 # DiscreteUniformSampler

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#### Details

• Improvement
• Status: Closed
• Minor
• Resolution: Implemented
• 1.3
• None

#### Description

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:

```// Favouring the least significant bits

// Pre-compute
int mask = n - 1;

// Or favouring the most significant bits

// Pre-compute
int shift = Integer.numberOfLeadingZeros(n) + 1;

return nextInt() >>> shift;
```

n is not a power of 2:

```// 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);
```

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.

```0, 1, 2, ..., 0, 1, 2, ...
```

Algorithm 2 returns sample values in a linear order, e.g.

```0, 0, 1, 1, 2, 2, ...
```

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.

#### People Alex Herbert Alex Herbert
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#### Dates

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Resolved:

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