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    * Licensed to the Apache Software Foundation (ASF) under one or more
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    * 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.commons.collections4.bloomfilter;
  
  /**
   * The definition of a Bloom filter shape.
   *
   * <p>This class contains the values for the filter configuration and is used to
   * convert a Hasher into a BloomFilter as well as verify that two Bloom filters are
   * compatible. (i.e. can be compared or merged)</p>
   *
   * <h2>Interrelatedness of values</h2>
   *
   * <dl>
   * <dt>Number of Items ({@code n})</dt>
   * <dd>{@code n = ceil(m / (-k / ln(1 - exp(ln(p) / k))))}</dd>
   * <dt>Probability of False Positives ({@code p})</dt>
   * <dd>{@code p = pow(1 - exp(-k / (m / n)), k)}</dd>
   * <dt>Number of Bits ({@code m})</dt>
   * <dd>{@code m = ceil((n * ln(p)) / ln(1 / pow(2, ln(2))))}</dd>
   * <dt>Number of Functions ({@code k})</dt>
   * <dd>{@code k = round((m / n) * ln(2))}</dd>
   * </dl>
   *
   * <h2>Estimations from cardinality based on shape</h2>
   *
   * <p>Several estimates can be calculated from the Shape and the cardinality of a Bloom filter.</p>
   *
   * <p>In the calculation below the following values are used:</p>
   * <ul>
   * <li>double c = the cardinality of the Bloom filter.</li>
   * <li>double m = numberOfBits as specified in the shape.</li>
   * <li>double k = numberOfHashFunctions as specified in the shape.</li>
   * </ul>
   *
   * <h3>Estimate N - n()</h3>
   *
   * <p>The calculation for the estimate of N is: {@code -(m/k) * ln(1 - (c/m))}.  This is the calculation
   * performed by the {@code Shape.estimateN(cardinality)} method below.  This estimate is roughly equivalent to the
   * number of hashers that have been merged into a filter to create the cardinality specified.</p>
   *
   * <p><em>Note:</em></p>
   * <ul>
   * <li>if cardinality == numberOfBits, then result is infinity.</li>
   * <li>if cardinality &gt; numberOfBits, then result is NaN.</li>
   * </ul>
   *
   * <h3>Estimate N of Union - n(A &cup; B)</h3>
   *
   * <p>To estimate the number of items in the union of two Bloom filters with the same shape, merge them together and
   * calculate the estimated N from the result.</p>
   *
   * <h3>Estimate N of the Intersection - n(A &cap; B)</h3>
   *
   * <p>To estimate the number of items in the intersection of two Bloom filters A and B with the same shape the calculation is:
   * n(A) + n(b) - n(A &cup; B).</p>
   *
   * <p>Care must be taken when any of the n(x) returns infinity.  In general the following assumptions are true:
   *
   * <ul>
   * <li>If n(A) = &infin; and n(B) &lt; &infin; then n(A &cap; B) = n(B)</li>
   * <li>If n(A) &lt; &infin; and n(B) = &infin; then n(A &cap; B) = n(A)</li>
   * <li>If n(A) = &infin; and n(B) = &infin; then n(A &cap; B) = &infin;</li>
   * <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>
   * </ul>
   *
   * @see <a href="https://hur.st/bloomfilter">Bloom Filter calculator</a>
   * @see <a href="https://en.wikipedia.org/wiki/Bloom_filter">Bloom filter
   * [Wikipedia]</a>
   * @since 4.5.0-M1
   */
  public final class Shape {
  
      /**
       * The natural logarithm of 2. Used in several calculations. Approximately 0.693147180559945.
       */
      private static final double LN_2 = Math.log(2.0);
  
      /**
       * ln(1 / 2^ln(2)). Used in calculating the number of bits. Approximately -0.480453013918201.
       *
       * <p>ln(1 / 2^ln(2)) = ln(1) - ln(2^ln(2)) = -ln(2) * ln(2)</p>
       */
      private static final double DENOMINATOR = -LN_2 * LN_2;
  
     /**
      * Calculates the number of hash functions given numberOfItems and numberOfBits.
      * This is a method so that the calculation is consistent across all constructors.
      *
      * @param numberOfItems the number of items in the filter.
      * @param numberOfBits the number of bits in the filter.
      * @return the optimal number of hash functions.
      * @throws IllegalArgumentException if the calculated number of hash function is {@code < 1}
      */
     private static int calculateNumberOfHashFunctions(final int numberOfItems, final int numberOfBits) {
         // k = round((m / n) * ln(2)) We change order so that we use real math rather
         // than integer math.
         final long k = Math.round(LN_2 * numberOfBits / numberOfItems);
         if (k < 1) {
             throw new IllegalArgumentException(String.format("Filter too small: Calculated number of hash functions (%s) was less than 1", k));
         }
         // Normally we would check that numberOfHashFunctions <= Integer.MAX_VALUE but
         // since numberOfBits is at most Integer.MAX_VALUE the numerator of
         // numberOfHashFunctions is ln(2) * Integer.MAX_VALUE = 646456992.9449 the
         // value of k cannot be above Integer.MAX_VALUE.
         return (int) k;
     }
 
     /**
      * Checks the calculated probability is {@code < 1.0}.
      *
      * <p>
      * 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
      * performed once upon construction.
      * </p>
      *
      * @param probability the probability
      * @throws IllegalArgumentException if the probability is {@code >= 1.0}.
      */
     private static void checkCalculatedProbability(final double probability) {
         // We do not need to check for p <= 0.0 since we only allow positive values for
         // parameters and the closest we can come to exp(-kn/m) == 1 is
         // exp(-1/Integer.MAX_INT) approx 0.9999999995343387 so Math.pow(x, y) will
         // always be 0<x<1 and y>0
         if (probability >= 1.0) {
             throw new IllegalArgumentException("Calculated probability is greater than or equal to 1: " + probability);
         }
     }
 
     /**
      * Checks number of bits is strictly positive.
      *
      * @param numberOfBits the number of bits
      * @return the number of bits
      * @throws IllegalArgumentException if the number of bits is {@code < 1}.
      */
     private static int checkNumberOfBits(final int numberOfBits) {
         if (numberOfBits < 1) {
             throw new IllegalArgumentException("Number of bits must be greater than 0: " + numberOfBits);
         }
         return numberOfBits;
     }
 
     /**
      * Checks number of hash functions is strictly positive.
      *
      * @param numberOfHashFunctions the number of hash functions
      * @return the number of hash functions
      * @throws IllegalArgumentException if the number of hash functions is {@code < 1}.
      */
     private static int checkNumberOfHashFunctions(final int numberOfHashFunctions) {
         if (numberOfHashFunctions < 1) {
             throw new IllegalArgumentException("Number of hash functions must be greater than 0: " + numberOfHashFunctions);
         }
         return numberOfHashFunctions;
     }
 
     /**
      * Checks number of items is strictly positive.
      *
      * @param numberOfItems the number of items
      * @return the number of items
      * @throws IllegalArgumentException if the number of items is {@code < 1}.
      */
     private static int checkNumberOfItems(final int numberOfItems) {
         if (numberOfItems < 1) {
             throw new IllegalArgumentException("Number of items must be greater than 0: " + numberOfItems);
         }
         return numberOfItems;
     }
 
     /**
      * Checks the probability is in the range 0.0, exclusive, to 1.0, exclusive.
      *
      * @param probability the probability
      * @throws IllegalArgumentException if the probability is not in the range {@code (0, 1)}
      */
     private static void checkProbability(final double probability) {
         // Using the negation of within the desired range will catch NaN
         if (!(probability > 0.0 && probability < 1.0)) {
             throw new IllegalArgumentException("Probability must be greater than 0 and less than 1: " + probability);
         }
     }
 
     /**
      * Constructs a filter configuration with the specified number of hashFunctions ({@code k}) and
      * bits ({@code m}).
      *
      * @param numberOfHashFunctions Number of hash functions to use for each item placed in the filter.
      * @param numberOfBits The number of bits in the filter
      * @return a valid Shape.
      * @throws IllegalArgumentException if {@code numberOfHashFunctions < 1} or {@code numberOfBits < 1}
      */
     public static Shape fromKM(final int numberOfHashFunctions, final int numberOfBits) {
         return new Shape(numberOfHashFunctions, numberOfBits);
     }
 
     /**
      * Constructs a filter configuration with the specified number of items ({@code n}) and
      * bits ({@code m}).
      *
      * <p>The optimal number of hash functions ({@code k}) is computed.
      * <pre>k = round((m / n) * ln(2))</pre>
      *
      * <p>The false-positive probability is computed using the number of items, bits and hash
      * functions. An exception is raised if this is greater than or equal to 1 (i.e. the
      * shape is invalid for use as a Bloom filter).
      *
      * @param numberOfItems Number of items to be placed in the filter
      * @param numberOfBits The number of bits in the filter
      * @return a valid Shape.
      * @throws IllegalArgumentException if {@code numberOfItems < 1}, {@code numberOfBits < 1},
      * the calculated number of hash function is {@code < 1}, or if the actual probability is {@code >= 1.0}
      */
     public static Shape fromNM(final int numberOfItems, final int numberOfBits) {
         checkNumberOfItems(numberOfItems);
         checkNumberOfBits(numberOfBits);
         final int numberOfHashFunctions = calculateNumberOfHashFunctions(numberOfItems, numberOfBits);
         final Shape shape = new Shape(numberOfHashFunctions, numberOfBits);
         // check that probability is within range
         checkCalculatedProbability(shape.getProbability(numberOfItems));
         return shape;
     }
 
     /**
      * Constructs a filter configuration with the specified number of items, bits
      * and hash functions.
      *
      * <p>The false-positive probability is computed using the number of items, bits and hash
      * functions. An exception is raised if this is greater than or equal to 1 (i.e. the
      * shape is invalid for use as a Bloom filter).
      *
      * @param numberOfItems Number of items to be placed in the filter
      * @param numberOfBits The number of bits in the filter.
      * @param numberOfHashFunctions The number of hash functions in the filter
      * @return a valid Shape.
      * @throws IllegalArgumentException if {@code numberOfItems < 1}, {@code numberOfBits < 1},
      * {@code numberOfHashFunctions < 1}, or if the actual probability is {@code >= 1.0}.
      */
     public static Shape fromNMK(final int numberOfItems, final int numberOfBits, final int numberOfHashFunctions) {
         checkNumberOfItems(numberOfItems);
         checkNumberOfBits(numberOfBits);
         checkNumberOfHashFunctions(numberOfHashFunctions);
         // check that probability is within range
         final Shape shape = new Shape(numberOfHashFunctions, numberOfBits);
         // check that probability is within range
         checkCalculatedProbability(shape.getProbability(numberOfItems));
         return shape;
     }
 
     /**
      * Constructs a filter configuration with the specified number of items ({@code n}) and
      * desired false-positive probability ({@code p}).
      *
      * <p>The number of bits ({@code m}) for the filter is computed.
      * <pre>m = ceil(n * ln(p) / ln(1 / 2^ln(2)))</pre>
      *
      * <p>The optimal number of hash functions ({@code k}) is computed.
      * <pre>k = round((m / n) * ln(2))</pre>
      *
      * <p>The actual probability will be approximately equal to the
      * desired probability but will be dependent upon the calculated number of bits and hash
      * functions. An exception is raised if this is greater than or equal to 1 (i.e. the
      * shape is invalid for use as a Bloom filter).
      *
      * @param numberOfItems Number of items to be placed in the filter
      * @param probability The desired false-positive probability in the range {@code (0, 1)}
      * @return a valid Shape
      * @throws IllegalArgumentException if {@code numberOfItems < 1}, if the desired probability
      * is not in the range {@code (0, 1)} or if the actual probability is {@code >= 1.0}.
      */
     public static Shape fromNP(final int numberOfItems, final double probability) {
         checkNumberOfItems(numberOfItems);
         checkProbability(probability);
 
         // Number of bits (m)
         final double m = Math.ceil(numberOfItems * Math.log(probability) / DENOMINATOR);
         if (m > Integer.MAX_VALUE) {
             throw new IllegalArgumentException("Resulting filter has more than " + Integer.MAX_VALUE + " bits: " + m);
         }
         final int numberOfBits = (int) m;
 
         final int numberOfHashFunctions = calculateNumberOfHashFunctions(numberOfItems, numberOfBits);
         final Shape shape = new Shape(numberOfHashFunctions, numberOfBits);
         // check that probability is within range
         checkCalculatedProbability(shape.getProbability(numberOfItems));
         return shape;
     }
 
     /**
      * Constructs a filter configuration with a desired false-positive probability ({@code p}) and the
      * specified number of bits ({@code m}) and hash functions ({@code k}).
      *
      * <p>The number of items ({@code n}) to be stored in the filter is computed.
      * <pre>n = ceil(m / (-k / ln(1 - exp(ln(p) / k))))</pre>
      *
      * <p>The actual probability will be approximately equal to the
      * desired probability but will be dependent upon the calculated Bloom filter capacity
      * (number of items). An exception is raised if this is greater than or equal to 1 (i.e. the
      * shape is invalid for use as a Bloom filter).
      *
      * @param probability The desired false-positive probability in the range {@code (0, 1)}
      * @param numberOfBits The number of bits in the filter
      * @param numberOfHashFunctions The number of hash functions in the filter
      * @return a valid Shape.
      * @throws IllegalArgumentException if the desired probability is not in the range {@code (0, 1)},
      * {@code numberOfBits < 1}, {@code numberOfHashFunctions < 1}, or the actual
      * probability is {@code >= 1.0}
      */
     public static Shape fromPMK(final double probability, final int numberOfBits, final int numberOfHashFunctions) {
         checkProbability(probability);
         checkNumberOfBits(numberOfBits);
         checkNumberOfHashFunctions(numberOfHashFunctions);
 
         // Number of items (n):
         // n = ceil(m / (-k / ln(1 - exp(ln(p) / k))))
         final double n = Math.ceil(numberOfBits / (-numberOfHashFunctions / Math.log(-Math.expm1(Math.log(probability) / numberOfHashFunctions))));
 
         // log of probability is always < 0
         // number of hash functions is >= 1
         // e^x where x < 0 = [0,1)
         // log 1-e^x = [log1, log0) = <0 with an effective lower limit of -53
         // numberOfBits/ (-numberOfHashFunctions / [-53,0) ) >0
         // ceil( >0 ) >= 1
         // so we cannot produce a negative value thus we don't check for it.
         //
         // similarly we cannot produce a number greater than numberOfBits so we
         // do not have to check for Integer.MAX_VALUE either.
 
         final Shape shape = new Shape(numberOfHashFunctions, numberOfBits);
         // check that probability is within range
         checkCalculatedProbability(shape.getProbability((int) n));
         return shape;
     }
 
     /**
      * Number of hash functions to create a filter ({@code k}).
      */
     private final int numberOfHashFunctions;
 
     /**
      * Number of bits in the filter ({@code m}).
      */
     private final int numberOfBits;
 
     /**
      * Constructs a filter configuration with the specified number of hashFunctions ({@code k}) and
      * bits ({@code m}).
      *
      * @param numberOfHashFunctions Number of hash functions to use for each item placed in the filter.
      * @param numberOfBits The number of bits in the filter
      * @throws IllegalArgumentException if {@code numberOfHashFunctions < 1} or {@code numberOfBits < 1}
      */
     private Shape(final int numberOfHashFunctions, final int numberOfBits) {
         this.numberOfHashFunctions = checkNumberOfHashFunctions(numberOfHashFunctions);
         this.numberOfBits = checkNumberOfBits(numberOfBits);
     }
 
     @Override
     public boolean equals(final Object obj) {
         // Shape is final so no check for the same class as inheritance is not possible
         if (obj instanceof Shape) {
             final Shape other = (Shape) obj;
             return numberOfBits == other.numberOfBits && numberOfHashFunctions == other.numberOfHashFunctions;
         }
         return false;
     }
 
     /**
      * Estimates the maximum number of elements that can be merged into a filter of
      * this shape before the false positive rate exceeds the desired rate. <p> The
      * formula for deriving {@code k} when {@code m} and {@code n} are known is:
      *
      * <p>{@code k = ln2 * m / n}</p>
      *
      * <p>Solving for {@code n} yields:</p>
      *
      * <p>{@code n = ln2 * m / k}</p>
      *
      * @return An estimate of max N.
      */
     public double estimateMaxN() {
         return numberOfBits * LN_2 / numberOfHashFunctions;
     }
 
     /**
      * Estimate the number of items in a Bloom filter with this shape and the specified number of bits enabled.
      *
      * <p><em>Note:</em></p>
      * <ul>
      * <li> if cardinality == numberOfBits, then result is infinity.</li>
      * <li> if cardinality &gt; numberOfBits, then result is NaN.</li>
      * </ul>
      *
      * @param cardinality the number of enabled  bits also known as the hamming value.
      * @return An estimate of the number of items in the Bloom filter.
      */
     public double estimateN(final int cardinality) {
         final double c = cardinality;
         final double m = numberOfBits;
         final double k = numberOfHashFunctions;
         return -(m / k) * Math.log1p(-c / m);
     }
 
     /**
      * Gets the number of bits in the Bloom filter.
      * This is also known as {@code m}.
      *
      * @return the number of bits in the Bloom filter ({@code m}).
      */
     public int getNumberOfBits() {
         return numberOfBits;
     }
 
     /**
      * Gets the number of hash functions used to construct the filter.
      * This is also known as {@code k}.
      *
      * @return the number of hash functions used to construct the filter ({@code k}).
      */
     public int getNumberOfHashFunctions() {
         return numberOfHashFunctions;
     }
 
     /**
      * Calculates the probability of false positives ({@code p}) given
      * numberOfItems ({@code n}), numberOfBits ({@code m}) and numberOfHashFunctions ({@code k}).
      * <pre>p = pow(1 - exp(-k / (m / n)), k)</pre>
      *
      * <p>This is the probability that a Bloom filter will return true for the presence of an item
      * when it does not contain the item.</p>
      *
      * <p>The probability assumes that the Bloom filter is filled with the expected number of
      * items. If the filter contains fewer items then the actual probability will be lower.
      * Thus, this returns the worst-case false positive probability for a filter that has not
      * exceeded its expected number of items.</p>
      *
      * @param numberOfItems the number of items hashed into the Bloom filter.
      * @return the probability of false positives.
      */
     public double getProbability(final int numberOfItems) {
         if (numberOfItems < 0) {
             throw new IllegalArgumentException("Number of items must be greater than or equal to 0: " + numberOfItems);
         }
         if (numberOfItems == 0) {
             return 0;
         }
         return Math.pow(-Math.expm1(-1.0 * numberOfHashFunctions * numberOfItems / numberOfBits), numberOfHashFunctions);
     }
 
     @Override
     public int hashCode() {
         // Match Arrays.hashCode(new int[] {numberOfBits, numberOfHashFunctions})
         return (31 + numberOfBits) * 31 + numberOfHashFunctions;
     }
 
     /**
      * Determines if a cardinality is sparse based on the shape.
      * <p>This method assumes that bit maps are 64bits and indexes are 32bits. If the memory
      * necessary to store the cardinality as indexes is less than the estimated memory for bit maps,
      * the cardinality is determined to be {@code sparse}.</p>
      *
      * @param cardinality the cardinality to check.
      * @return true if the cardinality is sparse within the shape.
      */
     public boolean isSparse(final int cardinality) {
 
         /*
          * Since the size of a bit map is a long and the size of an index is an int,
          * there can be 2 indexes for each bit map. In Bloom filters indexes are evenly
          * distributed across the range of possible values, Thus if the cardinality
          * (number of indexes) is less than or equal to 2*number of bit maps the
          * cardinality is sparse within the shape.
          */
         return cardinality <= BitMaps.numberOfBitMaps(this) * 2;
     }
 
     @Override
     public String toString() {
         return String.format("Shape[k=%s m=%s]", numberOfHashFunctions, numberOfBits);
     }
 }