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//! A fast generator of discrete, bounded
//! [Zipf-distributed](https://en.wikipedia.org/wiki/Zipf's_law) random numbers.
//!
//! For a random variable `X` whose values are distributed according to this distribution, the
//! probability mass function is given by
//!
//! ```ignore
//! P(X = k) = H(N,s) * 1 / k^s for k = 1,2,...,N
//! ```
//!
//! `H(N,s)` is the normalizing constant which corresponds to the generalized harmonic number
//! of order `N` of `s`.
//!
//! # Example
//!
//! ```
//! use rand::distributions::Distribution;
//!
//! let mut rng = rand::thread_rng();
//! let mut zipf = zipf::ZipfDistribution::new(1000, 1.03).unwrap();
//! let sample = zipf.sample(&mut rng);
//! ```
//!
//! This implementation is effectively a direct port of Apache Common's
//! [RejectionInversionZipfSampler][ref], written in Java. It is based on the method described by
//! Wolfgang Hörmann and Gerhard Derflinger in [*Rejection-inversion to generate variates from
//! monotone discrete distributions*](https://dl.acm.org/citation.cfm?id=235029) from *ACM
//! Transactions on Modeling and Computer Simulation (TOMACS) 6.3 (1996)*.
//!
//! [ref]: https://github.com/apache/commons-rng/blob/6a1b0c16090912e8fc5de2c1fb5bd8490ac14699/commons-rng-sampling/src/main/java/org/apache/commons/rng/sampling/distribution/RejectionInversionZipfSampler.java
#![warn(rust_2018_idioms)]
use rand::Rng;
/// Random number generator that generates Zipf-distributed random numbers using rejection
/// inversion.
#[derive(Clone, Copy)]
pub struct ZipfDistribution {
/// Number of elements
num_elements: f64,
/// Exponent parameter of the distribution
exponent: f64,
/// `hIntegral(1.5) - 1}`
h_integral_x1: f64,
/// `hIntegral(num_elements + 0.5)}`
h_integral_num_elements: f64,
/// `2 - hIntegralInverse(hIntegral(2.5) - h(2)}`
s: f64,
}
impl ZipfDistribution {
/// Creates a new [Zipf-distributed](https://en.wikipedia.org/wiki/Zipf's_law)
/// random number generator.
///
/// Note that both the number of elements and the exponent must be greater than 0.
pub fn new(num_elements: usize, exponent: f64) -> Result<Self, ()> {
if num_elements == 0 {
return Err(());
}
if exponent <= 0f64 {
return Err(());
}
let z = ZipfDistribution {
num_elements: num_elements as f64,
exponent,
h_integral_x1: ZipfDistribution::h_integral(1.5, exponent) - 1f64,
h_integral_num_elements: ZipfDistribution::h_integral(
num_elements as f64 + 0.5,
exponent,
),
s: 2f64
- ZipfDistribution::h_integral_inv(
ZipfDistribution::h_integral(2.5, exponent)
- ZipfDistribution::h(2f64, exponent),
exponent,
),
};
// populate cache
Ok(z)
}
}
impl ZipfDistribution {
fn next<R: Rng + ?Sized>(&self, rng: &mut R) -> usize {
// The paper describes an algorithm for exponents larger than 1 (Algorithm ZRI).
//
// The original method uses
// H(x) = (v + x)^(1 - q) / (1 - q)
// as the integral of the hat function.
//
// This function is undefined for q = 1, which is the reason for the limitation of the
// exponent.
//
// If instead the integral function
// H(x) = ((v + x)^(1 - q) - 1) / (1 - q)
// is used, for which a meaningful limit exists for q = 1, the method works for all
// positive exponents.
//
// The following implementation uses v = 0 and generates integral number in the range [1,
// num_elements]. This is different to the original method where v is defined to
// be positive and numbers are taken from [0, i_max]. This explains why the implementation
// looks slightly different.
let hnum = self.h_integral_num_elements;
loop {
use std::cmp;
let u: f64 = hnum + rng.gen::<f64>() * (self.h_integral_x1 - hnum);
// u is uniformly distributed in (h_integral_x1, h_integral_num_elements]
let x: f64 = ZipfDistribution::h_integral_inv(u, self.exponent);
// Limit k to the range [1, num_elements] if it would be outside
// due to numerical inaccuracies.
let k64 = x.max(1.0).min(self.num_elements);
// float -> integer rounds towards zero, so we add 0.5
// to prevent bias towards k == 1
let k = cmp::max(1, (k64 + 0.5) as usize);
// Here, the distribution of k is given by:
//
// P(k = 1) = C * (hIntegral(1.5) - h_integral_x1) = C
// P(k = m) = C * (hIntegral(m + 1/2) - hIntegral(m - 1/2)) for m >= 2
//
// where C = 1 / (h_integral_num_elements - h_integral_x1)
if k64 - x <= self.s
|| u >= ZipfDistribution::h_integral(k64 + 0.5, self.exponent)
- ZipfDistribution::h(k64, self.exponent)
{
// Case k = 1:
//
// The right inequality is always true, because replacing k by 1 gives
// u >= hIntegral(1.5) - h(1) = h_integral_x1 and u is taken from
// (h_integral_x1, h_integral_num_elements].
//
// Therefore, the acceptance rate for k = 1 is P(accepted | k = 1) = 1
// and the probability that 1 is returned as random value is
// P(k = 1 and accepted) = P(accepted | k = 1) * P(k = 1) = C = C / 1^exponent
//
// Case k >= 2:
//
// The left inequality (k - x <= s) is just a short cut
// to avoid the more expensive evaluation of the right inequality
// (u >= hIntegral(k + 0.5) - h(k)) in many cases.
//
// If the left inequality is true, the right inequality is also true:
// Theorem 2 in the paper is valid for all positive exponents, because
// the requirements h'(x) = -exponent/x^(exponent + 1) < 0 and
// (-1/hInverse'(x))'' = (1+1/exponent) * x^(1/exponent-1) >= 0
// are both fulfilled.
// Therefore, f(x) = x - hIntegralInverse(hIntegral(x + 0.5) - h(x))
// is a non-decreasing function. If k - x <= s holds,
// k - x <= s + f(k) - f(2) is obviously also true which is equivalent to
// -x <= -hIntegralInverse(hIntegral(k + 0.5) - h(k)),
// -hIntegralInverse(u) <= -hIntegralInverse(hIntegral(k + 0.5) - h(k)),
// and finally u >= hIntegral(k + 0.5) - h(k).
//
// Hence, the right inequality determines the acceptance rate:
// P(accepted | k = m) = h(m) / (hIntegrated(m+1/2) - hIntegrated(m-1/2))
// The probability that m is returned is given by
// P(k = m and accepted) = P(accepted | k = m) * P(k = m)
// = C * h(m) = C / m^exponent.
//
// In both cases the probabilities are proportional to the probability mass
// function of the Zipf distribution.
return k;
}
}
}
}
impl rand::distributions::Distribution<usize> for ZipfDistribution {
fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> usize {
self.next(rng)
}
}
use std::fmt;
impl fmt::Debug for ZipfDistribution {
fn fmt(&self, f: &mut fmt::Formatter<'_>) -> Result<(), fmt::Error> {
f.debug_struct("ZipfDistribution")
.field("e", &self.exponent)
.field("n", &self.num_elements)
.finish()
}
}
impl ZipfDistribution {
/// Computes `H(x)`, defined as
///
/// - `(x^(1 - exponent) - 1) / (1 - exponent)`, if `exponent != 1`
/// - `log(x)`, if `exponent == 1`
///
/// `H(x)` is an integral function of `h(x)`, the derivative of `H(x)` is `h(x)`.
fn h_integral(x: f64, exponent: f64) -> f64 {
let log_x = x.ln();
helper2((1f64 - exponent) * log_x) * log_x
}
/// Computes `h(x) = 1 / x^exponent`
fn h(x: f64, exponent: f64) -> f64 {
(-exponent * x.ln()).exp()
}
/// The inverse function of `H(x)`.
/// Returns the `y` for which `H(y) = x`.
fn h_integral_inv(x: f64, exponent: f64) -> f64 {
let mut t: f64 = x * (1f64 - exponent);
if t < -1f64 {
// Limit value to the range [-1, +inf).
// t could be smaller than -1 in some rare cases due to numerical errors.
t = -1f64;
}
(helper1(t) * x).exp()
}
}
/// Helper function that calculates `log(1 + x) / x`.
/// A Taylor series expansion is used, if x is close to 0.
fn helper1(x: f64) -> f64 {
if x.abs() > 1e-8 {
x.ln_1p() / x
} else {
1f64 - x * (0.5 - x * (1.0 / 3.0 - 0.25 * x))
}
}
/// Helper function to calculate `(exp(x) - 1) / x`.
/// A Taylor series expansion is used, if x is close to 0.
fn helper2(x: f64) -> f64 {
if x.abs() > 1e-8 {
x.exp_m1() / x
} else {
1f64 + x * 0.5 * (1f64 + x * 1.0 / 3.0 * (1f64 + 0.25 * x))
}
}
#[cfg(test)]
mod test {
use super::ZipfDistribution;
use rand::distributions::Distribution;
#[inline]
fn test(alpha: f64) {
const N: usize = 100;
// as the alpha increases, we need more samples, since the frequency in the tail grows so low
let samples = (2.0f64.powf(alpha) * 5000000.0) as usize;
let mut rng = rand::thread_rng();
let zipf = ZipfDistribution::new(N, alpha).unwrap();
let harmonic: f64 = (1..=N).map(|n| 1.0 / (n as f64).powf(alpha)).sum();
// sample a bunch
let mut buckets = vec![0; N];
for _ in 0..samples {
let sample = zipf.sample(&mut rng);
buckets[sample - 1] += 1;
}
// for each bucket, see that frequency roughly matches what we expect
// note that the ratios here are ratios _of fractions_, so 0.5 does not mean we're off by
// 50%, it means we're off by 50% _of the frequency_. in other words, if the frequency was
// supposed to be 0.10, and it was actually 0.05, that's a 50% difference.
for i in 0..N {
let freq = buckets[i] as f64 / samples as f64;
let expected = (1.0 / ((i + 1) as f64).powf(alpha)) / harmonic;
// println!("{} {} {}", i + 1, freq, expected);
let off_by = (expected - freq).abs();
assert!(off_by < 0.1); // never be off by more than 10% in absolute terms
let good = off_by < expected;
if !good {
panic!("got {}, expected {} for k = {}", freq, expected, i + 1);
}
}
}
#[test]
fn one() {
test(1.00);
}
#[test]
fn two() {
test(2.00);
}
#[test]
fn three() {
test(3.00);
}
#[test]
fn float() {
test(1.08);
}
#[test]
fn debug() {
eprintln!("{:?}", ZipfDistribution::new(100, 1.0).unwrap());
}
#[test]
fn errs() {
ZipfDistribution::new(0, 1.0).unwrap_err();
ZipfDistribution::new(100, 0.0).unwrap_err();
ZipfDistribution::new(100, -1.0).unwrap_err();
}
}