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package org.apache.commons.math3.random;

import org.apache.commons.math3.exception.DimensionMismatchException;
import org.apache.commons.math3.exception.NotPositiveException;
import org.apache.commons.math3.exception.NullArgumentException;
import org.apache.commons.math3.exception.OutOfRangeException;
import org.apache.commons.math3.util.MathUtils;

Implementation of a Halton sequence.

A Halton sequence is a low-discrepancy sequence generating points in the interval [0, 1] according to
  H(n) = d_0 / b + d_1 / b^2 .... d_j / b^j+1
  with
  n = d_j * b^j-1 + ... d_1 * b + d_0 * b^0

For higher dimensions, subsequent prime numbers are used as base, e.g. { 2, 3, 5 } for a Halton sequence in R^3.

Halton sequences are known to suffer from linear correlation for larger prime numbers, thus the individual digits
are usually scrambled. This implementation already comes with support for up to 40 dimensions with optimal weight
numbers from 
H. Chi: Scrambled quasirandom sequences and their applications.

The generator supports two modes:

  
sequential generation of points: nextVector()
  
random access to the i-th point in the sequence: skipTo(int)

See Also:
Halton sequence (Wikipedia)

 * On the Halton sequence and its scramblings
Since:
3.3/**
 * Implementation of a Halton sequence.
 * <p>
 * A Halton sequence is a low-discrepancy sequence generating points in the interval [0, 1] according to
 * <pre>
 *   H(n) = d_0 / b + d_1 / b^2 .... d_j / b^j+1
 *
 *   with
 *
 *   n = d_j * b^j-1 + ... d_1 * b + d_0 * b^0
 * </pre>
 * For higher dimensions, subsequent prime numbers are used as base, e.g. { 2, 3, 5 } for a Halton sequence in R^3.
 * <p>
 * Halton sequences are known to suffer from linear correlation for larger prime numbers, thus the individual digits
 * are usually scrambled. This implementation already comes with support for up to 40 dimensions with optimal weight
 * numbers from <a href="http://etd.lib.fsu.edu/theses/available/etd-07062004-140409/unrestricted/dissertation1.pdf">
 * H. Chi: Scrambled quasirandom sequences and their applications</a>.
 * <p>
 * The generator supports two modes:
 * <ul>
 *   <li>sequential generation of points: {@link #nextVector()}</li>
 *   <li>random access to the i-th point in the sequence: {@link #skipTo(int)}</li>
 * </ul>
 *
 * @see <a href="http://en.wikipedia.org/wiki/Halton_sequence">Halton sequence (Wikipedia)</a>
 * @see <a href="https://lirias.kuleuven.be/bitstream/123456789/131168/1/mcm2005_bartv.pdf">
 * On the Halton sequence and its scramblings</a>
 * @since 3.3
 */
public class HaltonSequenceGenerator implements RandomVectorGenerator {

    
The first 40 primes. /** The first 40 primes. */
    private static final int[] PRIMES = new int[] {
        2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67,
        71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139,
        149, 151, 157, 163, 167, 173
    };

    
The optimal weights used for scrambling of the first 40 dimension. /** The optimal weights used for scrambling of the first 40 dimension. */
    private static final int[] WEIGHTS = new int[] {
        1, 2, 3, 3, 8, 11, 12, 14, 7, 18, 12, 13, 17, 18, 29, 14, 18, 43, 41,
        44, 40, 30, 47, 65, 71, 28, 40, 60, 79, 89, 56, 50, 52, 61, 108, 56,
        66, 63, 60, 66
    };

    
Space dimension. /** Space dimension. */
    private final int dimension;

    
The current index in the sequence. /** The current index in the sequence. */
    private int count = 0;

    
The base numbers for each component. /** The base numbers for each component. */
    private final int[] base;

    
The scrambling weights for each component. /** The scrambling weights for each component. */
    private final int[] weight;

    
Construct a new Halton sequence generator for the given space dimension.
Params:
dimension – the space dimension
Throws:
OutOfRangeException – if the space dimension is outside the allowed range of [1, 40]/**
     * Construct a new Halton sequence generator for the given space dimension.
     *
     * @param dimension the space dimension
     * @throws OutOfRangeException if the space dimension is outside the allowed range of [1, 40]
     */
    public HaltonSequenceGenerator(final int dimension) throws OutOfRangeException {
        this(dimension, PRIMES, WEIGHTS);
    }

    
Construct a new Halton sequence generator with the given base numbers and weights for each dimension.
The length of the bases array defines the space dimension and is required to be > 0.
Params:
dimension – the space dimension
bases – the base number for each dimension, entries should be (pairwise) prime, may not be null
weights – the weights used during scrambling, may be null in which case no scrambling will be performed
Throws:
NullArgumentException – if base is null
OutOfRangeException – if the space dimension is outside the range [1, len], where
  len refers to the length of the bases array
DimensionMismatchException – if weights is non-null and the length of the input arrays differ/**
     * Construct a new Halton sequence generator with the given base numbers and weights for each dimension.
     * The length of the bases array defines the space dimension and is required to be &gt; 0.
     *
     * @param dimension the space dimension
     * @param bases the base number for each dimension, entries should be (pairwise) prime, may not be null
     * @param weights the weights used during scrambling, may be null in which case no scrambling will be performed
     * @throws NullArgumentException if base is null
     * @throws OutOfRangeException if the space dimension is outside the range [1, len], where
     *   len refers to the length of the bases array
     * @throws DimensionMismatchException if weights is non-null and the length of the input arrays differ
     */
    public HaltonSequenceGenerator(final int dimension, final int[] bases, final int[] weights)
            throws NullArgumentException, OutOfRangeException, DimensionMismatchException {

        MathUtils.checkNotNull(bases);

        if (dimension < 1 || dimension > bases.length) {
            throw new OutOfRangeException(dimension, 1, PRIMES.length);
        }

        if (weights != null && weights.length != bases.length) {
            throw new DimensionMismatchException(weights.length, bases.length);
        }

        this.dimension = dimension;
        this.base = bases.clone();
        this.weight = weights == null ? null : weights.clone();
        count = 0;
    }

    
{@inheritDoc} /** {@inheritDoc} */
    public double[] nextVector() {
        final double[] v = new double[dimension];
        for (int i = 0; i < dimension; i++) {
            int index = count;
            double f = 1.0 / base[i];

            int j = 0;
            while (index > 0) {
                final int digit = scramble(i, j, base[i], index % base[i]);
                v[i] += f * digit;
                index /= base[i]; // floor( index / base )
                f /= base[i];
            }
        }
        count++;
        return v;
    }

    
Performs scrambling of digit d_j according to the formula: 
  ( weight_i * d_j ) mod base

Implementations can override this method to do a different scrambling.
Params:
i – the dimension index
j – the digit index
b – the base for this dimension
digit – the j-th digit
Returns:
the scrambled digit/**
     * Performs scrambling of digit {@code d_j} according to the formula:
     * <pre>
     *   ( weight_i * d_j ) mod base
     * </pre>
     * Implementations can override this method to do a different scrambling.
     *
     * @param i the dimension index
     * @param j the digit index
     * @param b the base for this dimension
     * @param digit the j-th digit
     * @return the scrambled digit
     */
    protected int scramble(final int i, final int j, final int b, final int digit) {
        return weight != null ? (weight[i] * digit) % b : digit;
    }

    
Skip to the i-th point in the Halton sequence.

This operation can be performed in O(1).
Params:
index – the index in the sequence to skip to
Throws:
NotPositiveException – if index < 0
Returns:
the i-th point in the Halton sequence/**
     * Skip to the i-th point in the Halton sequence.
     * <p>
     * This operation can be performed in O(1).
     *
     * @param index the index in the sequence to skip to
     * @return the i-th point in the Halton sequence
     * @throws NotPositiveException if index &lt; 0
     */
    public double[] skipTo(final int index) throws NotPositiveException {
        count = index;
        return nextVector();
    }

    
Returns the index i of the next point in the Halton sequence that will be returned by calling nextVector(). 
Returns:
the index of the next point/**
     * Returns the index i of the next point in the Halton sequence that will be returned
     * by calling {@link #nextVector()}.
     *
     * @return the index of the next point
     */
    public int getNextIndex() {
        return count;
    }

}