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/* 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 "redis.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 don't
* limited to cardinalities up to 10^9, at the cost of just 1 additional
* bit per register.
* * The use of 16384 6-bit registers for a great level of accuracy, using
* a total of 12k per key.
* * The use of the Redis string data type. No new type is introduced.
* * No attempt is made to compress the data structure as in [1]. Also the
* algorithm used is the original HyperLogLog Algorithm as in [2], with
* the only difference that a 64 bit hash function is used, so no correction
* is performed for values near 2^32 as in [1].
*
* [1] Heule, Nunkesser, Hall: HyperLogLog in Practice: Algorithmic
* Engineering of a State of The Art Cardinality Estimation Algorithm.
*
* [2] P. Flajolet, Éric Fusy, O. Gandouet, and F. Meunier. Hyperloglog: The
* analysis of a near-optimal cardinality estimation algorithm.
*
* 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 HLL_SPARSE_MAX.
*/
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[0] |= (1<<7)
#define HLL_VALID_CACHE(hdr) (((hdr)->card[0] & (1<<7)) == 0)
#define HLL_P 14 /* The greater is P, the smaller the error. */
#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_MAX_ENCODING 1
#define HLL_SPARSE_MAX 3000
/* =========================== 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 affacted after the OR.
*
* -------------------------------------------------------------------------
*
* Setting the register is a bit more complex, let's assume that 'val'
* is the value we want to set, already in the right range.
*
* We need two steps, in one we need to clear the bits, and in the other
* we need to bitwise-OR the new bits.
*
* Let's try with 'pos' = 1, so our first byte at 'b' is 0,
*
* "fb" is 6 in this case.
*
* +--------+
* |11000000| <- Our byte at b0
* +--------+
*
* To create a AND-mask to clear the bits about this position, we just
* initialize the mask with the value 63, left shift it of "fs" bits,
* and finally invert the result.
*
* +--------+
* |00111111| <- "mask" starts at 63
* |11000000| <- "mask" after left shift of "ls" bits.
* |00111111| <- "mask" after invert.
* +--------+
*
* Now we can bitwise-AND the byte at "b" with the mask, and bitwise-OR
* it with "val" left-shifted of "ls" bits to set the new bits.
*
* Now let's focus on the next byte b1:
*
* +--------+
* |22221111| <- Initial value of b1
* +--------+
*
* To build the AND mask we start again with the 63 value, right shift
* it by 8-fb bits, and invert it.
*
* +--------+
* |00111111| <- "mask" set at 2&6-1
* |00001111| <- "mask" after the right shift by 8-fb = 2 bits
* |11110000| <- "mask" after bitwise not.
* +--------+
*
* Now we can mask it with b+1 to clear the old bits, and bitwise-OR
* with "val" left-shifted by "rs" bits to set the new value.
*/
/* Note: if we access the last counter, we will also access the b+1 byte
* that is out of the array, but sds strings always have an implicit null
* term, so the byte exists, and we can skip the conditional (or the need
* to allocate 1 byte more explicitly). */
/* Store the value of the register at position 'regnum' into variable 'target'.
* 'p' is an array of unsigned bytes. */
#define HLL_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)
/* ========================= HyperLogLog algorithm ========================= */
/* Our hash function is MurmurHash2, 64 bit version.
* It was modified for Redis in order to provide the same result in
* big and little endian archs (endian neutral). */
uint64_t MurmurHash64A (const void * key, int len, unsigned int seed) {
const uint64_t m = 0xc6a4a7935bd1e995;
const int r = 47;
uint64_t h = seed ^ (len * m);
const uint8_t *data = (const uint8_t *)key;
const uint8_t *end = data + (len-(len&7));
while(data != end) {
uint64_t k;
#if (BYTE_ORDER == LITTLE_ENDIAN)
k = *((uint64_t*)data);
#else
k = (uint64_t) data[0];
k |= (uint64_t) data[1] << 8;
k |= (uint64_t) data[2] << 16;
k |= (uint64_t) data[3] << 24;
k |= (uint64_t) data[4] << 32;
k |= (uint64_t) data[5] << 40;
k |= (uint64_t) data[6] << 48;
k |= (uint64_t) data[7] << 56;
#endif
k *= m;
k ^= k >> r;
k *= m;
h ^= k;
h *= m;
data += 8;
}
switch(len & 7) {
case 7: h ^= (uint64_t)data[6] << 48;
case 6: h ^= (uint64_t)data[5] << 40;
case 5: h ^= (uint64_t)data[4] << 32;
case 4: h ^= (uint64_t)data[3] << 24;
case 3: h ^= (uint64_t)data[2] << 16;
case 2: h ^= (uint64_t)data[1] << 8;
case 1: h ^= (uint64_t)data[0];
h *= m;
};
h ^= h >> r;
h *= m;
h ^= h >> r;
return h;
}
/* 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, int *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 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 |= ((uint64_t)1<<63); /* Make sure the loop terminates. */
bit = HLL_REGISTERS; /* First bit not used to address the register. */
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 ================== */
/* "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.
*
* '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 hllDenseAdd(uint8_t *registers, unsigned char *ele, size_t elesize) {
uint8_t oldcount, count;
int index;
/* Update the register if this element produced a longer run of zeroes. */
count = hllPatLen(ele,elesize,&index);
HLL_DENSE_GET_REGISTER(oldcount,registers,index);
if (count > oldcount) {
HLL_DENSE_SET_REGISTER(registers,index,count);
return 1;
} else {
return 0;
}
}
/* Compute SUM(2^-reg) in the dense representation.
* PE is an array with a pre-computer table of values 2^-reg indexed by reg.
* As a side effect the integer pointed by 'ezp' is set to the number
* of zero registers. */
double hllDenseSum(uint8_t *registers, double *PE, int *ezp) {
double E = 0;
int j, ez = 0;
/* 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; if (r0 == 0) ez++;
r1 = (r[0] >> 6 | r[1] << 2) & 63; if (r1 == 0) ez++;
r2 = (r[1] >> 4 | r[2] << 4) & 63; if (r2 == 0) ez++;
r3 = (r[2] >> 2) & 63; if (r3 == 0) ez++;
r4 = r[3] & 63; if (r4 == 0) ez++;
r5 = (r[3] >> 6 | r[4] << 2) & 63; if (r5 == 0) ez++;
r6 = (r[4] >> 4 | r[5] << 4) & 63; if (r6 == 0) ez++;
r7 = (r[5] >> 2) & 63; if (r7 == 0) ez++;
r8 = r[6] & 63; if (r8 == 0) ez++;
r9 = (r[6] >> 6 | r[7] << 2) & 63; if (r9 == 0) ez++;
r10 = (r[7] >> 4 | r[8] << 4) & 63; if (r10 == 0) ez++;
r11 = (r[8] >> 2) & 63; if (r11 == 0) ez++;
r12 = r[9] & 63; if (r12 == 0) ez++;
r13 = (r[9] >> 6 | r[10] << 2) & 63; if (r13 == 0) ez++;
r14 = (r[10] >> 4 | r[11] << 4) & 63; if (r14 == 0) ez++;
r15 = (r[11] >> 2) & 63; if (r15 == 0) ez++;
/* Additional parens will allow the compiler to optimize the
* code more with a loss of precision that is not very relevant
* here (floating point math is not commutative!). */
E += (PE[r0] + PE[r1]) + (PE[r2] + PE[r3]) + (PE[r4] + PE[r5]) +
(PE[r6] + PE[r7]) + (PE[r8] + PE[r9]) + (PE[r10] + PE[r11]) +
(PE[r12] + PE[r13]) + (PE[r14] + PE[r15]);
r += 12;
}
} else {
for (j = 0; j < HLL_REGISTERS; j++) {
unsigned long reg;
HLL_DENSE_GET_REGISTER(reg,registers,j);
if (reg == 0) {
ez++;
E += 1; /* 2^(-reg[j]) is 1 when m is 0. */
} else {
E += PE[reg]; /* Precomputed 2^(-reg[j]). */
}
}
}
*ezp = ez;
return E;
}
/* ================== 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. */
sds hllSparseToDense(sds sparse) {
sds dense;
struct hllhdr *hdr, *oldhdr = (struct hllhdr*)sparse;
int idx = 0, runlen, regval;
uint8_t *p = (uint8_t*)sparse, *end = p+sdslen(sparse);
/* 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);
while(runlen--) {
HLL_DENSE_SET_REGISTER(hdr->registers,idx,regval);
idx++;
}
p++;
}
}
/* Free the old representation and return the new one. */
sdsfree(sparse);
return dense;
}
/* "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.
*
* 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.
*
* 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 HLL_SPARSE_MAX. */
int hllSparseAdd(robj *o, unsigned char *ele, size_t elesize) {
struct hllhdr *hdr;
uint8_t oldcount, count, *sparse, *end, *p, *prev, *next;
int index, first, span;
int is_zero = 0, is_xzero = 0, is_val = 0, runlen = 0;
/* Update the register if this element produced a longer run of zeroes. */
count = hllPatLen(ele,elesize,&index);
/* 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 previos opcode at the end of the loop. */
next = NULL; /* Points to the next opcode at the end of the loop. */
while(p < end) {
/* Set span to the number of registers covered by this opcode. */
if (HLL_SPARSE_IS_ZERO(p)) span = HLL_SPARSE_ZERO_LEN(p);
else if (HLL_SPARSE_IS_XZERO(p)) span = HLL_SPARSE_XZERO_LEN(p);
else span = HLL_SPARSE_VAL_LEN(p);
/* Break if this opcode covers the register as 'index'. */
if (first+span >= index) break;
prev = p;
p += (HLL_SPARSE_IS_XZERO(p)) ? 2 : 1;
first += span;
}
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 muliple
* opcodes. The worst case is an XZERO split in the middle resuling 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 = seq-n;
int oldlen = is_xzero ? 2 : 1;
int deltalen = seqlen-oldlen;
if (deltalen > 0 && sdslen(o->ptr) > HLL_SPARSE_MAX) goto promote;
if (deltalen && next) {
memmove(next+deltalen,next,next-sparse);
sdsIncrLen(o->ptr,deltalen);
}
memcpy(p,seq,seqlen);
updated:
/* Step 4: Merge adjacent values if possible.
*
* The representation was updated, however the resulting representation
* may not be optimal: adjacent opcodes may be merged into a single one.
* We start from the opcode before the one we updated trying to merge
* opcodes up to the next 5 opcodes (since we need to consider the three
* opcodes resuling from the worst-case split of the updated opcode,
* plus the two opcodes at the left and right of the original one). */
hdr = o->ptr;
HLL_INVALIDATE_CACHE(hdr);
return 1;
promote: /* Promote to dense representation. */
o->ptr = hllSparseToDense(o->ptr);
hdr = o->ptr;
return hllDenseAdd(hdr->registers, ele, elesize);
}
/* Compute SUM(2^-reg) in the sparse representation.
* PE is an array with a pre-computer table of values 2^-reg indexed by reg.
* As a side effect the integer pointed by 'ezp' is set to the number
* of zero registers. */
double hllSparseSum(uint8_t *sparse, int sparselen, double *PE, int *ezp) {
double E = 0;
int ez = 0, 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;
ez += runlen;
E += 1*runlen; /* 2^(-reg[j]) is 1 when m is 0. */
p++;
} else if (HLL_SPARSE_IS_XZERO(p)) {
runlen = HLL_SPARSE_XZERO_LEN(p);
idx += runlen;
ez += runlen;
E += 1*runlen; /* 2^(-reg[j]) is 1 when m is 0. */
p += 2;
} else {
runlen = HLL_SPARSE_VAL_LEN(p);
regval = HLL_SPARSE_VAL_VALUE(p);
idx += runlen;
E += PE[regval]*runlen;
p++;
}
}
redisAssert(idx == HLL_REGISTERS);
*ezp = ez;
return E;
}
/* ========================= HyperLogLog Count ==============================
* This is the core of the algorithm where the approximated count is computed.
* The function uses the lower level hllDenseSum() and hllSparseSum() functions
* as helpers to compute the SUM(2^-reg) part of the computation, which is
* representation-specific, while all the rest is common. */
/* Return the approximated cardinality of the set based on the armonic
* mean of the registers values. 'hdr' points to the start of the SDS
* representing the String object holding the HLL representation. */
uint64_t hllCount(struct hllhdr *hdr) {
double m = HLL_REGISTERS;
double alpha = 0.7213/(1+1.079/m);
double E;
int ez; /* Number of registers equal to 0. */
int j;
/* We precompute 2^(-reg[j]) in a small table in order to
* speedup the computation of SUM(2^-register[0..i]). */
static int initialized = 0;
static double PE[64];
if (!initialized) {
PE[0] = 1; /* 2^(-reg[j]) is 1 when m is 0. */
for (j = 1; j < 64; j++) {
/* 2^(-reg[j]) is the same as 1/2^reg[j]. */
PE[j] = 1.0/(1ULL << j);
}
initialized = 1;
}
/* Compute SUM(2^-register[0..i]). */
if (hdr->encoding == HLL_DENSE) {
E = hllDenseSum(hdr->registers,PE,&ez);
} else {
E = hllSparseSum(hdr->registers,sdslen((sds)hdr)-HLL_HDR_SIZE,PE,&ez);
}
/* Muliply the inverse of E for alpha_m * m^2 to have the raw estimate. */
E = (1/E)*alpha*m*m;
/* Use the LINEARCOUNTING algorithm for small cardinalities.
* For larger values but up to 72000 HyperLogLog raw approximation is
* used since linear counting error starts to increase. However HyperLogLog
* shows a strong bias in the range 2.5*16384 - 72000, so we try to
* compensate for it. */
if (E < m*2.5 && ez != 0) {
E = m*log(m/ez); /* LINEARCOUNTING() */
} else if (m == 16384 && E < 72000) {
/* We did polynomial regression of the bias for this range, this
* way we can compute the bias for a given cardinality and correct
* according to it. Only apply the correction for P=14 that's what
* we use and the value the correction was verified with. */
double bias = 5.9119*1.0e-18*(E*E*E*E)
-1.4253*1.0e-12*(E*E*E)+
1.2940*1.0e-7*(E*E)
-5.2921*1.0e-3*E+
83.3216;
E -= E*(bias/100);
}
/* We don't apply the correction for E > 1/30 of 2^32 since we use
* a 64 bit function and 6 bit counters. To apply the correction for
* 1/30 of 2^64 is not needed since it would require a huge set
* to approach such a value. */
return (uint64_t) E;
}
/* ========================== HyperLogLog commands ========================== */
/* 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;
}
redisAssert((p-(uint8_t*)s) == sparselen);
/* Create the actual object. */
o = createObject(REDIS_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 REDIS_OK if this is true, otherwise reply to the client
* with an error and return REDIS_ERR. */
int isHLLObjectOrReply(redisClient *c, robj *o) {
struct hllhdr *hdr;
/* Key exists, check type */
if (checkType(c,o,REDIS_STRING))
return REDIS_ERR; /* Error already sent. */
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 REDIS_OK;
invalid:
addReplySds(c,
sdsnew("-WRONGTYPE Key is not a valid "
"HyperLogLog string value.\r\n"));
return REDIS_ERR;
}
/* PFADD var ele ele ele ... ele => :0 or :1 */
void pfaddCommand(redisClient *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) != REDIS_OK) return;
o = dbUnshareStringValue(c->db,c->argv[1],o);
}
/* Perform the low level ADD operation for every element. */
hdr = o->ptr;
for (j = 2; j < c->argc; j++) {
if (hllDenseAdd(hdr->registers, (unsigned char*)c->argv[j]->ptr,
sdslen(c->argv[j]->ptr)))
{
updated++;
}
}
if (updated) {
signalModifiedKey(c->db,c->argv[1]);
notifyKeyspaceEvent(REDIS_NOTIFY_STRING,"pfadd",c->argv[1],c->db->id);
server.dirty++;
HLL_INVALIDATE_CACHE(hdr);
}
addReply(c, updated ? shared.cone : shared.czero);
}
/* PFCOUNT var -> approximated cardinality of set. */
void pfcountCommand(redisClient *c) {
robj *o = lookupKeyRead(c->db,c->argv[1]);
struct hllhdr *hdr;
uint64_t card;
if (o == NULL) {
/* No key? Cardinality is zero since no element was added, otherwise
* we would have a key as HLLADD creates it as a side effect. */
addReply(c,shared.czero);
} else {
if (isHLLObjectOrReply(c,o) != REDIS_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 {
/* Recompute it and update the cached value. */
card = hllCount(hdr);
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 not considered a read-only command even if the
* data structure is not modified, since the cached value
* may be modified and given that the HLL is a Redis string
* we need to propagate the change. */
signalModifiedKey(c->db,c->argv[1]);
server.dirty++;
}
addReplyLongLong(c,card);
}
}
/* PFMERGE dest src1 src2 src3 ... srcN => OK */
void pfmergeCommand(redisClient *c) {
uint8_t max[HLL_REGISTERS];
struct hllhdr *hdr;
int j, i;
/* Compute an HLL with M[i] = MAX(M[i]_j).
* We we 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++) {
uint8_t val;
/* 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) != REDIS_OK) return;
/* Merge with this HLL with our 'max' HHL by setting max[i]
* to MAX(max[i],hll[i]). */
hdr = o->ptr;
for (i = 0; i < HLL_REGISTERS; i++) {
HLL_DENSE_GET_REGISTER(val,hdr->registers,i);
if (val > max[i]) max[i] = val;
}
}
/* Create / unshare the destination key's value if needed. */
robj *o = lookupKeyRead(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);
}
/* Write the resulting HLL to the destination HLL registers and
* invalidate the cached value. */
hdr = o->ptr;
for (j = 0; j < HLL_REGISTERS; j++) {
HLL_DENSE_SET_REGISTER(hdr->registers,j,max[j]);
}
HLL_INVALIDATE_CACHE(hdr);
signalModifiedKey(c->db,c->argv[1]);
/* We generate an HLLADD event for HLLMERGE for semantical simplicity
* since in theory this is a mass-add of elements. */
notifyKeyspaceEvent(REDIS_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(redisClient *c) {
int j, i;
sds bitcounters = sdsnewlen(NULL,HLL_DENSE_SIZE);
struct hllhdr *hdr = (struct hllhdr*) bitcounters;
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 is adds unique elements and check that the estimated value
* is always reasonable bounds.
*
* We check that the error is smaller than 4 times than the expected
* standard error, to make it very unlikely for the test to fail because
* of a "bad" run. */
memset(hdr->registers,0,HLL_DENSE_SIZE-HLL_HDR_SIZE);
double relerr = 1.04/sqrt(HLL_REGISTERS);
int64_t checkpoint = 1000;
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));
if (j == checkpoint) {
int64_t abserr = checkpoint- (int64_t)hllCount(hdr);
if (abserr < 0) abserr = -abserr;
if (abserr > (uint64_t)(relerr*4*checkpoint)) {
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);
}
/* PFGETREG
* Return the registers values of the specified HLL. */
void pfgetregCommand(redisClient *c) {
robj *o = lookupKeyRead(c->db,c->argv[1]);
struct hllhdr *hdr;
int j;
if (o == NULL) {
addReplyError(c,"The specified key does not exist");
return;
} else {
if (isHLLObjectOrReply(c,o) != REDIS_OK) return;
hdr = o->ptr;
addReplyMultiBulkLen(c,HLL_REGISTERS);
for (j = 0; j < HLL_REGISTERS; j++) {
uint8_t val;
HLL_DENSE_GET_REGISTER(val,hdr->registers,j);
addReplyLongLong(c,val);
}
}
}
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