.
 *
 * The function always succeed, however if as a result of the operation
 * the approximated cardinality changed, 1 is returned. Otherwise 0
 * is returned. */
int hllAdd(uint8_t *registers, unsigned char *ele, size_t elesize) {
    uint64_t hash, bit, index;
    uint8_t oldcount, count;

    /* Count the number of zeroes starting from bit REDIS_HLL_REGISTERS
     * (that is a power of two corresponding to the first bit we don't use
     * as index). The max run can be 64-P+1 bits.
     *
     * Note that the final "1" ending the sequence of zeroes must be
     * included in the count, so if we find "001" the count is 3, and
     * the smallest count possible is no zeroes at all, just a 1 bit
     * at the first position, that is a count of 1.
     *
     * This may sound like inefficient, but actually in the average case
     * there are high probabilities to find a 1 after a few iterations. */
    hash = MurmurHash64A(ele,elesize,0);
    bit = REDIS_HLL_REGISTERS;
    count = 1;
    while((hash & bit) == 0) {
        count++;
        /* Test the next bit. Note that if we run out of bits in the 64
         * bit integer, bit will be set to 0, and the while test will fail,
         * so we can save the explicit check and yet the algorithm will
         * terminate. */
        bit <<= 1;
    }

    /* Update the register if this element produced a longer run of zeroes. */
    index = hash & REDIS_HLL_P_MASK; /* Index a register inside registers. */
    HLL_GET_REGISTER(oldcount,registers,index);
    if (count > oldcount) {
        HLL_SET_REGISTER(registers,index,count);
        return 1;
    } else {
        return 0;
    }
}

/* Return the approximated cardinality of the set based on the armonic
 * mean of the registers values. */
uint64_t hllCount(uint8_t *registers) {
    double m = REDIS_HLL_REGISTERS;
    double alpha = 0.7213/(1+1.079/m);
    double E = 0, linearcounting_factor;
    int ez = 0; /* Number of registers equal to 0. */
    int j;

    /* We precompute 2^(-reg[j]) in a small table in order to
     * speedup the computation of SUM(2^-register[0..i]). */
    static int initialized = 0;
    static double PE[64];
    if (!initialized) {
        PE[0] = 1; /* 2^(-reg[j]) is 1 when m is 0. */
        for (j = 1; j < 64; j++) {
            /* 2^(-reg[j]) is the same as 1/2^reg[j]. */
            PE[j] = 1.0/(1ULL << j);
        }
        initialized = 1;
    }

    /* Compute SUM(2^-register[0..i]).
     * Redis default is to use 16384 registers 6 bits each. The code works
     * with other values by modifying the defines, but for our target value
     * we take a faster path with unrolled loops. */
    if (REDIS_HLL_REGISTERS == 16384 && REDIS_HLL_BITS == 6) {
        uint8_t *r = registers;
        unsigned long r0, r1, r2, r3, r4, r5, r6, r7, r8, r9,
                      r10, r11, r12, r13, r14, r15;
        for (j = 0; j < 1024; j++) {
            /* Handle 16 registers per iteration. */
            r0 = r[0] & 63; if (r0 == 0) ez++;
            r1 = (r[0] >> 6 | r[1] << 2) & 63; if (r1 == 0) ez++;
            r2 = (r[1] >> 4 | r[2] << 4) & 63; if (r2 == 0) ez++;
            r3 = (r[2] >> 2) & 63; if (r3 == 0) ez++;
            r4 = r[3] & 63; if (r4 == 0) ez++;
            r5 = (r[3] >> 6 | r[4] << 2) & 63; if (r5 == 0) ez++;
            r6 = (r[4] >> 4 | r[5] << 4) & 63; if (r6 == 0) ez++;
            r7 = (r[5] >> 2) & 63; if (r7 == 0) ez++;
            r8 = r[6] & 63; if (r8 == 0) ez++;
            r9 = (r[6] >> 6 | r[7] << 2) & 63; if (r9 == 0) ez++;
            r10 = (r[7] >> 4 | r[8] << 4) & 63; if (r10 == 0) ez++;
            r11 = (r[8] >> 2) & 63; if (r11 == 0) ez++;
            r12 = r[9] & 63; if (r12 == 0) ez++;
            r13 = (r[9] >> 6 | r[10] << 2) & 63; if (r13 == 0) ez++;
            r14 = (r[10] >> 4 | r[11] << 4) & 63; if (r14 == 0) ez++;
            r15 = (r[11] >> 2) & 63; if (r15 == 0) ez++;

            /* Additional parens will allow the compiler to optimize the
             * code more with a loss of precision that is not very relevant
             * here (floating point math is not commutative!). */
            E += (PE[r0] + PE[r1]) + (PE[r2] + PE[r3]) + (PE[r4] + PE[r5]) +
                 (PE[r6] + PE[r7]) + (PE[r8] + PE[r9]) + (PE[r10] + PE[r11]) +
                 (PE[r12] + PE[r13]) + (PE[r14] + PE[r15]);
            r += 12;
        }
    } else {
        for (j = 0; j < REDIS_HLL_REGISTERS; j++) {
            unsigned long reg;

            HLL_GET_REGISTER(reg,registers,j);
            if (reg == 0) {
                ez++;
                E += 1; /* 2^(-reg[j]) is 1 when m is 0. */
            } else {
                E += PE[reg]; /* Precomputed 2^(-reg[j]). */
            }
        }
    }
    /* Muliply the inverse of E for alpha_m * m^2 to have the raw estimate. */
    E = (1/E)*alpha*m*m;

    /* Use the LINEARCOUNTING algorithm for small cardinalities. Note that
     * the HyperLogLog paper suggests using this correction for E < m*2.5
     * while we are using it for E < m*3 since this was verified to have
     * better median / max error rate in the 40000 - 50000 cardinality
     * interval when P * is 14 (m = 16k).
     *
     * However for other values of P we resort to the paper's value of 2.5
     * since no test was performed for other values. */
    linearcounting_factor = (m == 16384) ? 3 : 2.5;
    if (E < m*linearcounting_factor && ez != 0) {
        E = m*log(m/ez); /* LINEARCOUNTING() */
    }
    /* We don't apply the correction for E > 1/30 of 2^32 since we use
     * a 64 bit function and 6 bit counters. To apply the correction for
     * 1/30 of 2^64 is not needed since it would require a huge set
     * to approach such a value. */
    return (uint64_t) E;
}

/* ========================== HyperLogLog commands ========================== */

/* HLLADD var ele ele ele ... ele => :0 or :1 */
void hllAddCommand(redisClient *c) {
    robj *o = lookupKeyWrite(c->db,c->argv[1]);
    uint8_t *registers;
    int updated = 0, j;

    if (o == NULL) {
        /* Create the key with a string value of the exact length to
         * hold our HLL data structure. sdsnewlen() when NULL is passed
         * is guaranteed to return bytes initialized to zero. */
        o = createObject(REDIS_STRING,sdsnewlen(NULL,REDIS_HLL_SIZE));
        dbAdd(c->db,c->argv[1],o);
        updated++;
    } else {
        /* Key exists, check type */
        if (checkType(c,o,REDIS_STRING))
            return;

        /* If this is a string representing an HLL, the size should match
         * exactly. */
        if (stringObjectLen(o) != REDIS_HLL_SIZE) {
            addReplyErrorFormat(c,
                "HLLADD target key must contain a %d bytes string.",
                REDIS_HLL_SIZE);
            return;
        }
        o = dbUnshareStringValue(c->db,c->argv[1],o);
    }
    /* Perform the low level ADD operation for every element. */
    registers = o->ptr;
    for (j = 2; j < c->argc; j++) {
        if (hllAdd(registers, (unsigned char*)c->argv[j]->ptr,
                sdslen(c->argv[j]->ptr)))
        {
            updated++;
        }
    }
    if (updated) {
        signalModifiedKey(c->db,c->argv[1]);
        notifyKeyspaceEvent(REDIS_NOTIFY_STRING,"hlladd",c->argv[1],c->db->id);
        server.dirty++;
        /* Invalidate the cached cardinality. */
        registers[REDIS_HLL_SIZE-1] |= (1<<7);
    }
    addReply(c, updated ? shared.cone : shared.czero);
}

/* HLLCOUNT var -> approximated cardinality of set. */
void hllCountCommand(redisClient *c) {
    robj *o = lookupKeyRead(c->db,c->argv[1]);
    uint8_t *registers;
    uint64_t card;

    if (o == NULL) {
        /* No key? Cardinality is zero since no element was added, otherwise
         * we would have a key as HLLADD creates it as a side effect. */
        addReply(c,shared.czero);
    } else {
        /* Key exists, check type */
        if (checkType(c,o,REDIS_STRING))
            return;

        /* If this is a string representing an HLL, the size should match
         * exactly. */
        if (stringObjectLen(o) != REDIS_HLL_SIZE) {
            addReplyErrorFormat(c,
                "HLLCOUNT target key must contain a %d bytes string.",
                REDIS_HLL_SIZE);
            return;
        }

        /* Check if the cached cardinality is valid. */
        registers = o->ptr;
        if ((registers[REDIS_HLL_SIZE-1] & (1<<7)) == 0) {
            /* Just return the cached value. */
            card = (uint64_t)registers[REDIS_HLL_SIZE-8];
            card |= (uint64_t)registers[REDIS_HLL_SIZE-7] << 8;
            card |= (uint64_t)registers[REDIS_HLL_SIZE-6] << 16;
            card |= (uint64_t)registers[REDIS_HLL_SIZE-5] << 24;
            card |= (uint64_t)registers[REDIS_HLL_SIZE-4] << 32;
            card |= (uint64_t)registers[REDIS_HLL_SIZE-3] << 40;
            card |= (uint64_t)registers[REDIS_HLL_SIZE-2] << 48;
            card |= (uint64_t)registers[REDIS_HLL_SIZE-1] << 56;
        } else {
            /* Recompute it and update the cached value. */
            card = hllCount(registers);
            registers[REDIS_HLL_SIZE-8] = card & 0xff;
            registers[REDIS_HLL_SIZE-7] = (card >> 8) & 0xff;
            registers[REDIS_HLL_SIZE-6] = (card >> 16) & 0xff;
            registers[REDIS_HLL_SIZE-5] = (card >> 24) & 0xff;
            registers[REDIS_HLL_SIZE-4] = (card >> 32) & 0xff;
            registers[REDIS_HLL_SIZE-3] = (card >> 40) & 0xff;
            registers[REDIS_HLL_SIZE-2] = (card >> 48) & 0xff;
            registers[REDIS_HLL_SIZE-1] = (card >> 56) & 0xff;
            /* This is not considered a read-only command even if the
             * data structure is not modified, since the cached value
             * may be modified and given that the HLL is a Redis string
             * we need to propagate the change. */
            signalModifiedKey(c->db,c->argv[1]);
            server.dirty++;
        }
        addReplyLongLong(c,card);
    }
}

/* This command performs a self-test of the HLL registers implementation.
 * Something that is not easy to test from within the outside.
 *
 * The test is conceived to test that the different counters of our data
 * structure are accessible and that setting their values both result in
 * the correct value to be retained and not affect adjacent values. */

#define REDIS_HLL_TEST_CYCLES 1000
void hllSelftestCommand(redisClient *c) {
    int j, i;
    sds bitcounters = sdsnewlen(NULL,REDIS_HLL_SIZE);
    uint8_t bytecounters[REDIS_HLL_REGISTERS];

    for (j = 0; j < REDIS_HLL_TEST_CYCLES; j++) {
        /* Set the HLL counters and an array of unsigned byes of the
         * same size to the same set of random values. */
        for (i = 0; i < REDIS_HLL_REGISTERS; i++) {
            unsigned int r = rand() & REDIS_HLL_REGISTER_MAX;

            bytecounters[i] = r;
            HLL_SET_REGISTER(bitcounters,i,r);
        }
        /* Check that we are able to retrieve the same values. */
        for (i = 0; i < REDIS_HLL_REGISTERS; i++) {
            unsigned int val;

            HLL_GET_REGISTER(val,bitcounters,i);
            if (val != bytecounters[i]) {
                addReplyErrorFormat(c,
                    "TESTFAILED Register %d should be %d but is %d",
                    i, (int) bytecounters[i], (int) val);
                goto cleanup;
            }
        }
    }

    /* Success! */
    addReply(c,shared.ok);

cleanup:
    sdsfree(bitcounters);
}