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https://github.com/fluencelabs/redis
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improved HyperLogLog cardinality estimation
based on method described in https://arxiv.org/abs/1702.01284 that does not rely on any magic constants
This commit is contained in:
Otmar Ertl
parent
6470b21f59
commit
1e9a774871
@ -192,6 +192,7 @@ struct hllhdr {
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#define HLL_VALID_CACHE(hdr) (((hdr)->card[7] & (1<<7)) == 0)
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#define HLL_P 14 /* The greater is P, the smaller the error. */
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#define HLL_Q (63-HLL_P)
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#define HLL_REGISTERS (1<<HLL_P) /* With P=14, 16384 registers. */
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#define HLL_P_MASK (HLL_REGISTERS-1) /* Mask to index register. */
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#define HLL_BITS 6 /* Enough to count up to 63 leading zeroes. */
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@ -510,13 +511,9 @@ int hllDenseAdd(uint8_t *registers, unsigned char *ele, size_t elesize) {
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return hllDenseSet(registers,index,count);
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}
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/* Compute SUM(2^-reg) in the dense representation.
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* PE is an array with a pre-computer table of values 2^-reg indexed by reg.
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* As a side effect the integer pointed by 'ezp' is set to the number
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* of zero registers. */
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double hllDenseSum(uint8_t *registers, double *PE, int *ezp) {
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double E = 0;
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int j, ez = 0;
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/* Compute the register histogram in the dense representation. */
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void hllDenseRegHisto(uint8_t *registers, int* regHisto) {
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int j;
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/* Redis default is to use 16384 registers 6 bits each. The code works
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* with other values by modifying the defines, but for our target value
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@ -527,47 +524,49 @@ double hllDenseSum(uint8_t *registers, double *PE, int *ezp) {
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r10, r11, r12, r13, r14, r15;
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for (j = 0; j < 1024; j++) {
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/* Handle 16 registers per iteration. */
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r0 = r[0] & 63; if (r0 == 0) ez++;
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r1 = (r[0] >> 6 | r[1] << 2) & 63; if (r1 == 0) ez++;
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r2 = (r[1] >> 4 | r[2] << 4) & 63; if (r2 == 0) ez++;
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r3 = (r[2] >> 2) & 63; if (r3 == 0) ez++;
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r4 = r[3] & 63; if (r4 == 0) ez++;
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r5 = (r[3] >> 6 | r[4] << 2) & 63; if (r5 == 0) ez++;
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r6 = (r[4] >> 4 | r[5] << 4) & 63; if (r6 == 0) ez++;
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r7 = (r[5] >> 2) & 63; if (r7 == 0) ez++;
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r8 = r[6] & 63; if (r8 == 0) ez++;
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r9 = (r[6] >> 6 | r[7] << 2) & 63; if (r9 == 0) ez++;
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r10 = (r[7] >> 4 | r[8] << 4) & 63; if (r10 == 0) ez++;
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r11 = (r[8] >> 2) & 63; if (r11 == 0) ez++;
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r12 = r[9] & 63; if (r12 == 0) ez++;
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r13 = (r[9] >> 6 | r[10] << 2) & 63; if (r13 == 0) ez++;
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r14 = (r[10] >> 4 | r[11] << 4) & 63; if (r14 == 0) ez++;
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r15 = (r[11] >> 2) & 63; if (r15 == 0) ez++;
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r0 = r[0] & 63;
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r1 = (r[0] >> 6 | r[1] << 2) & 63;
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r2 = (r[1] >> 4 | r[2] << 4) & 63;
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r3 = (r[2] >> 2) & 63;
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r4 = r[3] & 63;
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r5 = (r[3] >> 6 | r[4] << 2) & 63;
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r6 = (r[4] >> 4 | r[5] << 4) & 63;
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r7 = (r[5] >> 2) & 63;
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r8 = r[6] & 63;
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r9 = (r[6] >> 6 | r[7] << 2) & 63;
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r10 = (r[7] >> 4 | r[8] << 4) & 63;
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r11 = (r[8] >> 2) & 63;
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r12 = r[9] & 63;
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r13 = (r[9] >> 6 | r[10] << 2) & 63;
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r14 = (r[10] >> 4 | r[11] << 4) & 63;
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r15 = (r[11] >> 2) & 63;
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regHisto[r0] += 1;
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regHisto[r1] += 1;
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regHisto[r2] += 1;
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regHisto[r3] += 1;
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regHisto[r4] += 1;
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regHisto[r5] += 1;
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regHisto[r6] += 1;
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regHisto[r7] += 1;
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regHisto[r8] += 1;
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regHisto[r9] += 1;
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regHisto[r10] += 1;
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regHisto[r11] += 1;
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regHisto[r12] += 1;
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regHisto[r13] += 1;
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regHisto[r14] += 1;
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regHisto[r15] += 1;
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/* Additional parens will allow the compiler to optimize the
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* code more with a loss of precision that is not very relevant
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* here (floating point math is not commutative!). */
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E += (PE[r0] + PE[r1]) + (PE[r2] + PE[r3]) + (PE[r4] + PE[r5]) +
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(PE[r6] + PE[r7]) + (PE[r8] + PE[r9]) + (PE[r10] + PE[r11]) +
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(PE[r12] + PE[r13]) + (PE[r14] + PE[r15]);
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r += 12;
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}
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} else {
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for (j = 0; j < HLL_REGISTERS; j++) {
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for(j = 0; j < HLL_REGISTERS; j++) {
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unsigned long reg;
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HLL_DENSE_GET_REGISTER(reg,registers,j);
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if (reg == 0) {
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ez++;
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/* Increment E at the end of the loop. */
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} else {
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E += PE[reg]; /* Precomputed 2^(-reg[j]). */
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}
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regHisto[reg] += 1;
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}
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E += ez; /* Add 2^0 'ez' times. */
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}
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*ezp = ez;
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return E;
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}
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/* ================== Sparse representation implementation ================= */
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@ -903,76 +902,96 @@ int hllSparseAdd(robj *o, unsigned char *ele, size_t elesize) {
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return hllSparseSet(o,index,count);
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}
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/* Compute SUM(2^-reg) in the sparse representation.
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* PE is an array with a pre-computer table of values 2^-reg indexed by reg.
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* As a side effect the integer pointed by 'ezp' is set to the number
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* of zero registers. */
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double hllSparseSum(uint8_t *sparse, int sparselen, double *PE, int *ezp, int *invalid) {
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double E = 0;
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int ez = 0, idx = 0, runlen, regval;
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/* Compute the register histogram in the sparse representation. */
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void hllSparseRegHisto(uint8_t *sparse, int sparselen, int *invalid, int* regHisto) {
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int idx = 0, runlen, regval;
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uint8_t *end = sparse+sparselen, *p = sparse;
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while(p < end) {
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if (HLL_SPARSE_IS_ZERO(p)) {
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runlen = HLL_SPARSE_ZERO_LEN(p);
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idx += runlen;
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ez += runlen;
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/* Increment E at the end of the loop. */
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regHisto[0] += runlen;
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p++;
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} else if (HLL_SPARSE_IS_XZERO(p)) {
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runlen = HLL_SPARSE_XZERO_LEN(p);
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idx += runlen;
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ez += runlen;
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/* Increment E at the end of the loop. */
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regHisto[0] += runlen;
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p += 2;
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} else {
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runlen = HLL_SPARSE_VAL_LEN(p);
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regval = HLL_SPARSE_VAL_VALUE(p);
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idx += runlen;
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E += PE[regval]*runlen;
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regHisto[regval] += runlen;
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p++;
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}
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}
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if (idx != HLL_REGISTERS && invalid) *invalid = 1;
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E += ez; /* Add 2^0 'ez' times. */
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*ezp = ez;
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return E;
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}
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/* ========================= HyperLogLog Count ==============================
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* This is the core of the algorithm where the approximated count is computed.
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* The function uses the lower level hllDenseSum() and hllSparseSum() functions
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* as helpers to compute the SUM(2^-reg) part of the computation, which is
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* representation-specific, while all the rest is common. */
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* The function uses the lower level hllDenseRegHisto() and hllSparseRegHisto()
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* functions as helpers to compute histogram of register values part of the
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* computation, which is representation-specific, while all the rest is common. */
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/* Implements the SUM operation for uint8_t data type which is only used
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* internally as speedup for PFCOUNT with multiple keys. */
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double hllRawSum(uint8_t *registers, double *PE, int *ezp) {
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double E = 0;
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int j, ez = 0;
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/* Implements the register histogram calculation for uint8_t data type
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* which is only used internally as speedup for PFCOUNT with multiple keys. */
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void hllRawRegHisto(uint8_t *registers, int* regHisto) {
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uint64_t *word = (uint64_t*) registers;
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uint8_t *bytes;
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int j;
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for (j = 0; j < HLL_REGISTERS/8; j++) {
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if (*word == 0) {
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ez += 8;
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regHisto[0] += 8;
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} else {
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bytes = (uint8_t*) word;
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if (bytes[0]) E += PE[bytes[0]]; else ez++;
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if (bytes[1]) E += PE[bytes[1]]; else ez++;
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if (bytes[2]) E += PE[bytes[2]]; else ez++;
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if (bytes[3]) E += PE[bytes[3]]; else ez++;
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if (bytes[4]) E += PE[bytes[4]]; else ez++;
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if (bytes[5]) E += PE[bytes[5]]; else ez++;
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if (bytes[6]) E += PE[bytes[6]]; else ez++;
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if (bytes[7]) E += PE[bytes[7]]; else ez++;
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regHisto[bytes[0]] += 1;
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regHisto[bytes[1]] += 1;
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regHisto[bytes[2]] += 1;
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regHisto[bytes[3]] += 1;
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regHisto[bytes[4]] += 1;
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regHisto[bytes[5]] += 1;
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regHisto[bytes[6]] += 1;
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regHisto[bytes[7]] += 1;
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}
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word++;
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}
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E += ez; /* 2^(-reg[j]) is 1 when m is 0, add it 'ez' times for every
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zero register in the HLL. */
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*ezp = ez;
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return E;
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}
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/* Helper function sigma as defined in
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* "New cardinality estimation algorithms for HyperLogLog sketches"
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* Otmar Ertl, arXiv:1702.01284 */
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double hllSigma(double x) {
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if (x == 1.) return INFINITY;
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double zPrime;
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double y = 1;
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double z = x;
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do {
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x *= x;
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zPrime = z;
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z += x * y;
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y += y;
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} while(zPrime != z);
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return z;
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}
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/* Helper function tau as defined in
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* "New cardinality estimation algorithms for HyperLogLog sketches"
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* Otmar Ertl, arXiv:1702.01284 */
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double hllTau(double x) {
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if (x == 0. || x == 1.) return 0.;
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double zPrime;
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double y = 1.0;
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double z = 1 - x;
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do {
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x = sqrt(x);
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zPrime = z;
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y *= 0.5;
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z -= pow(1 - x, 2)*y;
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} while(zPrime != z);
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return z / 3;
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}
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/* Return the approximated cardinality of the set based on the harmonic
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@ -988,49 +1007,34 @@ double hllRawSum(uint8_t *registers, double *PE, int *ezp) {
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* keys (no need to work with 6-bit integers encoding). */
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uint64_t hllCount(struct hllhdr *hdr, int *invalid) {
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double m = HLL_REGISTERS;
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double E, alpha = 0.7213/(1+1.079/m);
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int j, ez; /* Number of registers equal to 0. */
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double E;
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int j;
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double alphaInf = 0.5 / log(2.);
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int regHisto[HLL_Q+2] = {0};
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/* We precompute 2^(-reg[j]) in a small table in order to
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* speedup the computation of SUM(2^-register[0..i]). */
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static int initialized = 0;
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static double PE[64];
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if (!initialized) {
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PE[0] = 1; /* 2^(-reg[j]) is 1 when m is 0. */
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for (j = 1; j < 64; j++) {
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/* 2^(-reg[j]) is the same as 1/2^reg[j]. */
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PE[j] = 1.0/(1ULL << j);
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}
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initialized = 1;
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}
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/* Compute SUM(2^-register[0..i]). */
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/* Compute register histogram */
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if (hdr->encoding == HLL_DENSE) {
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E = hllDenseSum(hdr->registers,PE,&ez);
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hllDenseRegHisto(hdr->registers,regHisto);
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} else if (hdr->encoding == HLL_SPARSE) {
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E = hllSparseSum(hdr->registers,
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sdslen((sds)hdr)-HLL_HDR_SIZE,PE,&ez,invalid);
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hllSparseRegHisto(hdr->registers,
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sdslen((sds)hdr)-HLL_HDR_SIZE,invalid,regHisto);
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} else if (hdr->encoding == HLL_RAW) {
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E = hllRawSum(hdr->registers,PE,&ez);
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hllRawRegHisto(hdr->registers,regHisto);
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} else {
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serverPanic("Unknown HyperLogLog encoding in hllCount()");
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}
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/* Apply loglog-beta to the raw estimate. See:
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* "LogLog-Beta and More: A New Algorithm for Cardinality Estimation
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* Based on LogLog Counting" Jason Qin, Denys Kim, Yumei Tung
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* arXiv:1612.02284 */
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double zl = log(ez + 1);
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double beta = -0.370393911*ez +
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0.070471823*zl +
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0.17393686*pow(zl,2) +
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0.16339839*pow(zl,3) +
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-0.09237745*pow(zl,4) +
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0.03738027*pow(zl,5) +
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-0.005384159*pow(zl,6) +
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0.00042419*pow(zl,7);
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/* Estimate cardinality form register histogram. See:
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* "New cardinality estimation algorithms for HyperLogLog sketches"
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* Otmar Ertl, arXiv:1702.01284 */
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double z = m * hllTau((m-regHisto[HLL_Q+1])/(double)m);
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for (j = HLL_Q; j >= 1; --j) {
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z += regHisto[j];
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z *= 0.5;
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}
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z += m * hllSigma(regHisto[0]/(double)m);
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E = llroundl(alphaInf*m*m/z);
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E = llroundl(alpha*m*(m-ez)*(1/(E+beta)));
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return (uint64_t) E;
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}
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