/*
* Licensed to the Apache Software Foundation (ASF) under one or more
* contributor license agreements. See the NOTICE file distributed with
* this work for additional information regarding copyright ownership.
* The ASF licenses this file to You under the Apache License, Version 2.0
* (the "License"); you may not use this file except in compliance with
* the License. You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package org.apache.solr.util.hll;
/**
* Static functions for computing constants and parameters used in the HLL
* algorithm.
*/
final class HLLUtil {
/**
* Precomputed <code>pwMaxMask</code> values indexed by <code>registerSizeInBits</code>.
* Calculated with this formula:
* <pre>
* int maxRegisterValue = (1 << registerSizeInBits) - 1;
* // Mask with all bits set except for (maxRegisterValue - 1) least significant bits (see #addRaw())
* return ~((1L << (maxRegisterValue - 1)) - 1);
* </pre>
*
* @see #pwMaxMask(int)
*/
private static final long[] PW_MASK = {
~((1L << (((1 << 0) - 1) - 1)) - 1),
~((1L << (((1 << 1) - 1) - 1)) - 1),
~((1L << (((1 << 2) - 1) - 1)) - 1),
~((1L << (((1 << 3) - 1) - 1)) - 1),
~((1L << (((1 << 4) - 1) - 1)) - 1),
~((1L << (((1 << 5) - 1) - 1)) - 1),
~((1L << (((1 << 6) - 1) - 1)) - 1),
~((1L << (((1 << 7) - 1) - 1)) - 1),
~((1L << (((1 << 8) - 1) - 1)) - 1)
};
/**
* Precomputed <code>twoToL</code> values indexed by a linear combination of
* <code>regWidth</code> and <code>log2m</code>.
*
* The array is one-dimensional and can be accessed by using index
* <code>(REG_WIDTH_INDEX_MULTIPLIER * regWidth) + log2m</code>
* for <code>regWidth</code> and <code>log2m</code> between the specified
* <code>HLL.{MINIMUM,MAXIMUM}_{REGWIDTH,LOG2M}_PARAM</code> constants.
*
* @see #largeEstimator(int, int, double)
* @see #largeEstimatorCutoff(int, int)
* @see "<a href='http://research.neustar.biz/2013/01/24/hyperloglog-googles-take-on-engineering-hll/'>Blog post with section on 2^L</a>"
*/
private static final double[] TWO_TO_L = new double[(HLL.MAXIMUM_REGWIDTH_PARAM + 1) * (HLL.MAXIMUM_LOG2M_PARAM + 1)];
/**
* Spacing constant used to compute offsets into {@link #TWO_TO_L}.
*/
private static final int REG_WIDTH_INDEX_MULTIPLIER = HLL.MAXIMUM_LOG2M_PARAM + 1;
static {
for(int regWidth = HLL.MINIMUM_REGWIDTH_PARAM; regWidth <= HLL.MAXIMUM_REGWIDTH_PARAM; regWidth++) {
for(int log2m = HLL.MINIMUM_LOG2M_PARAM ; log2m <= HLL.MAXIMUM_LOG2M_PARAM; log2m++) {
int maxRegisterValue = (1 << regWidth) - 1;
// Since 1 is added to p(w) in the insertion algorithm, only
// (maxRegisterValue - 1) bits are inspected hence the hash
// space is one power of two smaller.
final int pwBits = (maxRegisterValue - 1);
final int totalBits = (pwBits + log2m);
final double twoToL = Math.pow(2, totalBits);
TWO_TO_L[(REG_WIDTH_INDEX_MULTIPLIER * regWidth) + log2m] = twoToL;
}
}
}
// ************************************************************************
/**
* Computes the bit-width of HLL registers necessary to estimate a set of
* the specified cardinality.
*
* @param expectedUniqueElements an upper bound on the number of unique
* elements that are expected. This must be greater than zero.
* @return a register size in bits (i.e. <code>log2(log2(n))</code>)
*/
public static int registerBitSize(final long expectedUniqueElements) {
return Math.max(HLL.MINIMUM_REGWIDTH_PARAM,
(int)Math.ceil(NumberUtil.log2(NumberUtil.log2(expectedUniqueElements))));
}
// ========================================================================
/**
* Computes the 'alpha-m-squared' constant used by the HyperLogLog algorithm.
*
* @param m this must be a power of two, cannot be less than
* 16 (2<sup>4</sup>), and cannot be greater than 65536 (2<sup>16</sup>).
* @return gamma times <code>registerCount</code> squared where gamma is
* based on the value of <code>registerCount</code>.
* @throws IllegalArgumentException if <code>registerCount</code> is less
* than 16.
*/
public static double alphaMSquared(final int m) {
switch(m) {
case 1/*2^0*/:
case 2/*2^1*/:
case 4/*2^2*/:
case 8/*2^3*/:
throw new IllegalArgumentException("'m' cannot be less than 16 (" + m + " < 16).");
case 16/*2^4*/:
return 0.673 * m * m;
case 32/*2^5*/:
return 0.697 * m * m;
case 64/*2^6*/:
return 0.709 * m * m;
default/*>2^6*/:
return (0.7213 / (1.0 + 1.079 / m)) * m * m;
}
}
// ========================================================================
/**
* Computes a mask that prevents overflow of HyperLogLog registers.
*
* @param registerSizeInBits the size of the HLL registers, in bits.
* @return mask a <code>long</code> mask to prevent overflow of the registers
* @see #registerBitSize(long)
*/
public static long pwMaxMask(final int registerSizeInBits) {
return PW_MASK[registerSizeInBits];
}
// ========================================================================
/**
* The cutoff for using the "small range correction" formula, in the
* HyperLogLog algorithm.
*
* @param m the number of registers in the HLL. <em>m<em> in the paper.
* @return the cutoff for the small range correction.
* @see #smallEstimator(int, int)
*/
public static double smallEstimatorCutoff(final int m) {
return ((double)m * 5) / 2;
}
/**
* The "small range correction" formula from the HyperLogLog algorithm. Only
* appropriate if both the estimator is smaller than <pre>(5/2) * m</pre> and
* there are still registers that have the zero value.
*
* @param m the number of registers in the HLL. <em>m<em> in the paper.
* @param numberOfZeroes the number of registers with value zero. <em>V</em>
* in the paper.
* @return a corrected cardinality estimate.
*/
public static double smallEstimator(final int m, final int numberOfZeroes) {
return m * Math.log((double)m / numberOfZeroes);
}
/**
* The cutoff for using the "large range correction" formula, from the
* HyperLogLog algorithm, adapted for 64 bit hashes.
*
* @param log2m log-base-2 of the number of registers in the HLL. <em>b<em> in the paper.
* @param registerSizeInBits the size of the HLL registers, in bits.
* @return the cutoff for the large range correction.
* @see #largeEstimator(int, int, double)
* @see "<a href='http://research.neustar.biz/2013/01/24/hyperloglog-googles-take-on-engineering-hll/'>Blog post with section on 64 bit hashes and 'large range correction' cutoff</a>"
*/
public static double largeEstimatorCutoff(final int log2m, final int registerSizeInBits) {
return (TWO_TO_L[(REG_WIDTH_INDEX_MULTIPLIER * registerSizeInBits) + log2m]) / 30.0;
}
/**
* The "large range correction" formula from the HyperLogLog algorithm, adapted
* for 64 bit hashes. Only appropriate for estimators whose value exceeds
* the return of {@link #largeEstimatorCutoff(int, int)}.
*
* @param log2m log-base-2 of the number of registers in the HLL. <em>b<em> in the paper.
* @param registerSizeInBits the size of the HLL registers, in bits.
* @param estimator the original estimator ("E" in the paper).
* @return a corrected cardinality estimate.
* @see "<a href='http://research.neustar.biz/2013/01/24/hyperloglog-googles-take-on-engineering-hll/'>Blog post with section on 64 bit hashes and 'large range correction'</a>"
*/
public static double largeEstimator(final int log2m, final int registerSizeInBits, final double estimator) {
final double twoToL = TWO_TO_L[(REG_WIDTH_INDEX_MULTIPLIER * registerSizeInBits) + log2m];
return -1 * twoToL * Math.log(1.0 - (estimator/twoToL));
}
}