    001/*
    002 * Licensed to the Apache Software Foundation (ASF) under one or more
    003 * contributor license agreements.  See the NOTICE file distributed with
    004 * this work for additional information regarding copyright ownership.
    005 * The ASF licenses this file to You under the Apache License, Version 2.0
    006 * (the "License"); you may not use this file except in compliance with
    007 * the License.  You may obtain a copy of the License at
    008 *
    009 *      http://www.apache.org/licenses/LICENSE-2.0
    010 *
    011 * Unless required by applicable law or agreed to in writing, software
    012 * distributed under the License is distributed on an "AS IS" BASIS,
    013 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
    014 * See the License for the specific language governing permissions and
    015 * limitations under the License.
    016 */
    017package org.apache.commons.collections4.bloomfilter;
    018
    019/**
    020 * The definition of a Bloom filter shape.
    021 *
    022 * <p>This class contains the values for the filter configuration and is used to
    023 * convert a Hasher into a BloomFilter as well as verify that two Bloom filters are
    024 * compatible. (i.e. can be compared or merged)</p>
    025 *
    026 * <h2>Interrelatedness of values</h2>
    027 *
    028 * <dl>
    029 * <dt>Number of Items ({@code n})</dt>
    030 * <dd>{@code n = ceil(m / (-k / ln(1 - exp(ln(p) / k))))}</dd>
    031 * <dt>Probability of False Positives ({@code p})</dt>
    032 * <dd>{@code p = pow(1 - exp(-k / (m / n)), k)}</dd>
    033 * <dt>Number of Bits ({@code m})</dt>
    034 * <dd>{@code m = ceil((n * ln(p)) / ln(1 / pow(2, ln(2))))}</dd>
    035 * <dt>Number of Functions ({@code k})</dt>
    036 * <dd>{@code k = round((m / n) * ln(2))}</dd>
    037 * </dl>
    038 *
    039 * <h2>Estimations from cardinality based on shape</h2>
    040 *
    041 * <p>Several estimates can be calculated from the Shape and the cardinality of a Bloom filter.</p>
    042 *
    043 * <p>In the calculation below the following values are used:</p>
    044 * <ul>
    045 * <li>double c = the cardinality of the Bloom filter.</li>
    046 * <li>double m = numberOfBits as specified in the shape.</li>
    047 * <li>double k = numberOfHashFunctions as specified in the shape.</li>
    048 * </ul>
    049 *
    050 * <h3>Estimate N - n()</h3>
    051 *
    052 * <p>The calculation for the estimate of N is: {@code -(m/k) * ln(1 - (c/m))}.  This is the calculation
    053 * performed by the {@code Shape.estimateN(cardinality)} method below.  This estimate is roughly equivalent to the
    054 * number of hashers that have been merged into a filter to create the cardinality specified.</p>
    055 *
    056 * <p><em>Note:</em></p>
    057 * <ul>
    058 * <li>if cardinality == numberOfBits, then result is infinity.</li>
    059 * <li>if cardinality &gt; numberOfBits, then result is NaN.</li>
    060 * </ul>
    061 *
    062 * <h3>Estimate N of Union - n(A &cup; B)</h3>
    063 *
    064 * <p>To estimate the number of items in the union of two Bloom filters with the same shape, merge them together and
    065 * calculate the estimated N from the result.</p>
    066 *
    067 * <h3>Estimate N of the Intersection - n(A &cap; B)</h3>
    068 *
    069 * <p>To estimate the number of items in the intersection of two Bloom filters A and B with the same shape the calculation is:
    070 * n(A) + n(b) - n(A &cup; B).</p>
    071 *
    072 * <p>Care must be taken when any of the n(x) returns infinity.  In general the following assumptions are true:
    073 *
    074 * <ul>
    075 * <li>If n(A) = &infin; and n(B) &lt; &infin; then n(A &cap; B) = n(B)</li>
    076 * <li>If n(A) &lt; &infin; and n(B) = &infin; then n(A &cap; B) = n(A)</li>
    077 * <li>If n(A) = &infin; and n(B) = &infin; then n(A &cap; B) = &infin;</li>
    078 * <li>If n(A) &lt; &infin; and n(B) &lt; &infin; and n(A &cup; B) = &infin; then n(A &cap; B) is undefined.</li>
    079 * </ul>
    080 *
    081 * @see <a href="https://hur.st/bloomfilter">Bloom Filter calculator</a>
    082 * @see <a href="https://en.wikipedia.org/wiki/Bloom_filter">Bloom filter
    083 * [Wikipedia]</a>
    084 * @since 4.5.0-M1
    085 */
    086public final class Shape {
    087
    088    /**
    089     * The natural logarithm of 2. Used in several calculations. Approximately 0.693147180559945.
    090     */
    091    private static final double LN_2 = Math.log(2.0);
    092
    093    /**
    094     * ln(1 / 2^ln(2)). Used in calculating the number of bits. Approximately -0.480453013918201.
    095     *
    096     * <p>ln(1 / 2^ln(2)) = ln(1) - ln(2^ln(2)) = -ln(2) * ln(2)</p>
    097     */
    098    private static final double DENOMINATOR = -LN_2 * LN_2;
    099
    100    /**
    101     * Calculates the number of hash functions given numberOfItems and numberOfBits.
    102     * This is a method so that the calculation is consistent across all constructors.
    103     *
    104     * @param numberOfItems the number of items in the filter.
    105     * @param numberOfBits the number of bits in the filter.
    106     * @return the optimal number of hash functions.
    107     * @throws IllegalArgumentException if the calculated number of hash function is {@code < 1}
    108     */
    109    private static int calculateNumberOfHashFunctions(final int numberOfItems, final int numberOfBits) {
    110        // k = round((m / n) * ln(2)) We change order so that we use real math rather
    111        // than integer math.
    112        final long k = Math.round(LN_2 * numberOfBits / numberOfItems);
    113        if (k < 1) {
    114            throw new IllegalArgumentException(String.format("Filter too small: Calculated number of hash functions (%s) was less than 1", k));
    115        }
    116        // Normally we would check that numberOfHashFunctions <= Integer.MAX_VALUE but
    117        // since numberOfBits is at most Integer.MAX_VALUE the numerator of
    118        // numberOfHashFunctions is ln(2) * Integer.MAX_VALUE = 646456992.9449 the
    119        // value of k cannot be above Integer.MAX_VALUE.
    120        return (int) k;
    121    }
    122
    123    /**
    124     * Checks the calculated probability is {@code < 1.0}.
    125     *
    126     * <p>
    127     * This function is used to verify that the dynamically calculated probability for the Shape is in the valid range 0 to 1 exclusive. This need only be
    128     * performed once upon construction.
    129     * </p>
    130     *
    131     * @param probability the probability
    132     * @throws IllegalArgumentException if the probability is {@code >= 1.0}.
    133     */
    134    private static void checkCalculatedProbability(final double probability) {
    135        // We do not need to check for p <= 0.0 since we only allow positive values for
    136        // parameters and the closest we can come to exp(-kn/m) == 1 is
    137        // exp(-1/Integer.MAX_INT) approx 0.9999999995343387 so Math.pow(x, y) will
    138        // always be 0<x<1 and y>0
    139        if (probability >= 1.0) {
    140            throw new IllegalArgumentException("Calculated probability is greater than or equal to 1: " + probability);
    141        }
    142    }
    143
    144    /**
    145     * Checks number of bits is strictly positive.
    146     *
    147     * @param numberOfBits the number of bits
    148     * @return the number of bits
    149     * @throws IllegalArgumentException if the number of bits is {@code < 1}.
    150     */
    151    private static int checkNumberOfBits(final int numberOfBits) {
    152        if (numberOfBits < 1) {
    153            throw new IllegalArgumentException("Number of bits must be greater than 0: " + numberOfBits);
    154        }
    155        return numberOfBits;
    156    }
    157
    158    /**
    159     * Checks number of hash functions is strictly positive.
    160     *
    161     * @param numberOfHashFunctions the number of hash functions
    162     * @return the number of hash functions
    163     * @throws IllegalArgumentException if the number of hash functions is {@code < 1}.
    164     */
    165    private static int checkNumberOfHashFunctions(final int numberOfHashFunctions) {
    166        if (numberOfHashFunctions < 1) {
    167            throw new IllegalArgumentException("Number of hash functions must be greater than 0: " + numberOfHashFunctions);
    168        }
    169        return numberOfHashFunctions;
    170    }
    171
    172    /**
    173     * Checks number of items is strictly positive.
    174     *
    175     * @param numberOfItems the number of items
    176     * @return the number of items
    177     * @throws IllegalArgumentException if the number of items is {@code < 1}.
    178     */
    179    private static int checkNumberOfItems(final int numberOfItems) {
    180        if (numberOfItems < 1) {
    181            throw new IllegalArgumentException("Number of items must be greater than 0: " + numberOfItems);
    182        }
    183        return numberOfItems;
    184    }
    185
    186    /**
    187     * Checks the probability is in the range 0.0, exclusive, to 1.0, exclusive.
    188     *
    189     * @param probability the probability
    190     * @throws IllegalArgumentException if the probability is not in the range {@code (0, 1)}
    191     */
    192    private static void checkProbability(final double probability) {
    193        // Using the negation of within the desired range will catch NaN
    194        if (!(probability > 0.0 && probability < 1.0)) {
    195            throw new IllegalArgumentException("Probability must be greater than 0 and less than 1: " + probability);
    196        }
    197    }
    198
    199    /**
    200     * Constructs a filter configuration with the specified number of hashFunctions ({@code k}) and
    201     * bits ({@code m}).
    202     *
    203     * @param numberOfHashFunctions Number of hash functions to use for each item placed in the filter.
    204     * @param numberOfBits The number of bits in the filter
    205     * @return a valid Shape.
    206     * @throws IllegalArgumentException if {@code numberOfHashFunctions < 1} or {@code numberOfBits < 1}
    207     */
    208    public static Shape fromKM(final int numberOfHashFunctions, final int numberOfBits) {
    209        return new Shape(numberOfHashFunctions, numberOfBits);
    210    }
    211
    212    /**
    213     * Constructs a filter configuration with the specified number of items ({@code n}) and
    214     * bits ({@code m}).
    215     *
    216     * <p>The optimal number of hash functions ({@code k}) is computed.
    217     * <pre>k = round((m / n) * ln(2))</pre>
    218     *
    219     * <p>The false-positive probability is computed using the number of items, bits and hash
    220     * functions. An exception is raised if this is greater than or equal to 1 (i.e. the
    221     * shape is invalid for use as a Bloom filter).
    222     *
    223     * @param numberOfItems Number of items to be placed in the filter
    224     * @param numberOfBits The number of bits in the filter
    225     * @return a valid Shape.
    226     * @throws IllegalArgumentException if {@code numberOfItems < 1}, {@code numberOfBits < 1},
    227     * the calculated number of hash function is {@code < 1}, or if the actual probability is {@code >= 1.0}
    228     */
    229    public static Shape fromNM(final int numberOfItems, final int numberOfBits) {
    230        checkNumberOfItems(numberOfItems);
    231        checkNumberOfBits(numberOfBits);
    232        final int numberOfHashFunctions = calculateNumberOfHashFunctions(numberOfItems, numberOfBits);
    233        final Shape shape = new Shape(numberOfHashFunctions, numberOfBits);
    234        // check that probability is within range
    235        checkCalculatedProbability(shape.getProbability(numberOfItems));
    236        return shape;
    237    }
    238
    239    /**
    240     * Constructs a filter configuration with the specified number of items, bits
    241     * and hash functions.
    242     *
    243     * <p>The false-positive probability is computed using the number of items, bits and hash
    244     * functions. An exception is raised if this is greater than or equal to 1 (i.e. the
    245     * shape is invalid for use as a Bloom filter).
    246     *
    247     * @param numberOfItems Number of items to be placed in the filter
    248     * @param numberOfBits The number of bits in the filter.
    249     * @param numberOfHashFunctions The number of hash functions in the filter
    250     * @return a valid Shape.
    251     * @throws IllegalArgumentException if {@code numberOfItems < 1}, {@code numberOfBits < 1},
    252     * {@code numberOfHashFunctions < 1}, or if the actual probability is {@code >= 1.0}.
    253     */
    254    public static Shape fromNMK(final int numberOfItems, final int numberOfBits, final int numberOfHashFunctions) {
    255        checkNumberOfItems(numberOfItems);
    256        checkNumberOfBits(numberOfBits);
    257        checkNumberOfHashFunctions(numberOfHashFunctions);
    258        // check that probability is within range
    259        final Shape shape = new Shape(numberOfHashFunctions, numberOfBits);
    260        // check that probability is within range
    261        checkCalculatedProbability(shape.getProbability(numberOfItems));
    262        return shape;
    263    }
    264
    265    /**
    266     * Constructs a filter configuration with the specified number of items ({@code n}) and
    267     * desired false-positive probability ({@code p}).
    268     *
    269     * <p>The number of bits ({@code m}) for the filter is computed.
    270     * <pre>m = ceil(n * ln(p) / ln(1 / 2^ln(2)))</pre>
    271     *
    272     * <p>The optimal number of hash functions ({@code k}) is computed.
    273     * <pre>k = round((m / n) * ln(2))</pre>
    274     *
    275     * <p>The actual probability will be approximately equal to the
    276     * desired probability but will be dependent upon the calculated number of bits and hash
    277     * functions. An exception is raised if this is greater than or equal to 1 (i.e. the
    278     * shape is invalid for use as a Bloom filter).
    279     *
    280     * @param numberOfItems Number of items to be placed in the filter
    281     * @param probability The desired false-positive probability in the range {@code (0, 1)}
    282     * @return a valid Shape
    283     * @throws IllegalArgumentException if {@code numberOfItems < 1}, if the desired probability
    284     * is not in the range {@code (0, 1)} or if the actual probability is {@code >= 1.0}.
    285     */
    286    public static Shape fromNP(final int numberOfItems, final double probability) {
    287        checkNumberOfItems(numberOfItems);
    288        checkProbability(probability);
    289
    290        // Number of bits (m)
    291        final double m = Math.ceil(numberOfItems * Math.log(probability) / DENOMINATOR);
    292        if (m > Integer.MAX_VALUE) {
    293            throw new IllegalArgumentException("Resulting filter has more than " + Integer.MAX_VALUE + " bits: " + m);
    294        }
    295        final int numberOfBits = (int) m;
    296
    297        final int numberOfHashFunctions = calculateNumberOfHashFunctions(numberOfItems, numberOfBits);
    298        final Shape shape = new Shape(numberOfHashFunctions, numberOfBits);
    299        // check that probability is within range
    300        checkCalculatedProbability(shape.getProbability(numberOfItems));
    301        return shape;
    302    }
    303
    304    /**
    305     * Constructs a filter configuration with a desired false-positive probability ({@code p}) and the
    306     * specified number of bits ({@code m}) and hash functions ({@code k}).
    307     *
    308     * <p>The number of items ({@code n}) to be stored in the filter is computed.
    309     * <pre>n = ceil(m / (-k / ln(1 - exp(ln(p) / k))))</pre>
    310     *
    311     * <p>The actual probability will be approximately equal to the
    312     * desired probability but will be dependent upon the calculated Bloom filter capacity
    313     * (number of items). An exception is raised if this is greater than or equal to 1 (i.e. the
    314     * shape is invalid for use as a Bloom filter).
    315     *
    316     * @param probability The desired false-positive probability in the range {@code (0, 1)}
    317     * @param numberOfBits The number of bits in the filter
    318     * @param numberOfHashFunctions The number of hash functions in the filter
    319     * @return a valid Shape.
    320     * @throws IllegalArgumentException if the desired probability is not in the range {@code (0, 1)},
    321     * {@code numberOfBits < 1}, {@code numberOfHashFunctions < 1}, or the actual
    322     * probability is {@code >= 1.0}
    323     */
    324    public static Shape fromPMK(final double probability, final int numberOfBits, final int numberOfHashFunctions) {
    325        checkProbability(probability);
    326        checkNumberOfBits(numberOfBits);
    327        checkNumberOfHashFunctions(numberOfHashFunctions);
    328
    329        // Number of items (n):
    330        // n = ceil(m / (-k / ln(1 - exp(ln(p) / k))))
    331        final double n = Math.ceil(numberOfBits / (-numberOfHashFunctions / Math.log(-Math.expm1(Math.log(probability) / numberOfHashFunctions))));
    332
    333        // log of probability is always < 0
    334        // number of hash functions is >= 1
    335        // e^x where x < 0 = [0,1)
    336        // log 1-e^x = [log1, log0) = <0 with an effective lower limit of -53
    337        // numberOfBits/ (-numberOfHashFunctions / [-53,0) ) >0
    338        // ceil( >0 ) >= 1
    339        // so we cannot produce a negative value thus we don't check for it.
    340        //
    341        // similarly we cannot produce a number greater than numberOfBits so we
    342        // do not have to check for Integer.MAX_VALUE either.
    343
    344        final Shape shape = new Shape(numberOfHashFunctions, numberOfBits);
    345        // check that probability is within range
    346        checkCalculatedProbability(shape.getProbability((int) n));
    347        return shape;
    348    }
    349
    350    /**
    351     * Number of hash functions to create a filter ({@code k}).
    352     */
    353    private final int numberOfHashFunctions;
    354
    355    /**
    356     * Number of bits in the filter ({@code m}).
    357     */
    358    private final int numberOfBits;
    359
    360    /**
    361     * Constructs a filter configuration with the specified number of hashFunctions ({@code k}) and
    362     * bits ({@code m}).
    363     *
    364     * @param numberOfHashFunctions Number of hash functions to use for each item placed in the filter.
    365     * @param numberOfBits The number of bits in the filter
    366     * @throws IllegalArgumentException if {@code numberOfHashFunctions < 1} or {@code numberOfBits < 1}
    367     */
    368    private Shape(final int numberOfHashFunctions, final int numberOfBits) {
    369        this.numberOfHashFunctions = checkNumberOfHashFunctions(numberOfHashFunctions);
    370        this.numberOfBits = checkNumberOfBits(numberOfBits);
    371    }
    372
    373    @Override
    374    public boolean equals(final Object obj) {
    375        // Shape is final so no check for the same class as inheritance is not possible
    376        if (obj instanceof Shape) {
    377            final Shape other = (Shape) obj;
    378            return numberOfBits == other.numberOfBits && numberOfHashFunctions == other.numberOfHashFunctions;
    379        }
    380        return false;
    381    }
    382
    383    /**
    384     * Estimates the maximum number of elements that can be merged into a filter of
    385     * this shape before the false positive rate exceeds the desired rate. <p> The
    386     * formula for deriving {@code k} when {@code m} and {@code n} are known is:
    387     *
    388     * <p>{@code k = ln2 * m / n}</p>
    389     *
    390     * <p>Solving for {@code n} yields:</p>
    391     *
    392     * <p>{@code n = ln2 * m / k}</p>
    393     *
    394     * @return An estimate of max N.
    395     */
    396    public double estimateMaxN() {
    397        return numberOfBits * LN_2 / numberOfHashFunctions;
    398    }
    399
    400    /**
    401     * Estimate the number of items in a Bloom filter with this shape and the specified number of bits enabled.
    402     *
    403     * <p><em>Note:</em></p>
    404     * <ul>
    405     * <li> if cardinality == numberOfBits, then result is infinity.</li>
    406     * <li> if cardinality &gt; numberOfBits, then result is NaN.</li>
    407     * </ul>
    408     *
    409     * @param cardinality the number of enabled  bits also known as the hamming value.
    410     * @return An estimate of the number of items in the Bloom filter.
    411     */
    412    public double estimateN(final int cardinality) {
    413        final double c = cardinality;
    414        final double m = numberOfBits;
    415        final double k = numberOfHashFunctions;
    416        return -(m / k) * Math.log1p(-c / m);
    417    }
    418
    419    /**
    420     * Gets the number of bits in the Bloom filter.
    421     * This is also known as {@code m}.
    422     *
    423     * @return the number of bits in the Bloom filter ({@code m}).
    424     */
    425    public int getNumberOfBits() {
    426        return numberOfBits;
    427    }
    428
    429    /**
    430     * Gets the number of hash functions used to construct the filter.
    431     * This is also known as {@code k}.
    432     *
    433     * @return the number of hash functions used to construct the filter ({@code k}).
    434     */
    435    public int getNumberOfHashFunctions() {
    436        return numberOfHashFunctions;
    437    }
    438
    439    /**
    440     * Calculates the probability of false positives ({@code p}) given
    441     * numberOfItems ({@code n}), numberOfBits ({@code m}) and numberOfHashFunctions ({@code k}).
    442     * <pre>p = pow(1 - exp(-k / (m / n)), k)</pre>
    443     *
    444     * <p>This is the probability that a Bloom filter will return true for the presence of an item
    445     * when it does not contain the item.</p>
    446     *
    447     * <p>The probability assumes that the Bloom filter is filled with the expected number of
    448     * items. If the filter contains fewer items then the actual probability will be lower.
    449     * Thus, this returns the worst-case false positive probability for a filter that has not
    450     * exceeded its expected number of items.</p>
    451     *
    452     * @param numberOfItems the number of items hashed into the Bloom filter.
    453     * @return the probability of false positives.
    454     */
    455    public double getProbability(final int numberOfItems) {
    456        if (numberOfItems < 0) {
    457            throw new IllegalArgumentException("Number of items must be greater than or equal to 0: " + numberOfItems);
    458        }
    459        if (numberOfItems == 0) {
    460            return 0;
    461        }
    462        return Math.pow(-Math.expm1(-1.0 * numberOfHashFunctions * numberOfItems / numberOfBits), numberOfHashFunctions);
    463    }
    464
    465    @Override
    466    public int hashCode() {
    467        // Match Arrays.hashCode(new int[] {numberOfBits, numberOfHashFunctions})
    468        return (31 + numberOfBits) * 31 + numberOfHashFunctions;
    469    }
    470
    471    /**
    472     * Determines if a cardinality is sparse based on the shape.
    473     * <p>This method assumes that bit maps are 64bits and indexes are 32bits. If the memory
    474     * necessary to store the cardinality as indexes is less than the estimated memory for bit maps,
    475     * the cardinality is determined to be {@code sparse}.</p>
    476     *
    477     * @param cardinality the cardinality to check.
    478     * @return true if the cardinality is sparse within the shape.
    479     */
    480    public boolean isSparse(final int cardinality) {
    481
    482        /*
    483         * Since the size of a bit map is a long and the size of an index is an int,
    484         * there can be 2 indexes for each bit map. In Bloom filters indexes are evenly
    485         * distributed across the range of possible values, Thus if the cardinality
    486         * (number of indexes) is less than or equal to 2*number of bit maps the
    487         * cardinality is sparse within the shape.
    488         */
    489        return cardinality <= BitMaps.numberOfBitMaps(this) * 2;
    490    }
    491
    492    @Override
    493    public String toString() {
    494        return String.format("Shape[k=%s m=%s]", numberOfHashFunctions, numberOfBits);
    495    }
    496}