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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.rng.sampling.distribution;
  
  import org.apache.commons.rng.UniformRandomProvider;
  
  import java.util.Arrays;
  
  /**
   * Distribution sampler that uses the <a
   * href="https://en.wikipedia.org/wiki/Alias_method">Alias method</a>. It can be used to
   * sample from {@code n} values each with an associated probability. If all unique items
   * are assigned the same probability it is more efficient to use the {@link DiscreteUniformSampler}.
   *
   * <p>This implementation is based on the detailed explanation of the alias method by
   * Keith Schartz and implements Vose's algorithm.</p>
   *
   * <ul>
   *  <li>
   *   <blockquote>
   *    Vose, M.D.,
   *    <i>A linear algorithm for generating random numbers with a given distribution,</i>
   *     IEEE Transactions on Software Engineering, 17, 972-975, 1991.
   *   </blockquote>
   *  </li>
   * </ul>
   *
   * <p>The algorithm will sample values in {@code O(1)} time after a pre-processing step of
   * {@code O(n)} time.</p>
   *
   * <p>The alias tables are constructed using fraction probabilities with an assumed denominator
   * of 2<sup>53</sup>. In the generic case sampling uses {@link UniformRandomProvider#nextInt(int)}
   * and the upper 53-bits from {@link UniformRandomProvider#nextLong()}.</p>
   *
   * <p>Zero padding the input probabilities can be used to make more sampling more efficient.
   * Any zero entry will always be aliased removing the requirement to compute a {@code long}.
   * Increased sampling speed comes at the cost of increased storage space. The algorithm requires
   * approximately 12 bytes of storage per input probability, that is {@code n * 12} for size
   * {@code n}. Zero-padding only requires 4 bytes of storage per padded value as the probability is
   * known to be zero. A table can be padded to a power of 2 using the utility function
   * {@link #of(UniformRandomProvider, double[], int)} to construct the sampler.</p>
   *
   * <p>An optimisation is performed for small table sizes that are a power of 2. In this case the
   * sampling uses 1 or 2 calls from {@link UniformRandomProvider#nextInt()} to generate up to
   * 64-bits for creation of an 11-bit index and 53-bits for the {@code long}. This optimisation
   * requires a generator with a high cycle length for the lower order bits.</p>
   *
   * <p>Larger table sizes that are a power of 2 will benefit from fast algorithms for
   * {@link UniformRandomProvider#nextInt(int)} that exploit the power of 2.</p>
   *
   * @see <a href="https://en.wikipedia.org/wiki/Alias_method">Alias Method</a>
   * @see <a href="http://www.keithschwarz.com/darts-dice-coins/">Darts, Dice, and Coins:
   * Sampling from a Discrete Distribution by Keith Schwartz</a>
   * @see <a href="https://ieeexplore.ieee.org/document/92917">Vose (1991) IEEE Transactions
   * on Software Engineering 17, 972-975.</a>
   * @since 1.3
   */
  public class AliasMethodDiscreteSampler
      implements SharedStateDiscreteSampler {
      /**
       * The default alpha factor for zero-padding an input probability table. The default
       * value will pad the probabilities by to the next power-of-2.
       */
      private static final int DEFAULT_ALPHA = 0;
      /** The value zero for a {@code double}. */
      private static final double ZERO = 0.0;
      /** The value 1.0 represented as the numerator of a fraction with denominator 2<sup>53</sup>. */
      private static final long ONE_AS_NUMERATOR = 1L << 53;
      /**
       * The multiplier to convert a {@code double} probability in the range {@code [0, 1]}
       * to the numerator of a fraction with denominator 2<sup>53</sup>.
       */
      private static final double CONVERT_TO_NUMERATOR = ONE_AS_NUMERATOR;
      /**
       * The maximum size of the small alias table. This is 2<sup>11</sup>.
       */
      private static final int MAX_SMALL_POWER_2_SIZE = 1 << 11;
  
      /** Underlying source of randomness. */
      protected final UniformRandomProvider rng;
  
      /**
       * The probability table. During sampling a random index into this table is selected.
       * A random probability is compared to the value at this index: if lower then the sample is the
       * index; if higher then the sample uses the corresponding entry in the alias table.
      *
      * <p>This has entries up to the last non-zero element since there is no need to store
      * probabilities of zero. This is an optimisation for zero-padded input. Any zero value will
      * always be aliased so any look-up index outside this table always uses the alias.</p>
      *
      * <p>Note that a uniform double in the range [0,1) can be generated using 53-bits from a long
      * to sample all the dyadic rationals with a denominator of 2<sup>53</sup>
      * (e.g. see org.apache.commons.rng.core.utils.NumberFactory.makeDouble(long)). To avoid
      * computation of a double and comparison to the probability as a double the probabilities are
      * stored as 53-bit longs to use integer arithmetic. This is the equivalent of storing the
      * numerator of a fraction with the denominator of 2<sup>53</sup>.</p>
      *
      * <p>During conversion of the probability to a double it is rounded up to the next integer
      * value. This ensures the functionality of comparing a uniform deviate distributed evenly on
      * the interval 1/2^53 to the unevenly distributed probability is equivalent, i.e. a uniform
      * deviate is either below the probability or above it:
      *
      * <pre>
      * Uniform deviate
      *  1/2^53    2/2^53    3/2^53    4/2^53
      * --|---------|---------|---------|---
      *      ^
      *      |
      *  probability
      *             ^
      *             |
      *         rounded up
      * </pre>
      *
      * <p>Round-up ensures a non-zero probability is always non-zero and zero probability remains
      * zero. Thus any item with a non-zero input probability can always be sampled, and a zero
      * input probability cannot be sampled.</p>
      *
      * @see <a href="https://en.wikipedia.org/wiki/Dyadic_rational">Dyadic rational</a>
      */
     protected final long[] probability;
 
     /**
      * The alias table. During sampling if the random probability is not below the entry in the
      * probability table then the sample is the alias.
      */
     protected final int[] alias;
 
     /**
      * Sample from the computed tables exploiting the small power-of-two table size.
      * This implements a variant of the optimised algorithm as per Vose (1991):
      *
      * <pre>
      * bits = obtained required number of random bits
      * v = (some of the bits) * constant1
      * j = (rest of the bits) * constant2
      * if v &lt; prob[j] then
      *   return j
      * else
      *   return alias[j]
      * </pre>
      *
      * <p>This is a variant because the bits are not multiplied by constants. In the case of
      * {@code v} the constant is a scale that is pre-applied to the probability table. In the
      * case of {@code j} the constant is not used to scale a deviate to an index; the index is
      * from a power-of-2 range and so the bits are used directly.</p>
      *
      * <p>This is implemented using up to 64 bits from the random generator.
      * The index for the table is computed using a mask to extract up to 11 of the lower bits
      * from an integer. The probability is computed using a second integer combined with the
      * remaining bits to create 53-bits for the numerator of a fraction with denominator
      * 2<sup>53</sup>. This is only computed on demand.</p>
      *
      * <p>Note: This supports a table size of up to 2^11, or 2048, exclusive. Any larger requires
      * consuming more than 64-bits and the algorithm is not more efficient than the
      * {@link AliasMethodDiscreteSampler}.</p>
      *
      * <p>Sampling uses 1 or 2 calls to {@link UniformRandomProvider#nextInt()}.</p>
      */
     private static class SmallTableAliasMethodDiscreteSampler extends AliasMethodDiscreteSampler {
         /** The mask to isolate the lower bits. */
         private final int mask;
 
         /**
          * Create a new instance.
          *
          * @param rng Generator of uniformly distributed random numbers.
          * @param probability Probability table.
          * @param alias Alias table.
          */
         SmallTableAliasMethodDiscreteSampler(final UniformRandomProvider rng,
                                              final long[] probability,
                                              final int[] alias) {
             super(rng, probability, alias);
             // Assume the table size is a power of 2 and create the mask
             mask = alias.length - 1;
         }
 
         @Override
         public int sample() {
             final int bits = rng.nextInt();
             // Isolate lower bits
             final int j = bits & mask;
 
             // Optimisation for zero-padded input tables
             if (j >= probability.length) {
                 // No probability must use the alias
                 return alias[j];
             }
 
             // Create a uniform random deviate as a long.
             // This replicates functionality from the o.a.c.rng.core.utils.NumberFactory.makeLong
             final long longBits = (((long) rng.nextInt()) << 32) | (bits & 0xffffffffL);
 
             // Choose between the two. Use a 53-bit long for the probability.
             return (longBits >>> 11) < probability[j] ? j : alias[j];
         }
 
         /** {@inheritDoc} */
         @Override
         public SharedStateDiscreteSampler withUniformRandomProvider(UniformRandomProvider rng) {
             return new SmallTableAliasMethodDiscreteSampler(rng, probability, alias);
         }
     }
 
     /**
      * Creates a sampler.
      *
      * <p>The input parameters are not validated and must be correctly computed alias tables.</p>
      *
      * @param rng Generator of uniformly distributed random numbers.
      * @param probability Probability table.
      * @param alias Alias table.
      */
     private AliasMethodDiscreteSampler(final UniformRandomProvider rng,
                                        final long[] probability,
                                        final int[] alias) {
         this.rng = rng;
         // Deliberate direct storage of input arrays
         this.probability = probability;
         this.alias = alias;
     }
 
     /** {@inheritDoc} */
     @Override
     public int sample() {
         // This implements the algorithm as per Vose (1991):
         // v = uniform()  in [0, 1)
         // j = uniform(n) in [0, n)
         // if v < prob[j] then
         //   return j
         // else
         //   return alias[j]
 
         final int j = rng.nextInt(alias.length);
 
         // Optimisation for zero-padded input tables
         if (j >= probability.length) {
             // No probability must use the alias
             return alias[j];
         }
 
         // Note: We could check the probability before computing a deviate.
         // p(j) == 0  => alias[j]
         // p(j) == 1  => j
         // However it is assumed these edge cases are rare:
         //
         // The probability table will be 1 for approximately 1/n samples, i.e. only the
         // last unpaired probability. This is only worth checking for when the table size (n)
         // is small. But in that case the user should zero-pad the table for performance.
         //
         // The probability table will be 0 when an input probability was zero. We
         // will assume this is also rare if modelling a discrete distribution where
         // all samples are possible. The edge case for zero-padded tables is handled above.
 
         // Choose between the two. Use a 53-bit long for the probability.
         return (rng.nextLong() >>> 11) < probability[j] ? j : alias[j];
     }
 
     /** {@inheritDoc} */
     @Override
     public String toString() {
         return "Alias method [" + rng.toString() + "]";
     }
 
     /** {@inheritDoc} */
     @Override
     public SharedStateDiscreteSampler withUniformRandomProvider(UniformRandomProvider rng) {
         return new AliasMethodDiscreteSampler(rng, probability, alias);
     }
 
     /**
      * Creates a sampler.
      *
      * <p>The probabilities will be normalised using their sum. The only requirement
      * is the sum is strictly positive.</p>
      *
      * <p>Where possible this method zero-pads the probabilities so the length is the next
      * power-of-two. Padding is bounded by the upper limit on the size of an array.</p>
      *
      * <p>To avoid zero-padding use the
      * {@link #of(UniformRandomProvider, double[], int)} method with a negative
      * {@code alpha} factor.</p>
      *
      * @param rng Generator of uniformly distributed random numbers.
      * @param probabilities The list of probabilities.
      * @return the sampler
      * @throws IllegalArgumentException if {@code probabilities} is null or empty, a
      * probability is negative, infinite or {@code NaN}, or the sum of all
      * probabilities is not strictly positive.
      * @see #of(UniformRandomProvider, double[], int)
      */
     public static SharedStateDiscreteSampler of(final UniformRandomProvider rng,
                                                 final double[] probabilities) {
         return of(rng, probabilities, DEFAULT_ALPHA);
     }
 
     /**
      * Creates a sampler.
      *
      * <p>The probabilities will be normalised using their sum. The only requirement
      * is the sum is strictly positive.</p>
      *
      * <p>Where possible this method zero-pads the probabilities to improve sampling
      * efficiency. Padding is bounded by the upper limit on the size of an array and
      * controlled by the {@code alpha} argument. Set to negative to disable
      * padding.</p>
      *
      * <p>For each zero padded value an entry is added to the tables which is always
      * aliased. This can be sampled with fewer bits required from the
      * {@link UniformRandomProvider}. Increasing the padding of zeros increases the
      * chance of using this fast path to selecting a sample. The penalty is
      * two-fold: initialisation is bounded by {@code O(n)} time with {@code n} the
      * size <strong>after</strong> padding; an additional memory cost of 4 bytes per
      * padded value.</p>
      *
      * <p>Zero padding to any length improves performance; using a power of 2 allows
      * the index into the tables to be more efficiently generated. The argument
      * {@code alpha} controls the level of padding. Positive values of {@code alpha}
      * represent a scale factor in powers of 2. The size of the input array will be
      * increased by a factor of 2<sup>alpha</sup> and then rounded-up to the next
      * power of 2. Padding is bounded by the upper limit on the size of an
      * array.</p>
      *
      * <p>The chance of executing the slow path is upper bounded at
      * 2<sup>-alpha</sup> when padding is enabled. Each successive doubling of
      * padding will have diminishing performance gains.</p>
      *
      * @param rng Generator of uniformly distributed random numbers.
      * @param probabilities The list of probabilities.
      * @param alpha The alpha factor controlling the zero padding.
      * @return the sampler
      * @throws IllegalArgumentException if {@code probabilities} is null or empty, a
      * probability is negative, infinite or {@code NaN}, or the sum of all
      * probabilities is not strictly positive.
      */
     public static SharedStateDiscreteSampler of(final UniformRandomProvider rng,
                                                 final double[] probabilities,
                                                 int alpha) {
         // The Alias method balances N categories with counts around the mean into N sections,
         // each allocated 'mean' observations.
         //
         // Consider 4 categories with counts 6,3,2,1. The histogram can be balanced into a
         // 2D array as 4 sections with a height of the mean:
         //
         // 6
         // 6
         // 6
         // 63   => 6366   --
         // 632     6326    |-- mean
         // 6321    6321   --
         //
         // section abcd
         //
         // Each section is divided as:
         // a: 6=1/1
         // b: 3=1/1
         // c: 2=2/3; 6=1/3   (6 is the alias)
         // d: 1=1/3; 6=2/3   (6 is the alias)
         //
         // The sample is obtained by randomly selecting a section, then choosing which category
         // from the pair based on a uniform random deviate.
 
         final double sumProb = InternalUtils.validateProbabilities(probabilities);
 
         // Allow zero-padding
         final int n = computeSize(probabilities.length, alpha);
 
         // Partition into small and large by splitting on the average.
         final double mean = sumProb / n;
         // The cardinality of smallSize + largeSize = n.
         // So fill the same array from either end.
         final int[] indices = new int[n];
         int large = n;
         int small = 0;
         for (int i = 0; i < probabilities.length; i++) {
             if (probabilities[i] >= mean) {
                 indices[--large] = i;
             } else {
                 indices[small++] = i;
             }
         }
 
         small = fillRemainingIndices(probabilities.length, indices, small);
 
         // This may be smaller than the input length if the probabilities were already padded.
         final int nonZeroIndex = findLastNonZeroIndex(probabilities);
 
         // The probabilities are modified so use a copy.
         // Note: probabilities are required only up to last nonZeroIndex
         final double[] remainingProbabilities = Arrays.copyOf(probabilities, nonZeroIndex + 1);
 
         // Allocate the final tables.
         // Probability table may be truncated (when zero padded).
         // The alias table is full length.
         final long[] probability = new long[remainingProbabilities.length];
         final int[] alias = new int[n];
 
         // This loop uses each large in turn to fill the alias table for small probabilities that
         // do not reach the requirement to fill an entire section alone (i.e. p < mean).
         // Since the sum of the small should be less than the sum of the large it should use up
         // all the small first. However floating point round-off can result in
         // misclassification of items as small or large. The Vose algorithm handles this using
         // a while loop conditioned on the size of both sets and a subsequent loop to use
         // unpaired items.
         while (large != n && small != 0) {
             // Index of the small and the large probabilities.
             final int j = indices[--small];
             final int k = indices[large++];
 
             // Optimisation for zero-padded input:
             // p(j) = 0 above the last nonZeroIndex
             if (j > nonZeroIndex) {
                 // The entire amount for the section is taken from the alias.
                 remainingProbabilities[k] -= mean;
             } else {
                 final double pj = remainingProbabilities[j];
 
                 // Item j is a small probability that is below the mean.
                 // Compute the weight of the section for item j: pj / mean.
                 // This is scaled by 2^53 and the ceiling function used to round-up
                 // the probability to a numerator of a fraction in the range [1,2^53].
                 // Ceiling ensures non-zero values.
                 probability[j] = (long) Math.ceil(CONVERT_TO_NUMERATOR * (pj / mean));
 
                 // The remaining amount for the section is taken from the alias.
                 // Effectively: probabilities[k] -= (mean - pj)
                 remainingProbabilities[k] += pj - mean;
             }
 
             // If not j then the alias is k
             alias[j] = k;
 
             // Add the remaining probability from large to the appropriate list.
             if (remainingProbabilities[k] >= mean) {
                 indices[--large] = k;
             } else {
                 indices[small++] = k;
             }
         }
 
         // Final loop conditions to consume unpaired items.
         // Note: The large set should never be non-empty but this can occur due to round-off
         // error so consume from both.
         fillTable(probability, alias, indices, 0, small);
         fillTable(probability, alias, indices, large, n);
 
         // Change the algorithm for small power of 2 sized tables
         return isSmallPowerOf2(n) ?
             new SmallTableAliasMethodDiscreteSampler(rng, probability, alias) :
             new AliasMethodDiscreteSampler(rng, probability, alias);
     }
 
     /**
      * Allocate the remaining indices from zero padding as small probabilities. The
      * number to add is from the length of the probability array to the length of
      * the padded probability array (which is the same length as the indices array).
      *
      * @param length Length of probability array.
      * @param indices Indices.
      * @param small Number of small indices.
      * @return the updated number of small indices
      */
     private static int fillRemainingIndices(final int length, final int[] indices, int small) {
         int updatedSmall = small;
         for (int i = length; i < indices.length; i++) {
             indices[updatedSmall++] = i;
         }
         return updatedSmall;
     }
 
     /**
      * Find the last non-zero index in the probabilities. This may be smaller than
      * the input length if the probabilities were already padded.
      *
      * @param probabilities The list of probabilities.
      * @return the index
      */
     private static int findLastNonZeroIndex(final double[] probabilities) {
         // No bounds check is performed when decrementing as the array contains at least one
         // value above zero.
         int nonZeroIndex = probabilities.length - 1;
         while (probabilities[nonZeroIndex] == ZERO) {
             nonZeroIndex--;
         }
         return nonZeroIndex;
     }
 
     /**
      * Compute the size after padding. A value of {@code alpha < 0} disables
      * padding. Otherwise the length will be increased by 2<sup>alpha</sup>
      * rounded-up to the next power of 2.
      *
      * @param length Length of probability array.
      * @param alpha The alpha factor controlling the zero padding.
      * @return the padded size
      */
     private static int computeSize(int length, int alpha) {
         if (alpha < 0) {
             // No padding
             return length;
         }
         // Use the number of leading zeros function to find the next power of 2,
         // i.e. ceil(log2(x))
         int pow2 = 32 - Integer.numberOfLeadingZeros(length - 1);
         // Increase by the alpha. Clip this to limit to a positive integer (2^30)
         pow2 = Math.min(30, pow2 + alpha);
         // Use max to handle a length above the highest possible power of 2
         return Math.max(length, 1 << pow2);
     }
 
     /**
      * Fill the tables using unpaired items that are in the range between {@code start} inclusive
      * and {@code end} exclusive.
      *
      * <p>Anything left must fill the entire section so the probability table is set
      * to 1 and there is no alias. This will occur for 1/n samples, i.e. the last
      * remaining unpaired probability. Note: When the tables are zero-padded the
      * remaining indices are from an input probability that is above zero so the
      * index will be allowed in the truncated probability array and no
      * index-out-of-bounds exception will occur.
      *
      * @param probability Probability table.
      * @param alias Alias table.
      * @param indices Unpaired indices.
      * @param start Start position.
      * @param end End position.
      */
     private static void fillTable(long[] probability, int[] alias, int[] indices, int start, int end) {
         for (int i = start; i < end; i++) {
             final int index = indices[i];
             probability[index] = ONE_AS_NUMERATOR;
             alias[index] = index;
         }
     }
 
     /**
      * Checks if the size is a small power of 2 so can be supported by the
      * {@link SmallTableAliasMethodDiscreteSampler}.
      *
      * @param n Size of the alias table.
      * @return true if supported by {@link SmallTableAliasMethodDiscreteSampler}
      */
     private static boolean isSmallPowerOf2(int n) {
         return n <= MAX_SMALL_POWER_2_SIZE && (n & (n - 1)) == 0;
     }
 }