AliasMethodDiscreteSampler.java
- /*
- * 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.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 < 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 final 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.
- */
- 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;
- }
- }