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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`. //! //! //! This implementation is effectively a direct port of Apache Common's //! [RejectionInversionZipfSampler](https://github.com/apache/commons-rng/blob/6a1b0c16090912e8fc5de2c1fb5bd8490ac14699/commons-rng-sampling/src/main/java/org/apache/commons/rng/sampling/distribution/RejectionInversionZipfSampler.java), //! 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)*. extern crate rand; use rand::Rng; /// Random number generator that generates Zipf-distributed random numbers using rejection /// inversion. pub struct ZipfDistribution<R> { /// Number of elements num_elements: isize, /// Exponent parameter of the distribution exponent: f64, /// `hIntegral(1.5) - 1}` h_integral_x1: Option<f64>, /// `hIntegral(num_elements + 0.5)}` h_integral_num_elements: Option<f64>, /// `2 - hIntegralInverse(hIntegral(2.5) - h(2)}` s: Option<f64>, /// Feeding random number generator sampler: R, } impl<R: Rng> ZipfDistribution<R> { /// 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(sampler: R, num_elements: usize, exponent: f64) -> Result<Self, ()> { if num_elements == 0 { return Err(()); } if exponent <= 0f64 { return Err(()); } let mut z = ZipfDistribution { num_elements: num_elements as isize, exponent: exponent, h_integral_x1: None, h_integral_num_elements: None, s: None, sampler: sampler, }; // populate cache let h_integral_x1 = z.h_integral(1.5) - 1f64; z.h_integral_x1 = Some(h_integral_x1); let h_integral_num_elements = z.h_integral(num_elements as f64 + 0.5); z.h_integral_num_elements = Some(h_integral_num_elements); let s = 2f64 - z.h_integral_inv(z.h_integral(2.5) - z.h(2f64)); z.s = Some(s); Ok(z) } } impl<R: Rng> Rng for ZipfDistribution<R> { fn next_u32(&mut self) -> u32 { // 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. // We know these were computed in new() let hnum = self.h_integral_num_elements.unwrap(); let h_x1 = self.h_integral_x1.unwrap(); let s = self.s.unwrap(); loop { let u: f64 = hnum + self.sampler.next_f64() * (h_x1 - hnum); // u is uniformly distributed in (h_integral_x1, h_integral_num_elements] let x: f64 = self.h_integral_inv(u); let mut k: isize = (x + 0.5) as isize; // Limit k to the range [1, num_elements] if it would be outside // due to numerical inaccuracies. if k < 1 { k = 1; } else if k > self.num_elements { k = self.num_elements; } // 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) let k64 = k as f64; if k64 - x <= s || u >= self.h_integral(k64 + 0.5) - self.h(k64) { // 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 as u32; } } } } use std::fmt; impl<R: fmt::Debug> fmt::Debug for ZipfDistribution<R> { fn fmt(&self, f: &mut fmt::Formatter) -> Result<(), fmt::Error> { write!(f, "Rejection inversion Zipf deviate [{:?}]", self.sampler) } } impl<R: Rng> ZipfDistribution<R> { /// 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(&self, x: f64) -> f64 { let log_x = x.ln(); helper2((1f64 - self.exponent) * log_x) * log_x } /// Computes `h(x) = 1 / x^exponent` fn h(&self, x: f64) -> f64 { (-self.exponent * x.ln()).exp() } /// The inverse function of `H(x)`. /// Returns the `y` for which `H(y) = x`. fn h_integral_inv(&self, x: f64) -> f64 { let mut t: f64 = x * (1f64 - self.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 * (0.33333333333333333 - 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 * 0.33333333333333333 * (1f64 + 0.25 * x)) } } #[cfg(test)] mod test { #[test] fn generate() { use rand::{self, Rng}; use super::ZipfDistribution; let rng = rand::thread_rng(); let mut g = ZipfDistribution::new(rng, 100000, 1.07).unwrap(); for _ in 0..100000 { println!("{}", g.next_f64()); } } }