hyperloglog.cpp

/* 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 "server.h"

#include <stdint.h>
#include <math.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 estimate
 *   cardinalities larger than 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.
 *
 * Redis uses two representations:
 *
 * 1) A "dense" representation where every entry is represented by
 *    a 6-bit integer.
 * 2) A "sparse" representation using run length compression suitable
 *    for representing HyperLogLogs with many registers set to 0 in
 *    a memory efficient way.
 *
 *
 * HLL header
 * ===
 *
 * Both the dense and sparse representation have a 16 byte header as follows:
 *
 * +------+---+-----+----------+
 * | HYLL | E | N/U | Cardin.  |
 * +------+---+-----+----------+
 *
 * The first 4 bytes are a magic string set to the bytes "HYLL".
 * "E" is one byte encoding, currently set to HLL_DENSE or
 * HLL_SPARSE. N/U are three not used bytes.
 *
 * The "Cardin." field is a 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.
 *
 * Dense representation
 * ===
 *
 * The dense representation used by Redis is the following:
 *
 * +--------+--------+--------+------//      //--+
 * |11000000|22221111|33333322|55444444 ....     |
 * +--------+--------+--------+------//      //--+
 *
 * The 6 bits counters are encoded one after the other starting from the
 * LSB to the MSB, and using the next bytes as needed.
 *
 * Sparse representation
 * ===
 *
 * The sparse representation encodes registers using a run length
 * encoding composed of three opcodes, two using one byte, and one using
 * of two bytes. The opcodes are called ZERO, XZERO and VAL.
 *
 * ZERO opcode is represented as 00xxxxxx. The 6-bit integer represented
 * by the six bits 'xxxxxx', plus 1, means that there are N registers set
 * to 0. This opcode can represent from 1 to 64 contiguous registers set
 * to the value of 0.
 *
 * XZERO opcode is represented by two bytes 01xxxxxx yyyyyyyy. The 14-bit
 * integer represented by the bits 'xxxxxx' as most significant bits and
 * 'yyyyyyyy' as least significant bits, plus 1, means that there are N
 * registers set to 0. This opcode can represent from 0 to 16384 contiguous
 * registers set to the value of 0.
 *
 * VAL opcode is represented as 1vvvvvxx. It contains a 5-bit integer
 * representing the value of a register, and a 2-bit integer representing
 * the number of contiguous registers set to that value 'vvvvv'.
 * To obtain the value and run length, the integers vvvvv and xx must be
 * incremented by one. This opcode can represent values from 1 to 32,
 * repeated from 1 to 4 times.
 *
 * The sparse representation can't represent registers with a value greater
 * than 32, however it is very unlikely that we find such a register in an
 * HLL with a cardinality where the sparse representation is still more
 * memory efficient than the dense representation. When this happens the
 * HLL is converted to the dense representation.
 *
 * The sparse representation is purely positional. For example a sparse
 * representation of an empty HLL is just: XZERO:16384.
 *
 * An HLL having only 3 non-zero registers at position 1000, 1020, 1021
 * respectively set to 2, 3, 3, is represented by the following three
 * opcodes:
 *
 * XZERO:1000 (Registers 0-999 are set to 0)
 * VAL:2,1    (1 register set to value 2, that is register 1000)
 * ZERO:19    (Registers 1001-1019 set to 0)
 * VAL:3,2    (2 registers set to value 3, that is registers 1020,1021)
 * XZERO:15362 (Registers 1022-16383 set to 0)
 *
 * In the example the sparse representation used just 7 bytes instead
 * of 12k in order to represent the HLL registers. In general for low
 * cardinality there is a big win in terms of space efficiency, traded
 * with CPU time since the sparse representation is slower to access:
 *
 * The following table shows average cardinality vs bytes used, 100
 * samples per cardinality (when the set was not representable because
 * of registers with too big value, the dense representation size was used
 * as a sample).
 *
 * 100 267
 * 200 485
 * 300 678
 * 400 859
 * 500 1033
 * 600 1205
 * 700 1375
 * 800 1544
 * 900 1713
 * 1000 1882
 * 2000 3480
 * 3000 4879
 * 4000 6089
 * 5000 7138
 * 6000 8042
 * 7000 8823
 * 8000 9500
 * 9000 10088
 * 10000 10591
 *
 * The dense representation uses 12288 bytes, so there is a big win up to
 * a cardinality of ~2000-3000. For bigger cardinalities the constant times
 * involved in updating the sparse representation is not justified by the
 * memory savings. The exact maximum length of the sparse representation
 * when this implementation switches to the dense representation is
 * configured via the define g_pserver->hll_sparse_max_bytes.
 */

struct hllhdr {
    char magic[4];      /* "HYLL" */
    uint8_t encoding;   /* HLL_DENSE or HLL_SPARSE. */
    uint8_t notused[3]; /* Reserved for future use, must be zero. */
    uint8_t card[8];    /* Cached cardinality, little endian. */
    uint8_t *registers() { /* Data bytes. */
        return reinterpret_cast<uint8_t*>(this+1);
    }
};

/* The cached cardinality MSB is used to signal validity of the cached value. */
#define HLL_INVALIDATE_CACHE(hdr) (hdr)->card[7] |= (1<<7)
#define HLL_VALID_CACHE(hdr) (((hdr)->card[7] & (1<<7)) == 0)

#define HLL_P 14 /* The greater is P, the smaller the error. */
#define HLL_Q (64-HLL_P) /* The number of bits of the hash value used for
                            determining the number of leading zeros. */
#define HLL_REGISTERS (1<<HLL_P) /* With P=14, 16384 registers. */
#define HLL_P_MASK (HLL_REGISTERS-1) /* Mask to index register. */
#define HLL_BITS 6 /* Enough to count up to 63 leading zeroes. */
#define HLL_REGISTER_MAX ((1<<HLL_BITS)-1)
#define HLL_HDR_SIZE sizeof(struct hllhdr)
#define HLL_DENSE_SIZE (HLL_HDR_SIZE+((HLL_REGISTERS*HLL_BITS+7)/8))
#define HLL_DENSE 0 /* Dense encoding. */
#define HLL_SPARSE 1 /* Sparse encoding. */
#define HLL_RAW 255 /* Only used internally, never exposed. */
#define HLL_MAX_ENCODING 1

static const char *invalid_hll_err = "-INVALIDOBJ Corrupted HLL object detected";

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

/* Macros to access the dense representation.
 *
 * 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 affected 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 an 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_DENSE_GET_REGISTER(target,p,regnum) do { \
    uint8_t *_p = (uint8_t*) p; \
    unsigned long _byte = regnum*HLL_BITS/8; \
    unsigned long _fb = regnum*HLL_BITS&7; \
    unsigned long _fb8 = 8 - _fb; \
    unsigned long b0 = _p[_byte]; \
    unsigned long b1 = _p[_byte+1]; \
    target = ((b0 >> _fb) | (b1 << _fb8)) & 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_DENSE_SET_REGISTER(p,regnum,val) do { \
    uint8_t *_p = (uint8_t*) p; \
    unsigned long _byte = regnum*HLL_BITS/8; \
    unsigned long _fb = regnum*HLL_BITS&7; \
    unsigned long _fb8 = 8 - _fb; \
    unsigned long _v = val; \
    _p[_byte] &= ~(HLL_REGISTER_MAX << _fb); \
    _p[_byte] |= _v << _fb; \
    _p[_byte+1] &= ~(HLL_REGISTER_MAX >> _fb8); \
    _p[_byte+1] |= _v >> _fb8; \
} while(0)

/* Macros to access the sparse representation.
 * The macros parameter is expected to be an uint8_t pointer. */
#define HLL_SPARSE_XZERO_BIT 0x40 /* 01xxxxxx */
#define HLL_SPARSE_VAL_BIT 0x80 /* 1vvvvvxx */
#define HLL_SPARSE_IS_ZERO(p) (((*(p)) & 0xc0) == 0) /* 00xxxxxx */
#define HLL_SPARSE_IS_XZERO(p) (((*(p)) & 0xc0) == HLL_SPARSE_XZERO_BIT)
#define HLL_SPARSE_IS_VAL(p) ((*(p)) & HLL_SPARSE_VAL_BIT)
#define HLL_SPARSE_ZERO_LEN(p) (((*(p)) & 0x3f)+1)
#define HLL_SPARSE_XZERO_LEN(p) (((((*(p)) & 0x3f) << 8) | (*((p)+1)))+1)
#define HLL_SPARSE_VAL_VALUE(p) ((((*(p)) >> 2) & 0x1f)+1)
#define HLL_SPARSE_VAL_LEN(p) (((*(p)) & 0x3)+1)
#define HLL_SPARSE_VAL_MAX_VALUE 32
#define HLL_SPARSE_VAL_MAX_LEN 4
#define HLL_SPARSE_ZERO_MAX_LEN 64
#define HLL_SPARSE_XZERO_MAX_LEN 16384
#define HLL_SPARSE_VAL_SET(p,val,len) do { \
    *(p) = (((val)-1)<<2|((len)-1))|HLL_SPARSE_VAL_BIT; \
} while(0)
#define HLL_SPARSE_ZERO_SET(p,len) do { \
    *(p) = (len)-1; \
} while(0)
#define HLL_SPARSE_XZERO_SET(p,len) do { \
    int _l = (len)-1; \
    *(p) = (_l>>8) | HLL_SPARSE_XZERO_BIT; \
    *((p)+1) = (_l&0xff); \
} while(0)
#define HLL_ALPHA_INF 0.721347520444481703680 /* constant for 0.5/ln(2) */

/* ========================= 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)
    #ifdef USE_ALIGNED_ACCESS
        memcpy(&k,data,sizeof(uint64_t));
    #else
        k = *((uint64_t*)data);
    #endif
#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; /* fall-thru */
    case 6: h ^= (uint64_t)data[5] << 40; /* fall-thru */
    case 5: h ^= (uint64_t)data[4] << 32; /* fall-thru */
    case 4: h ^= (uint64_t)data[3] << 24; /* fall-thru */
    case 3: h ^= (uint64_t)data[2] << 16; /* fall-thru */
    case 2: h ^= (uint64_t)data[1] << 8; /* fall-thru */
    case 1: h ^= (uint64_t)data[0];
            h *= m; /* fall-thru */
    };

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

/* Given a string element to add to the HyperLogLog, returns the length
 * of the pattern 000..1 of the element hash. As a side effect 'regp' is
 * set to the register index this element hashes to. */
int hllPatLen(unsigned char *ele, size_t elesize, long *regp) {
    uint64_t hash, bit, index;
    int count;

    /* Count the number of zeroes starting from bit 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 = Q+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,0xadc83b19ULL);
    index = hash & HLL_P_MASK; /* Register index. */
    hash >>= HLL_P; /* Remove bits used to address the register. */
    hash |= ((uint64_t)1<<HLL_Q); /* Make sure the loop terminates
                                     and count will be <= Q+1. */
    bit = 1;
    count = 1; /* Initialized to 1 since we count the "00000...1" pattern. */
    while((hash & bit) == 0) {
        count++;
        bit <<= 1;
    }
    *regp = (int) index;
    return count;
}

/* ================== Dense representation implementation  ================== */

/* Low level function to set the dense HLL register at 'index' to the
 * specified value if the current value is smaller than 'count'.
 *
 * 'registers' is expected to have room for 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 hllDenseSet(uint8_t *registers, long index, uint8_t count) {
    uint8_t oldcount;

    HLL_DENSE_GET_REGISTER(oldcount,registers,index);
    if (count > oldcount) {
        HLL_DENSE_SET_REGISTER(registers,index,count);
        return 1;
    } else {
        return 0;
    }
}

/* "Add" the element in the dense hyperloglog data structure.
 * Actually nothing is added, but the max 0 pattern counter of the subset
 * the element belongs to is incremented if needed.
 *
 * This is just a wrapper to hllDenseSet(), performing the hashing of the
 * element in order to retrieve the index and zero-run count. */
int hllDenseAdd(uint8_t *registers, unsigned char *ele, size_t elesize) {
    long index;
    uint8_t count = hllPatLen(ele,elesize,&index);
    /* Update the register if this element produced a longer run of zeroes. */
    return hllDenseSet(registers,index,count);
}

/* Compute the register histogram in the dense representation. */
void hllDenseRegHisto(uint8_t *registers, int* reghisto) {
    int j;

    /* 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 (HLL_REGISTERS == 16384 && 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;
            r1 = (r[0] >> 6 | r[1] << 2) & 63;
            r2 = (r[1] >> 4 | r[2] << 4) & 63;
            r3 = (r[2] >> 2) & 63;
            r4 = r[3] & 63;
            r5 = (r[3] >> 6 | r[4] << 2) & 63;
            r6 = (r[4] >> 4 | r[5] << 4) & 63;
            r7 = (r[5] >> 2) & 63;
            r8 = r[6] & 63;
            r9 = (r[6] >> 6 | r[7] << 2) & 63;
            r10 = (r[7] >> 4 | r[8] << 4) & 63;
            r11 = (r[8] >> 2) & 63;
            r12 = r[9] & 63;
            r13 = (r[9] >> 6 | r[10] << 2) & 63;
            r14 = (r[10] >> 4 | r[11] << 4) & 63;
            r15 = (r[11] >> 2) & 63;

            reghisto[r0]++;
            reghisto[r1]++;
            reghisto[r2]++;
            reghisto[r3]++;
            reghisto[r4]++;
            reghisto[r5]++;
            reghisto[r6]++;
            reghisto[r7]++;
            reghisto[r8]++;
            reghisto[r9]++;
            reghisto[r10]++;
            reghisto[r11]++;
            reghisto[r12]++;
            reghisto[r13]++;
            reghisto[r14]++;
            reghisto[r15]++;

            r += 12;
        }
    } else {
        for(j = 0; j < HLL_REGISTERS; j++) {
            unsigned long reg;
            HLL_DENSE_GET_REGISTER(reg,registers,j);
            reghisto[reg]++;
        }
    }
}

/* ================== Sparse representation implementation  ================= */

/* Convert the HLL with sparse representation given as input in its dense
 * representation. Both representations are represented by SDS strings, and
 * the input representation is freed as a side effect.
 *
 * The function returns C_OK if the sparse representation was valid,
 * otherwise C_ERR is returned if the representation was corrupted. */
int hllSparseToDense(robj *o) {
    sds sparse = szFromObj(o), dense;
    struct hllhdr *hdr, *oldhdr = (struct hllhdr*)sparse;
    int idx = 0, runlen, regval;
    uint8_t *p = (uint8_t*)sparse, *end = p+sdslen(sparse);

    /* If the representation is already the right one return ASAP. */
    hdr = (struct hllhdr*) sparse;
    if (hdr->encoding == HLL_DENSE) return C_OK;

    /* Create a string of the right size filled with zero bytes.
     * Note that the cached cardinality is set to 0 as a side effect
     * that is exactly the cardinality of an empty HLL. */
    dense = sdsnewlen(NULL,HLL_DENSE_SIZE);
    hdr = (struct hllhdr*) dense;
    *hdr = *oldhdr; /* This will copy the magic and cached cardinality. */
    hdr->encoding = HLL_DENSE;

    /* Now read the sparse representation and set non-zero registers
     * accordingly. */
    p += HLL_HDR_SIZE;
    while(p < end) {
        if (HLL_SPARSE_IS_ZERO(p)) {
            runlen = HLL_SPARSE_ZERO_LEN(p);
            idx += runlen;
            p++;
        } else if (HLL_SPARSE_IS_XZERO(p)) {
            runlen = HLL_SPARSE_XZERO_LEN(p);
            idx += runlen;
            p += 2;
        } else {
            runlen = HLL_SPARSE_VAL_LEN(p);
            regval = HLL_SPARSE_VAL_VALUE(p);
            if ((runlen + idx) > HLL_REGISTERS) break; /* Overflow. */
            while(runlen--) {
                HLL_DENSE_SET_REGISTER(hdr->registers(),idx,regval);
                idx++;
            }
            p++;
        }
    }

    /* If the sparse representation was valid, we expect to find idx
     * set to HLL_REGISTERS. */
    if (idx != HLL_REGISTERS) {
        sdsfree(dense);
        return C_ERR;
    }

    /* Free the old representation and set the new one. */
    sdsfree(szFromObj(o));
    o->m_ptr = dense;
    return C_OK;
}

/* Low level function to set the sparse HLL register at 'index' to the
 * specified value if the current value is smaller than 'count'.
 *
 * The object 'o' is the String object holding the HLL. The function requires
 * a reference to the object in order to be able to enlarge the string if
 * needed.
 *
 * On success, the function returns 1 if the cardinality changed, or 0
 * if the register for this element was not updated.
 * On error (if the representation is invalid) -1 is returned.
 *
 * As a side effect the function may promote the HLL representation from
 * sparse to dense: this happens when a register requires to be set to a value
 * not representable with the sparse representation, or when the resulting
 * size would be greater than g_pserver->hll_sparse_max_bytes. */
int hllSparseSet(robj *o, long index, uint8_t count) {
    struct hllhdr *hdr;
    uint8_t oldcount, *sparse, *end, *p, *prev, *next;
    long first, span;
    long is_zero = 0, is_xzero = 0, is_val = 0, runlen = 0;
    uint8_t seq[5], *n = nullptr;
    int last;
    int len;
    int seqlen = 0;
    int oldlen = 0;
    int deltalen = 0;
    int scanlen = 0;

    /* If the count is too big to be representable by the sparse representation
     * switch to dense representation. */
    if (count > HLL_SPARSE_VAL_MAX_VALUE) goto promote;

    /* When updating a sparse representation, sometimes we may need to
     * enlarge the buffer for up to 3 bytes in the worst case (XZERO split
     * into XZERO-VAL-XZERO). Make sure there is enough space right now
     * so that the pointers we take during the execution of the function
     * will be valid all the time. */
    o->m_ptr = sdsMakeRoomFor(szFromObj(o),3);

    /* Step 1: we need to locate the opcode we need to modify to check
     * if a value update is actually needed. */
    sparse = p = ((uint8_t*)ptrFromObj(o)) + HLL_HDR_SIZE;
    end = p + sdslen(szFromObj(o)) - HLL_HDR_SIZE;

    first = 0;
    prev = NULL; /* Points to previous opcode at the end of the loop. */
    next = NULL; /* Points to the next opcode at the end of the loop. */
    span = 0;
    while(p < end) {
        long oplen;

        /* Set span to the number of registers covered by this opcode.
         *
         * This is the most performance critical loop of the sparse
         * representation. Sorting the conditionals from the most to the
         * least frequent opcode in many-bytes sparse HLLs is faster. */
        oplen = 1;
        if (HLL_SPARSE_IS_ZERO(p)) {
            span = HLL_SPARSE_ZERO_LEN(p);
        } else if (HLL_SPARSE_IS_VAL(p)) {
            span = HLL_SPARSE_VAL_LEN(p);
        } else { /* XZERO. */
            span = HLL_SPARSE_XZERO_LEN(p);
            oplen = 2;
        }
        /* Break if this opcode covers the register as 'index'. */
        if (index <= first+span-1) break;
        prev = p;
        p += oplen;
        first += span;
    }
    if (span == 0 || p >= end) return -1; /* Invalid format. */

    next = HLL_SPARSE_IS_XZERO(p) ? p+2 : p+1;
    if (next >= end) next = NULL;

    /* Cache current opcode type to avoid using the macro again and
     * again for something that will not change.
     * Also cache the run-length of the opcode. */
    if (HLL_SPARSE_IS_ZERO(p)) {
        is_zero = 1;
        runlen = HLL_SPARSE_ZERO_LEN(p);
    } else if (HLL_SPARSE_IS_XZERO(p)) {
        is_xzero = 1;
        runlen = HLL_SPARSE_XZERO_LEN(p);
    } else {
        is_val = 1;
        runlen = HLL_SPARSE_VAL_LEN(p);
    }

    /* Step 2: After the loop:
     *
     * 'first' stores to the index of the first register covered
     *  by the current opcode, which is pointed by 'p'.
     *
     * 'next' ad 'prev' store respectively the next and previous opcode,
     *  or NULL if the opcode at 'p' is respectively the last or first.
     *
     * 'span' is set to the number of registers covered by the current
     *  opcode.
     *
     * There are different cases in order to update the data structure
     * in place without generating it from scratch:
     *
     * A) If it is a VAL opcode already set to a value >= our 'count'
     *    no update is needed, regardless of the VAL run-length field.
     *    In this case PFADD returns 0 since no changes are performed.
     *
     * B) If it is a VAL opcode with len = 1 (representing only our
     *    register) and the value is less than 'count', we just update it
     *    since this is a trivial case. */
    if (is_val) {
        oldcount = HLL_SPARSE_VAL_VALUE(p);
        /* Case A. */
        if (oldcount >= count) return 0;

        /* Case B. */
        if (runlen == 1) {
            HLL_SPARSE_VAL_SET(p,count,1);
            goto updated;
        }
    }

    /* C) Another trivial to handle case is a ZERO opcode with a len of 1.
     * We can just replace it with a VAL opcode with our value and len of 1. */
    if (is_zero && runlen == 1) {
        HLL_SPARSE_VAL_SET(p,count,1);
        goto updated;
    }

    /* D) General case.
     *
     * The other cases are more complex: our register requires to be updated
     * and is either currently represented by a VAL opcode with len > 1,
     * by a ZERO opcode with len > 1, or by an XZERO opcode.
     *
     * In those cases the original opcode must be split into multiple
     * opcodes. The worst case is an XZERO split in the middle resulting into
     * XZERO - VAL - XZERO, so the resulting sequence max length is
     * 5 bytes.
     *
     * We perform the split writing the new sequence into the 'new' buffer
     * with 'newlen' as length. Later the new sequence is inserted in place
     * of the old one, possibly moving what is on the right a few bytes
     * if the new sequence is longer than the older one. */
    n = seq;
    last = first+span-1; /* Last register covered by the sequence. */

    if (is_zero || is_xzero) {
        /* Handle splitting of ZERO / XZERO. */
        if (index != first) {
            len = index-first;
            if (len > HLL_SPARSE_ZERO_MAX_LEN) {
                HLL_SPARSE_XZERO_SET(n,len);
                n += 2;
            } else {
                HLL_SPARSE_ZERO_SET(n,len);
                n++;
            }
        }
        HLL_SPARSE_VAL_SET(n,count,1);
        n++;
        if (index != last) {
            len = last-index;
            if (len > HLL_SPARSE_ZERO_MAX_LEN) {
                HLL_SPARSE_XZERO_SET(n,len);
                n += 2;
            } else {
                HLL_SPARSE_ZERO_SET(n,len);
                n++;
            }
        }
    } else {
        /* Handle splitting of VAL. */
        int curval = HLL_SPARSE_VAL_VALUE(p);

        if (index != first) {
            len = index-first;
            HLL_SPARSE_VAL_SET(n,curval,len);
            n++;
        }
        HLL_SPARSE_VAL_SET(n,count,1);
        n++;
        if (index != last) {
            len = last-index;
            HLL_SPARSE_VAL_SET(n,curval,len);
            n++;
        }
    }

    /* Step 3: substitute the new sequence with the old one.
     *
     * Note that we already allocated space on the sds string
     * calling sdsMakeRoomFor(). */
     seqlen = n-seq;
     oldlen = is_xzero ? 2 : 1;
     deltalen = seqlen-oldlen;

     if (deltalen > 0 &&
         sdslen(szFromObj(o))+deltalen > g_pserver->hll_sparse_max_bytes) goto promote;
     if (deltalen && next) memmove(next+deltalen,next,end-next);
     sdsIncrLen(szFromObj(o),deltalen);
     memcpy(p,seq,seqlen);
     end += deltalen;

updated:
    /* Step 4: Merge adjacent values if possible.
     *
     * The representation was updated, however the resulting representation
     * may not be optimal: adjacent VAL opcodes can sometimes be merged into
     * a single one. */
    p = prev ? prev : sparse;
    scanlen = 5; /* Scan up to 5 upcodes starting from prev. */
    while (p < end && scanlen--) {
        if (HLL_SPARSE_IS_XZERO(p)) {
            p += 2;
            continue;
        } else if (HLL_SPARSE_IS_ZERO(p)) {
            p++;
            continue;
        }
        /* We need two adjacent VAL opcodes to try a merge, having
         * the same value, and a len that fits the VAL opcode max len. */
        if (p+1 < end && HLL_SPARSE_IS_VAL(p+1)) {
            int v1 = HLL_SPARSE_VAL_VALUE(p);
            int v2 = HLL_SPARSE_VAL_VALUE(p+1);
            if (v1 == v2) {
                int len = HLL_SPARSE_VAL_LEN(p)+HLL_SPARSE_VAL_LEN(p+1);
                if (len <= HLL_SPARSE_VAL_MAX_LEN) {
                    HLL_SPARSE_VAL_SET(p+1,v1,len);
                    memmove(p,p+1,end-p);
                    sdsIncrLen(szFromObj(o),-1);
                    end--;
                    /* After a merge we reiterate without incrementing 'p'
                     * in order to try to merge the just merged value with
                     * a value on its right. */
                    continue;
                }
            }
        }
        p++;
    }

    /* Invalidate the cached cardinality. */
    hdr = (hllhdr*)ptrFromObj(o);
    HLL_INVALIDATE_CACHE(hdr);
    return 1;

promote: /* Promote to dense representation. */
    if (hllSparseToDense(o) == C_ERR) return -1; /* Corrupted HLL. */
    hdr = (hllhdr*)ptrFromObj(o);

    /* We need to call hllDenseAdd() to perform the operation after the
     * conversion. However the result must be 1, since if we need to
     * convert from sparse to dense a register requires to be updated.
     *
     * Note that this in turn means that PFADD will make sure the command
     * is propagated to slaves / AOF, so if there is a sparse -> dense
     * conversion, it will be performed in all the slaves as well. */
    int dense_retval = hllDenseSet(hdr->registers(),index,count);
    serverAssert(dense_retval == 1);
    return dense_retval;
}

/* "Add" the element in the sparse hyperloglog data structure.
 * Actually nothing is added, but the max 0 pattern counter of the subset
 * the element belongs to is incremented if needed.
 *
 * This function is actually a wrapper for hllSparseSet(), it only performs
 * the hashshing of the element to obtain the index and zeros run length. */
int hllSparseAdd(robj *o, unsigned char *ele, size_t elesize) {
    long index;
    uint8_t count = hllPatLen(ele,elesize,&index);
    /* Update the register if this element produced a longer run of zeroes. */
    return hllSparseSet(o,index,count);
}

/* Compute the register histogram in the sparse representation. */
void hllSparseRegHisto(uint8_t *sparse, int sparselen, int *invalid, int* reghisto) {
    int idx = 0, runlen, regval;
    uint8_t *end = sparse+sparselen, *p = sparse;

    while(p < end) {
        if (HLL_SPARSE_IS_ZERO(p)) {
            runlen = HLL_SPARSE_ZERO_LEN(p);
            idx += runlen;
            reghisto[0] += runlen;
            p++;
        } else if (HLL_SPARSE_IS_XZERO(p)) {
            runlen = HLL_SPARSE_XZERO_LEN(p);
            idx += runlen;
            reghisto[0] += runlen;
            p += 2;
        } else {
            runlen = HLL_SPARSE_VAL_LEN(p);
            regval = HLL_SPARSE_VAL_VALUE(p);
            idx += runlen;
            reghisto[regval] += runlen;
            p++;
        }
    }
    if (idx != HLL_REGISTERS && invalid) *invalid = 1;
}

/* ========================= HyperLogLog Count ==============================
 * This is the core of the algorithm where the approximated count is computed.
 * The function uses the lower level hllDenseRegHisto() and hllSparseRegHisto()
 * functions as helpers to compute histogram of register values part of the
 * computation, which is representation-specific, while all the rest is common. */

/* Implements the register histogram calculation for uint8_t data type
 * which is only used internally as speedup for PFCOUNT with multiple keys. */
void hllRawRegHisto(uint8_t *registers, int* reghisto) {
    uint64_t *word = (uint64_t*) registers;
    uint8_t *bytes;
    int j;

    for (j = 0; j < HLL_REGISTERS/8; j++) {
        if (*word == 0) {
            reghisto[0] += 8;
        } else {
            bytes = (uint8_t*) word;
            reghisto[bytes[0]]++;
            reghisto[bytes[1]]++;
            reghisto[bytes[2]]++;
            reghisto[bytes[3]]++;
            reghisto[bytes[4]]++;
            reghisto[bytes[5]]++;
            reghisto[bytes[6]]++;
            reghisto[bytes[7]]++;
        }
        word++;
    }
}

/* Helper function sigma as defined in
 * "New cardinality estimation algorithms for HyperLogLog sketches"
 * Otmar Ertl, arXiv:1702.01284 */
double hllSigma(double x) {
    if (x == 1.) return INFINITY;
    double zPrime;
    double y = 1;
    double z = x;
    do {
        x *= x;
        zPrime = z;
        z += x * y;
        y += y;
    } while(zPrime != z);
    return z;
}

/* Helper function tau as defined in
 * "New cardinality estimation algorithms for HyperLogLog sketches"
 * Otmar Ertl, arXiv:1702.01284 */
double hllTau(double x) {
    if (x == 0. || x == 1.) return 0.;
    double zPrime;
    double y = 1.0;
    double z = 1 - x;