From 1e9a7748716e1cd234893dd858d07ffa77920e41 Mon Sep 17 00:00:00 2001
From: Otmar Ertl <otmar.ertl@gmail.com>
Date: Sat, 10 Mar 2018 20:13:21 +0100
Subject: [PATCH] improved HyperLogLog cardinality estimation

based on method described in https://arxiv.org/abs/1702.01284
that does not rely on any magic constants
---
 src/hyperloglog.c | 230 +++++++++++++++++++++++-----------------------
 1 file changed, 117 insertions(+), 113 deletions(-)

diff --git a/src/hyperloglog.c b/src/hyperloglog.c
index 8ab9d2a30c0..7f5f62445c9 100644
--- a/src/hyperloglog.c
+++ b/src/hyperloglog.c
@@ -192,6 +192,7 @@ struct hllhdr {
 #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 (63-HLL_P)
 #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. */
@@ -510,13 +511,9 @@ int hllDenseAdd(uint8_t *registers, unsigned char *ele, size_t elesize) {
     return hllDenseSet(registers,index,count);
 }
 
-/* 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;
+/* 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
@@ -527,47 +524,49 @@ double hllDenseSum(uint8_t *registers, double *PE, int *ezp) {
                       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]);
+            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] += 1;
+            regHisto[r1] += 1;
+            regHisto[r2] += 1;
+            regHisto[r3] += 1;
+            regHisto[r4] += 1;
+            regHisto[r5] += 1;
+            regHisto[r6] += 1;
+            regHisto[r7] += 1;
+            regHisto[r8] += 1;
+            regHisto[r9] += 1;
+            regHisto[r10] += 1;
+            regHisto[r11] += 1;
+            regHisto[r12] += 1;
+            regHisto[r13] += 1;
+            regHisto[r14] += 1;
+            regHisto[r15] += 1;
+
             r += 12;
         }
     } else {
-        for (j = 0; j < HLL_REGISTERS; j++) {
+        for(j = 0; j < HLL_REGISTERS; j++) {
             unsigned long reg;
-
             HLL_DENSE_GET_REGISTER(reg,registers,j);
-            if (reg == 0) {
-                ez++;
-                /* Increment E at the end of the loop. */
-            } else {
-                E += PE[reg]; /* Precomputed 2^(-reg[j]). */
-            }
+            regHisto[reg] += 1;
         }
-        E += ez; /* Add 2^0 'ez' times. */
     }
-    *ezp = ez;
-    return E;
 }
 
 /* ================== Sparse representation implementation  ================= */
@@ -903,76 +902,96 @@ int hllSparseAdd(robj *o, unsigned char *ele, size_t elesize) {
     return hllSparseSet(o,index,count);
 }
 
-/* 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, int *invalid) {
-    double E = 0;
-    int ez = 0, idx = 0, runlen, regval;
+/* 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;
-            ez += runlen;
-            /* Increment E at the end of the loop. */
+            regHisto[0] += runlen;
             p++;
         } else if (HLL_SPARSE_IS_XZERO(p)) {
             runlen = HLL_SPARSE_XZERO_LEN(p);
             idx += runlen;
-            ez += runlen;
-            /* Increment E at the end of the loop. */
+            regHisto[0] += runlen;
             p += 2;
         } else {
             runlen = HLL_SPARSE_VAL_LEN(p);
             regval = HLL_SPARSE_VAL_VALUE(p);
             idx += runlen;
-            E += PE[regval]*runlen;
+            regHisto[regval] += runlen;
             p++;
         }
     }
     if (idx != HLL_REGISTERS && invalid) *invalid = 1;
-    E += ez; /* Add 2^0 'ez' times. */
-    *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. */
-
-/* Implements the SUM operation for uint8_t data type which is only used
- * internally as speedup for PFCOUNT with multiple keys. */
-double hllRawSum(uint8_t *registers, double *PE, int *ezp) {
-    double E = 0;
-    int j, ez = 0;
+ * 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) {
-            ez += 8;
+            regHisto[0] += 8;
         } else {
             bytes = (uint8_t*) word;
-            if (bytes[0]) E += PE[bytes[0]]; else ez++;
-            if (bytes[1]) E += PE[bytes[1]]; else ez++;
-            if (bytes[2]) E += PE[bytes[2]]; else ez++;
-            if (bytes[3]) E += PE[bytes[3]]; else ez++;
-            if (bytes[4]) E += PE[bytes[4]]; else ez++;
-            if (bytes[5]) E += PE[bytes[5]]; else ez++;
-            if (bytes[6]) E += PE[bytes[6]]; else ez++;
-            if (bytes[7]) E += PE[bytes[7]]; else ez++;
+            regHisto[bytes[0]] += 1;
+            regHisto[bytes[1]] += 1;
+            regHisto[bytes[2]] += 1;
+            regHisto[bytes[3]] += 1;
+            regHisto[bytes[4]] += 1;
+            regHisto[bytes[5]] += 1;
+            regHisto[bytes[6]] += 1;
+            regHisto[bytes[7]] += 1;
         }
         word++;
     }
-    E += ez; /* 2^(-reg[j]) is 1 when m is 0, add it 'ez' times for every
-                zero register in the HLL. */
-    *ezp = ez;
-    return E;
+}
+
+/* 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
@@ -988,49 +1007,34 @@ double hllRawSum(uint8_t *registers, double *PE, int *ezp) {
  * keys (no need to work with 6-bit integers encoding). */
 uint64_t hllCount(struct hllhdr *hdr, int *invalid) {
     double m = HLL_REGISTERS;
-    double E, alpha = 0.7213/(1+1.079/m);
-    int j, ez; /* Number of registers equal to 0. */
-
-    /* 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;
-    }
+    double E;
+    int j;
+    double alphaInf = 0.5 / log(2.);
+    int regHisto[HLL_Q+2] = {0};
 
-    /* Compute SUM(2^-register[0..i]). */
+    /* Compute register histogram */
     if (hdr->encoding == HLL_DENSE) {
-        E = hllDenseSum(hdr->registers,PE,&ez);
+        hllDenseRegHisto(hdr->registers,regHisto);
     } else if (hdr->encoding == HLL_SPARSE) {
-        E = hllSparseSum(hdr->registers,
-                         sdslen((sds)hdr)-HLL_HDR_SIZE,PE,&ez,invalid);
+        hllSparseRegHisto(hdr->registers,
+                         sdslen((sds)hdr)-HLL_HDR_SIZE,invalid,regHisto);
     } else if (hdr->encoding == HLL_RAW) {
-        E = hllRawSum(hdr->registers,PE,&ez);
+        hllRawRegHisto(hdr->registers,regHisto);
     } else {
         serverPanic("Unknown HyperLogLog encoding in hllCount()");
     }
 
-    /* Apply loglog-beta to the raw estimate. See:
-     * "LogLog-Beta and More: A New Algorithm for Cardinality Estimation
-     * Based on LogLog Counting" Jason Qin, Denys Kim, Yumei Tung
-     * arXiv:1612.02284 */
-    double zl = log(ez + 1);
-    double beta = -0.370393911*ez +
-                   0.070471823*zl +
-                   0.17393686*pow(zl,2) +
-                   0.16339839*pow(zl,3) +
-                  -0.09237745*pow(zl,4) +
-                   0.03738027*pow(zl,5) +
-                  -0.005384159*pow(zl,6) +
-                   0.00042419*pow(zl,7);
-
-    E  = llroundl(alpha*m*(m-ez)*(1/(E+beta)));
+    /* Estimate cardinality form 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(alphaInf*m*m/z);
+
     return (uint64_t) E;
 }