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https://issues.apache.org/jira/browse/RNG-95?page=com.atlassian.jira.plugin.system.issuetabpanels:comment-tabpanel&focusedCommentId=16886537#comment-16886537
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Alex D Herbert commented on RNG-95:
-----------------------------------
This method has been documented by Danial Lamire:
[Fast Random Integer Generation in an Interval|https://arxiv.org/abs/1805.10941]
> DiscreteUniformSampler
> ----------------------
>
> Key: RNG-95
> URL: https://issues.apache.org/jira/browse/RNG-95
> Project: Commons RNG
> Issue Type: Improvement
> Components: sampling
> Affects Versions: 1.3
> Reporter: Alex D Herbert
> Assignee: Alex D Herbert
> Priority: Minor
>
> The {{DiscreteUniformSampler}} delegates the creation of an integer in the
> range {{[0, n)}} to the {{UniformRandomProvider}}.
> This sampler will be repeatedly used to sample the same range. The default
> method in {{BaseProvider}} uses a dynamic algorithm that handles {{n}}
> differently when a power of 2.
> When the range is a power of 2 the method can use a series of bits from a
> random integer to generate a uniform integer in the range. This is fast.
> When the range is not a power of 2 the algorithm must reject samples when the
> sample would result in an over-representation of a particular value in the
> uniform range. This is necessary as {{n}} does not exactly fit into the
> number of possible values {{[0, 2^31)}} that can be produced by the generator
> (when using 31-bit signed integers). The rejection method uses integer
> arithmetic to determine the number of samples that fit into the range:
> {{samples = 2^31 / n}}. Extra samples that lead to over-representation are
> rejected: {{extra = 2^31 % n}}.
> Since {{n}} will not change a pre-computation step is possible to select the
> best algorithm.
> n is a power of 2:
> {code:java}
> // Favouring the least significant bits
> // Pre-compute
> int mask = n - 1;
> return nextInt() & mask;
> // Or favouring the most significant bits
> // Pre-compute
> int shift = Integer.numberOfLeadingZeros(n) + 1;
> return nextInt() >>> shift;
> {code}
> n is not a power of 2:
> {code:java}
> // Sample using modulus
> // Pre-compute
> final int fence = (int)(0x80000000L - 0x80000000L % n - 1);
> int bits;
> do {
> bits = rng.nextInt() >>> 1;
> } while (bits > fence);
> return bits % n;
> // Or using 32-bit unsigned arithmetic avoiding modulus
> // Pre-compute
> final long fence = (1L << 32) % n;
> long result;
> do {
> // Compute 64-bit unsigned product of n * [0, 2^32 - 1)
> result = n * (rng.nextInt() & 0xffffffffL);
> // Test the sample uniformity.
> } while ((result & 0xffffffffL) < fence);
> // Divide by 2^32 to get the sample
> return (int)(result >>> 32);
> {code}
> The second method uses a range of 2^32 instead of 2^31 so reducing the
> rejection probability and avoids the modulus operator; these both increase
> speed.
> Note algorithm 1 returns sample values in a repeat cycle from all values in
> the range {{[0, 2^31)}} due to the use of modulus, e.g.
> {noformat}
> 0, 1, 2, ..., 0, 1, 2, ...
> {noformat}
> Algorithm 2 returns sample values in a linear order, e.g.
> {noformat}
> 0, 0, 1, 1, 2, 2, ...
> {noformat}
> The suggested change is to implement smart pre-computation in the
> {{DiscreteUniformSampler}} based on the range and use the algorithms that
> favour the most significant bits from the generator.
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