Adding upstream version 5:7.0.15.upstream/5%7.0.15upstream

Signed-off-by: Daniel Baumann <daniel.baumann@progress-linux.org>

1 files changed, 1607 insertions, 0 deletions

diff --git a/src/hyperloglog.c b/src/hyperloglog.c
new file mode 100644
index 0000000..84fd6b6
--- /dev/null
+++ b/src/hyperloglog.c
@@ -0,0 +1,1607 @@
+/* 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 server.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. */
+};
+
+/* 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 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). */
+REDIS_NO_SANITIZE("alignment")
+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 = o->ptr, 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(o->ptr);
+    o->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 server.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;
+
+    /* 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->ptr = sdsMakeRoomFor(o->ptr,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*)o->ptr) + HLL_HDR_SIZE;
+    end = p + sdslen(o->ptr) - 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. */
+    uint8_t seq[5], *n = seq;
+    int last = first+span-1; /* Last register covered by the sequence. */
+    int len;
+
+    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(). */
+     int seqlen = n-seq;
+     int oldlen = is_xzero ? 2 : 1;
+     int deltalen = seqlen-oldlen;
+
+     if (deltalen > 0 &&
+         sdslen(o->ptr)+deltalen > server.hll_sparse_max_bytes) goto promote;
+     if (deltalen && next) memmove(next+deltalen,next,end-next);
+     sdsIncrLen(o->ptr,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;
+    int 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(o->ptr,-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 = o->ptr;
+    HLL_INVALIDATE_CACHE(hdr);
+    return 1;
+
+promote: /* Promote to dense representation. */
+    if (hllSparseToDense(o) == C_ERR) return -1; /* Corrupted HLL. */
+    hdr = o->ptr;
+
+    /* 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 hashing 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;
+    do {
+        x = sqrt(x);
+        zPrime = z;
+        y *= 0.5;
+        z -= pow(1 - x, 2)*y;
+    } while(zPrime != z);
+    return z / 3;
+}
+
+/* Return the approximated cardinality of the set based on the harmonic
+ * mean of the registers values. 'hdr' points to the start of the SDS
+ * representing the String object holding the HLL representation.
+ *
+ * If the sparse representation of the HLL object is not valid, the integer
+ * pointed by 'invalid' is set to non-zero, otherwise it is left untouched.
+ *
+ * hllCount() supports a special internal-only encoding of HLL_RAW, that
+ * is, hdr->registers will point to an uint8_t array of HLL_REGISTERS element.
+ * This is useful in order to speedup PFCOUNT when called against multiple
+ * keys (no need to work with 6-bit integers encoding). */
+uint64_t hllCount(struct hllhdr *hdr, int *invalid) {
+    double m = HLL_REGISTERS;
+    double E;
+    int j;
+    /* Note that reghisto size could be just HLL_Q+2, because HLL_Q+1 is
+     * the maximum frequency of the "000...1" sequence the hash function is
+     * able to return. However it is slow to check for sanity of the
+     * input: instead we history array at a safe size: overflows will
+     * just write data to wrong, but correctly allocated, places. */
+    int reghisto[64] = {0};
+
+    /* Compute register histogram */
+    if (hdr->encoding == HLL_DENSE) {
+        hllDenseRegHisto(hdr->registers,reghisto);
+    } else if (hdr->encoding == HLL_SPARSE) {
+        hllSparseRegHisto(hdr->registers,
+                         sdslen((sds)hdr)-HLL_HDR_SIZE,invalid,reghisto);
+    } else if (hdr->encoding == HLL_RAW) {
+        hllRawRegHisto(hdr->registers,reghisto);
+    } else {
+        serverPanic("Unknown HyperLogLog encoding in hllCount()");
+    }
+
+    /* Estimate cardinality from register histogram. See:
+     * "New cardinality estimation algorithms for HyperLogLog sketches"
+     * Otmar Ertl, arXiv:1702.01284 */
+    double z = m * hllTau((m-reghisto[HLL_Q+1])/(double)m);
+    for (j = HLL_Q; j >= 1; --j) {
+        z += reghisto[j];
+        z *= 0.5;
+    }
+    z += m * hllSigma(reghisto[0]/(double)m);
+    E = llroundl(HLL_ALPHA_INF*m*m/z);
+
+    return (uint64_t) E;
+}
+
+/* Call hllDenseAdd() or hllSparseAdd() according to the HLL encoding. */
+int hllAdd(robj *o, unsigned char *ele, size_t elesize) {
+    struct hllhdr *hdr = o->ptr;
+    switch(hdr->encoding) {
+    case HLL_DENSE: return hllDenseAdd(hdr->registers,ele,elesize);
+    case HLL_SPARSE: return hllSparseAdd(o,ele,elesize);
+    default: return -1; /* Invalid representation. */
+    }
+}
+
+/* Merge by computing MAX(registers[i],hll[i]) the HyperLogLog 'hll'
+ * with an array of uint8_t HLL_REGISTERS registers pointed by 'max'.
+ *
+ * The hll object must be already validated via isHLLObjectOrReply()
+ * or in some other way.
+ *
+ * If the HyperLogLog is sparse and is found to be invalid, C_ERR
+ * is returned, otherwise the function always succeeds. */
+int hllMerge(uint8_t *max, robj *hll) {
+    struct hllhdr *hdr = hll->ptr;
+    int i;
+
+    if (hdr->encoding == HLL_DENSE) {
+        uint8_t val;
+
+        for (i = 0; i < HLL_REGISTERS; i++) {
+            HLL_DENSE_GET_REGISTER(val,hdr->registers,i);
+            if (val > max[i]) max[i] = val;
+        }
+    } else {
+        uint8_t *p = hll->ptr, *end = p + sdslen(hll->ptr);
+        long runlen, regval;
+
+        p += HLL_HDR_SIZE;
+        i = 0;
+        while(p < end) {
+            if (HLL_SPARSE_IS_ZERO(p)) {
+                runlen = HLL_SPARSE_ZERO_LEN(p);
+                i += runlen;
+                p++;
+            } else if (HLL_SPARSE_IS_XZERO(p)) {
+                runlen = HLL_SPARSE_XZERO_LEN(p);
+                i += runlen;
+                p += 2;
+            } else {
+                runlen = HLL_SPARSE_VAL_LEN(p);
+                regval = HLL_SPARSE_VAL_VALUE(p);
+                if ((runlen + i) > HLL_REGISTERS) break; /* Overflow. */
+                while(runlen--) {
+                    if (regval > max[i]) max[i] = regval;
+                    i++;
+                }
+                p++;
+            }
+        }
+        if (i != HLL_REGISTERS) return C_ERR;
+    }
+    return C_OK;
+}
+
+/* ========================== HyperLogLog commands ========================== */
+
+/* Create an HLL object. We always create the HLL using sparse encoding.
+ * This will be upgraded to the dense representation as needed. */
+robj *createHLLObject(void) {
+    robj *o;
+    struct hllhdr *hdr;
+    sds s;
+    uint8_t *p;
+    int sparselen = HLL_HDR_SIZE +
+                    (((HLL_REGISTERS+(HLL_SPARSE_XZERO_MAX_LEN-1)) /
+                     HLL_SPARSE_XZERO_MAX_LEN)*2);
+    int aux;
+
+    /* Populate the sparse representation with as many XZERO opcodes as
+     * needed to represent all the registers. */
+    aux = HLL_REGISTERS;
+    s = sdsnewlen(NULL,sparselen);
+    p = (uint8_t*)s + HLL_HDR_SIZE;
+    while(aux) {
+        int xzero = HLL_SPARSE_XZERO_MAX_LEN;
+        if (xzero > aux) xzero = aux;
+        HLL_SPARSE_XZERO_SET(p,xzero);
+        p += 2;
+        aux -= xzero;
+    }
+    serverAssert((p-(uint8_t*)s) == sparselen);
+
+    /* Create the actual object. */
+    o = createObject(OBJ_STRING,s);
+    hdr = o->ptr;
+    memcpy(hdr->magic,"HYLL",4);
+    hdr->encoding = HLL_SPARSE;
+    return o;
+}
+
+/* Check if the object is a String with a valid HLL representation.
+ * Return C_OK if this is true, otherwise reply to the client
+ * with an error and return C_ERR. */
+int isHLLObjectOrReply(client *c, robj *o) {
+    struct hllhdr *hdr;
+
+    /* Key exists, check type */
+    if (checkType(c,o,OBJ_STRING))
+        return C_ERR; /* Error already sent. */
+
+    if (!sdsEncodedObject(o)) goto invalid;
+    if (stringObjectLen(o) < sizeof(*hdr)) goto invalid;
+    hdr = o->ptr;
+
+    /* Magic should be "HYLL". */
+    if (hdr->magic[0] != 'H' || hdr->magic[1] != 'Y' ||
+        hdr->magic[2] != 'L' || hdr->magic[3] != 'L') goto invalid;
+
+    if (hdr->encoding > HLL_MAX_ENCODING) goto invalid;
+
+    /* Dense representation string length should match exactly. */
+    if (hdr->encoding == HLL_DENSE &&
+        stringObjectLen(o) != HLL_DENSE_SIZE) goto invalid;
+
+    /* All tests passed. */
+    return C_OK;
+
+invalid:
+    addReplyError(c,"-WRONGTYPE Key is not a valid "
+               "HyperLogLog string value.");
+    return C_ERR;
+}
+
+/* PFADD var ele ele ele ... ele => :0 or :1 */
+void pfaddCommand(client *c) {
+    robj *o = lookupKeyWrite(c->db,c->argv[1]);
+    struct hllhdr *hdr;
+    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 = createHLLObject();
+        dbAdd(c->db,c->argv[1],o);
+        updated++;
+    } else {
+        if (isHLLObjectOrReply(c,o) != C_OK) return;
+        o = dbUnshareStringValue(c->db,c->argv[1],o);
+    }
+    /* Perform the low level ADD operation for every element. */
+    for (j = 2; j < c->argc; j++) {
+        int retval = hllAdd(o, (unsigned char*)c->argv[j]->ptr,
+                               sdslen(c->argv[j]->ptr));
+        switch(retval) {
+        case 1:
+            updated++;
+            break;
+        case -1:
+            addReplyError(c,invalid_hll_err);
+            return;
+        }
+    }
+    hdr = o->ptr;
+    if (updated) {
+        signalModifiedKey(c,c->db,c->argv[1]);
+        notifyKeyspaceEvent(NOTIFY_STRING,"pfadd",c->argv[1],c->db->id);
+        server.dirty += updated;
+        HLL_INVALIDATE_CACHE(hdr);
+    }
+    addReply(c, updated ? shared.cone : shared.czero);
+}
+
+/* PFCOUNT var -> approximated cardinality of set. */
+void pfcountCommand(client *c) {
+    robj *o;
+    struct hllhdr *hdr;
+    uint64_t card;
+
+    /* Case 1: multi-key keys, cardinality of the union.
+     *
+     * When multiple keys are specified, PFCOUNT actually computes
+     * the cardinality of the merge of the N HLLs specified. */
+    if (c->argc > 2) {
+        uint8_t max[HLL_HDR_SIZE+HLL_REGISTERS], *registers;
+        int j;
+
+        /* Compute an HLL with M[i] = MAX(M[i]_j). */
+        memset(max,0,sizeof(max));
+        hdr = (struct hllhdr*) max;
+        hdr->encoding = HLL_RAW; /* Special internal-only encoding. */
+        registers = max + HLL_HDR_SIZE;
+        for (j = 1; j < c->argc; j++) {
+            /* Check type and size. */
+            robj *o = lookupKeyRead(c->db,c->argv[j]);
+            if (o == NULL) continue; /* Assume empty HLL for non existing var.*/
+            if (isHLLObjectOrReply(c,o) != C_OK) return;
+
+            /* Merge with this HLL with our 'max' HLL by setting max[i]
+             * to MAX(max[i],hll[i]). */
+            if (hllMerge(registers,o) == C_ERR) {
+                addReplyError(c,invalid_hll_err);
+                return;
+            }
+        }
+
+        /* Compute cardinality of the resulting set. */
+        addReplyLongLong(c,hllCount(hdr,NULL));
+        return;
+    }
+
+    /* Case 2: cardinality of the single HLL.
+     *
+     * The user specified a single key. Either return the cached value
+     * or compute one and update the cache.
+     *
+     * Since a HLL is a regular Redis string type value, updating the cache does
+     * modify the value. We do a lookupKeyRead anyway since this is flagged as a
+     * read-only command. The difference is that with lookupKeyWrite, a
+     * logically expired key on a replica is deleted, while with lookupKeyRead
+     * it isn't, but the lookup returns NULL either way if the key is logically
+     * expired, which is what matters here. */
+    o = lookupKeyRead(c->db,c->argv[1]);
+    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 {
+        if (isHLLObjectOrReply(c,o) != C_OK) return;
+        o = dbUnshareStringValue(c->db,c->argv[1],o);
+
+        /* Check if the cached cardinality is valid. */
+        hdr = o->ptr;
+        if (HLL_VALID_CACHE(hdr)) {
+            /* Just return the cached value. */
+            card = (uint64_t)hdr->card[0];
+            card |= (uint64_t)hdr->card[1] << 8;
+            card |= (uint64_t)hdr->card[2] << 16;
+            card |= (uint64_t)hdr->card[3] << 24;
+            card |= (uint64_t)hdr->card[4] << 32;
+            card |= (uint64_t)hdr->card[5] << 40;
+            card |= (uint64_t)hdr->card[6] << 48;
+            card |= (uint64_t)hdr->card[7] << 56;
+        } else {
+            int invalid = 0;
+            /* Recompute it and update the cached value. */
+            card = hllCount(hdr,&invalid);
+            if (invalid) {
+                addReplyError(c,invalid_hll_err);
+                return;
+            }
+            hdr->card[0] = card & 0xff;
+            hdr->card[1] = (card >> 8) & 0xff;
+            hdr->card[2] = (card >> 16) & 0xff;
+            hdr->card[3] = (card >> 24) & 0xff;
+            hdr->card[4] = (card >> 32) & 0xff;
+            hdr->card[5] = (card >> 40) & 0xff;
+            hdr->card[6] = (card >> 48) & 0xff;
+            hdr->card[7] = (card >> 56) & 0xff;
+            /* This is considered a read-only command even if the cached value
+             * may be modified and given that the HLL is a Redis string
+             * we need to propagate the change. */
+            signalModifiedKey(c,c->db,c->argv[1]);
+            server.dirty++;
+        }
+        addReplyLongLong(c,card);
+    }
+}
+
+/* PFMERGE dest src1 src2 src3 ... srcN => OK */
+void pfmergeCommand(client *c) {
+    uint8_t max[HLL_REGISTERS];
+    struct hllhdr *hdr;
+    int j;
+    int use_dense = 0; /* Use dense representation as target? */
+
+    /* Compute an HLL with M[i] = MAX(M[i]_j).
+     * We store the maximum into the max array of registers. We'll write
+     * it to the target variable later. */
+    memset(max,0,sizeof(max));
+    for (j = 1; j < c->argc; j++) {
+        /* Check type and size. */
+        robj *o = lookupKeyRead(c->db,c->argv[j]);
+        if (o == NULL) continue; /* Assume empty HLL for non existing var. */
+        if (isHLLObjectOrReply(c,o) != C_OK) return;
+
+        /* If at least one involved HLL is dense, use the dense representation
+         * as target ASAP to save time and avoid the conversion step. */
+        hdr = o->ptr;
+        if (hdr->encoding == HLL_DENSE) use_dense = 1;
+
+        /* Merge with this HLL with our 'max' HLL by setting max[i]
+         * to MAX(max[i],hll[i]). */
+        if (hllMerge(max,o) == C_ERR) {
+            addReplyError(c,invalid_hll_err);
+            return;
+        }
+    }
+
+    /* Create / unshare the destination key's value if needed. */
+    robj *o = lookupKeyWrite(c->db,c->argv[1]);
+    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 = createHLLObject();
+        dbAdd(c->db,c->argv[1],o);
+    } else {
+        /* If key exists we are sure it's of the right type/size
+         * since we checked when merging the different HLLs, so we
+         * don't check again. */
+        o = dbUnshareStringValue(c->db,c->argv[1],o);
+    }
+
+    /* Convert the destination object to dense representation if at least
+     * one of the inputs was dense. */
+    if (use_dense && hllSparseToDense(o) == C_ERR) {
+        addReplyError(c,invalid_hll_err);
+        return;
+    }
+
+    /* Write the resulting HLL to the destination HLL registers and
+     * invalidate the cached value. */
+    for (j = 0; j < HLL_REGISTERS; j++) {
+        if (max[j] == 0) continue;
+        hdr = o->ptr;
+        switch(hdr->encoding) {
+        case HLL_DENSE: hllDenseSet(hdr->registers,j,max[j]); break;
+        case HLL_SPARSE: hllSparseSet(o,j,max[j]); break;
+        }
+    }
+    hdr = o->ptr; /* o->ptr may be different now, as a side effect of
+                     last hllSparseSet() call. */
+    HLL_INVALIDATE_CACHE(hdr);
+
+    signalModifiedKey(c,c->db,c->argv[1]);
+    /* We generate a PFADD event for PFMERGE for semantical simplicity
+     * since in theory this is a mass-add of elements. */
+    notifyKeyspaceEvent(NOTIFY_STRING,"pfadd",c->argv[1],c->db->id);
+    server.dirty++;
+    addReply(c,shared.ok);
+}
+
+/* ========================== Testing / Debugging  ========================== */
+
+/* PFSELFTEST
+ * This command performs a self-test of the HLL registers implementation.
+ * Something that is not easy to test from within the outside. */
+#define HLL_TEST_CYCLES 1000
+void pfselftestCommand(client *c) {
+    unsigned int j, i;
+    sds bitcounters = sdsnewlen(NULL,HLL_DENSE_SIZE);
+    struct hllhdr *hdr = (struct hllhdr*) bitcounters, *hdr2;
+    robj *o = NULL;
+    uint8_t bytecounters[HLL_REGISTERS];
+
+    /* Test 1: access registers.
+     * 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. */
+    for (j = 0; j < 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 < HLL_REGISTERS; i++) {
+            unsigned int r = rand() & HLL_REGISTER_MAX;
+
+            bytecounters[i] = r;
+            HLL_DENSE_SET_REGISTER(hdr->registers,i,r);
+        }
+        /* Check that we are able to retrieve the same values. */
+        for (i = 0; i < HLL_REGISTERS; i++) {
+            unsigned int val;
+
+            HLL_DENSE_GET_REGISTER(val,hdr->registers,i);
+            if (val != bytecounters[i]) {
+                addReplyErrorFormat(c,
+                    "TESTFAILED Register %d should be %d but is %d",
+                    i, (int) bytecounters[i], (int) val);
+                goto cleanup;
+            }
+        }
+    }
+
+    /* Test 2: approximation error.
+     * The test adds unique elements and check that the estimated value
+     * is always reasonable bounds.
+     *
+     * We check that the error is smaller than a few times than the expected
+     * standard error, to make it very unlikely for the test to fail because
+     * of a "bad" run.
+     *
+     * The test is performed with both dense and sparse HLLs at the same
+     * time also verifying that the computed cardinality is the same. */
+    memset(hdr->registers,0,HLL_DENSE_SIZE-HLL_HDR_SIZE);
+    o = createHLLObject();
+    double relerr = 1.04/sqrt(HLL_REGISTERS);
+    int64_t checkpoint = 1;
+    uint64_t seed = (uint64_t)rand() | (uint64_t)rand() << 32;
+    uint64_t ele;
+    for (j = 1; j <= 10000000; j++) {
+        ele = j ^ seed;
+        hllDenseAdd(hdr->registers,(unsigned char*)&ele,sizeof(ele));
+        hllAdd(o,(unsigned char*)&ele,sizeof(ele));
+
+        /* Make sure that for small cardinalities we use sparse
+         * encoding. */
+        if (j == checkpoint && j < server.hll_sparse_max_bytes/2) {
+            hdr2 = o->ptr;
+            if (hdr2->encoding != HLL_SPARSE) {
+                addReplyError(c, "TESTFAILED sparse encoding not used");
+                goto cleanup;
+            }
+        }
+
+        /* Check that dense and sparse representations agree. */
+        if (j == checkpoint && hllCount(hdr,NULL) != hllCount(o->ptr,NULL)) {
+                addReplyError(c, "TESTFAILED dense/sparse disagree");
+                goto cleanup;
+        }
+
+        /* Check error. */
+        if (j == checkpoint) {
+            int64_t abserr = checkpoint - (int64_t)hllCount(hdr,NULL);
+            uint64_t maxerr = ceil(relerr*6*checkpoint);
+
+            /* Adjust the max error we expect for cardinality 10
+             * since from time to time it is statistically likely to get
+             * much higher error due to collision, resulting into a false
+             * positive. */
+            if (j == 10) maxerr = 1;
+
+            if (abserr < 0) abserr = -abserr;
+            if (abserr > (int64_t)maxerr) {
+                addReplyErrorFormat(c,
+                    "TESTFAILED Too big error. card:%llu abserr:%llu",
+                    (unsigned long long) checkpoint,
+                    (unsigned long long) abserr);
+                goto cleanup;
+            }
+            checkpoint *= 10;
+        }
+    }
+
+    /* Success! */
+    addReply(c,shared.ok);
+
+cleanup:
+    sdsfree(bitcounters);
+    if (o) decrRefCount(o);
+}
+
+/* Different debugging related operations about the HLL implementation.
+ *
+ * PFDEBUG GETREG <key>
+ * PFDEBUG DECODE <key>
+ * PFDEBUG ENCODING <key>
+ * PFDEBUG TODENSE <key>
+ */
+void pfdebugCommand(client *c) {
+    char *cmd = c->argv[1]->ptr;
+    struct hllhdr *hdr;
+    robj *o;
+    int j;
+
+    o = lookupKeyWrite(c->db,c->argv[2]);
+    if (o == NULL) {
+        addReplyError(c,"The specified key does not exist");
+        return;
+    }
+    if (isHLLObjectOrReply(c,o) != C_OK) return;
+    o = dbUnshareStringValue(c->db,c->argv[2],o);
+    hdr = o->ptr;
+
+    /* PFDEBUG GETREG <key> */
+    if (!strcasecmp(cmd,"getreg")) {
+        if (c->argc != 3) goto arityerr;
+
+        if (hdr->encoding == HLL_SPARSE) {
+            if (hllSparseToDense(o) == C_ERR) {
+                addReplyError(c,invalid_hll_err);
+                return;
+            }
+            server.dirty++; /* Force propagation on encoding change. */
+        }
+
+        hdr = o->ptr;
+        addReplyArrayLen(c,HLL_REGISTERS);
+        for (j = 0; j < HLL_REGISTERS; j++) {
+            uint8_t val;
+
+            HLL_DENSE_GET_REGISTER(val,hdr->registers,j);
+            addReplyLongLong(c,val);
+        }
+    }
+    /* PFDEBUG DECODE <key> */
+    else if (!strcasecmp(cmd,"decode")) {
+        if (c->argc != 3) goto arityerr;
+
+        uint8_t *p = o->ptr, *end = p+sdslen(o->ptr);
+        sds decoded = sdsempty();
+
+        if (hdr->encoding != HLL_SPARSE) {
+            sdsfree(decoded);
+            addReplyError(c,"HLL encoding is not sparse");
+            return;
+        }
+
+        p += HLL_HDR_SIZE;
+        while(p < end) {
+            int runlen, regval;
+
+            if (HLL_SPARSE_IS_ZERO(p)) {
+                runlen = HLL_SPARSE_ZERO_LEN(p);
+                p++;
+                decoded = sdscatprintf(decoded,"z:%d ",runlen);
+            } else if (HLL_SPARSE_IS_XZERO(p)) {
+                runlen = HLL_SPARSE_XZERO_LEN(p);
+                p += 2;
+                decoded = sdscatprintf(decoded,"Z:%d ",runlen);
+            } else {
+                runlen = HLL_SPARSE_VAL_LEN(p);
+                regval = HLL_SPARSE_VAL_VALUE(p);
+                p++;
+                decoded = sdscatprintf(decoded,"v:%d,%d ",regval,runlen);
+            }
+        }
+        decoded = sdstrim(decoded," ");
+        addReplyBulkCBuffer(c,decoded,sdslen(decoded));
+        sdsfree(decoded);
+    }
+    /* PFDEBUG ENCODING <key> */
+    else if (!strcasecmp(cmd,"encoding")) {
+        char *encodingstr[2] = {"dense","sparse"};
+        if (c->argc != 3) goto arityerr;
+
+        addReplyStatus(c,encodingstr[hdr->encoding]);
+    }
+    /* PFDEBUG TODENSE <key> */
+    else if (!strcasecmp(cmd,"todense")) {
+        int conv = 0;
+        if (c->argc != 3) goto arityerr;
+
+        if (hdr->encoding == HLL_SPARSE) {
+            if (hllSparseToDense(o) == C_ERR) {
+                addReplyError(c,invalid_hll_err);
+                return;
+            }
+            conv = 1;
+            server.dirty++; /* Force propagation on encoding change. */
+        }
+        addReply(c,conv ? shared.cone : shared.czero);
+    } else {
+        addReplyErrorFormat(c,"Unknown PFDEBUG subcommand '%s'", cmd);
+    }
+    return;
+
+arityerr:
+    addReplyErrorFormat(c,
+        "Wrong number of arguments for the '%s' subcommand",cmd);
+}
+