Class AliasMethodDiscreteSampler

java.lang.Object
org.apache.commons.rng.sampling.distribution.AliasMethodDiscreteSampler
All Implemented Interfaces:
DiscreteSampler, SharedStateDiscreteSampler, SharedStateSampler<SharedStateDiscreteSampler>
public class AliasMethodDiscreteSampler extends Object implements SharedStateDiscreteSampler
Distribution sampler that uses the Alias method. It can be used to sample from n values each with an associated probability. If all unique items are assigned the same probability it is more efficient to use the DiscreteUniformSampler.

This implementation is based on the detailed explanation of the alias method by Keith Schartz and implements Vose's algorithm.

  • Vose, M.D., A linear algorithm for generating random numbers with a given distribution, IEEE Transactions on Software Engineering, 17, 972-975, 1991.

The algorithm will sample values in O(1) time after a pre-processing step of O(n) time.

The alias tables are constructed using fraction probabilities with an assumed denominator of 253. In the generic case sampling uses UniformRandomProvider.nextInt(int) and the upper 53-bits from UniformRandomProvider.nextLong().

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 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 n * 12 for size 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 of(UniformRandomProvider, double[], int) to construct the sampler.

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 UniformRandomProvider.nextInt() to generate up to 64-bits for creation of an 11-bit index and 53-bits for the long. This optimisation requires a generator with a high cycle length for the lower order bits.

Larger table sizes that are a power of 2 will benefit from fast algorithms for UniformRandomProvider.nextInt(int) that exploit the power of 2.

Since:
1.3
See Also:

  • Field Details

    • rng

      protected final UniformRandomProvider rng
      Underlying source of randomness.

    • probability

      protected final long[] probability
      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.

      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.

      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 253 (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 253.

      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: 

      Uniform deviate
 1/2^53    2/2^53    3/2^53    4/2^53
--|---------|---------|---------|---
     ^
     |
 probability
            ^
            |
        rounded up

      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.

      See Also:

    • alias

      protected final int[] alias
      The alias table. During sampling if the random probability is not below the entry in the probability table then the sample is the alias.

  • Method Details

    • sample

      public int sample()
      Creates an int sample.
      Specified by:
      sample in interface DiscreteSampler
      Returns:
      a sample.

    • toString

      public String toString()
      Overrides:
      toString in class Object

    • withUniformRandomProvider

      public SharedStateDiscreteSampler withUniformRandomProvider(UniformRandomProvider rng)
      Create a new instance of the sampler with the same underlying state using the given uniform random provider as the source of randomness.
      Specified by:
      withUniformRandomProvider in interface SharedStateSampler<SharedStateDiscreteSampler>
      Parameters:
      rng - Generator of uniformly distributed random numbers.
      Returns:
      the sampler

    • of

      public static SharedStateDiscreteSampler of(UniformRandomProvider rng, double[] probabilities)
      Creates a sampler.

      The probabilities will be normalised using their sum. The only requirement is the sum is strictly positive.

      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.

      To avoid zero-padding use the of(UniformRandomProvider, double[], int) method with a negative alpha factor.

      Parameters:
      rng - Generator of uniformly distributed random numbers.
      probabilities - The list of probabilities.
      Returns:
      the sampler
      Throws:
      IllegalArgumentException - if probabilities is null or empty, a probability is negative, infinite or NaN, or the sum of all probabilities is not strictly positive.
      See Also:

    • of

      public static SharedStateDiscreteSampler of(UniformRandomProvider rng, double[] probabilities, int alpha)
      Creates a sampler.

      The probabilities will be normalised using their sum. The only requirement is the sum is strictly positive.

      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 alpha argument. Set to negative to disable padding.

      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 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 O(n) time with n the size after padding; an additional memory cost of 4 bytes per padded value.

      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 alpha controls the level of padding. Positive values of alpha represent a scale factor in powers of 2. The size of the input array will be increased by a factor of 2alpha and then rounded-up to the next power of 2. Padding is bounded by the upper limit on the size of an array.

      The chance of executing the slow path is upper bounded at 2-alpha when padding is enabled. Each successive doubling of padding will have diminishing performance gains.

      Parameters:
      rng - Generator of uniformly distributed random numbers.
      probabilities - The list of probabilities.
      alpha - The alpha factor controlling the zero padding.
      Returns:
      the sampler
      Throws:
      IllegalArgumentException - if probabilities is null or empty, a probability is negative, infinite or NaN, or the sum of all probabilities is not strictly positive.