// Copyright 2017 The Abseil Authors.
//
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
// https://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.
#ifndef ABSL_RANDOM_POISSON_DISTRIBUTION_H_
#define ABSL_RANDOM_POISSON_DISTRIBUTION_H_
#include <cassert>
#include <cmath>
#include <istream>
#include <limits>
#include <ostream>
#include <type_traits>
#include "absl/random/internal/fast_uniform_bits.h"
#include "absl/random/internal/fastmath.h"
#include "absl/random/internal/generate_real.h"
#include "absl/random/internal/iostream_state_saver.h"
namespace absl {
ABSL_NAMESPACE_BEGIN
// absl::poisson_distribution:
// Generates discrete variates conforming to a Poisson distribution.
// p(n) = (mean^n / n!) exp(-mean)
//
// Depending on the parameter, the distribution selects one of the following
// algorithms:
// * The standard algorithm, attributed to Knuth, extended using a split method
// for larger values
// * The "Ratio of Uniforms as a convenient method for sampling from classical
// discrete distributions", Stadlober, 1989.
// http://www.sciencedirect.com/science/article/pii/0377042790903495
//
// NOTE: param_type.mean() is a double, which permits values larger than
// poisson_distribution<IntType>::max(), however this should be avoided and
// the distribution results are limited to the max() value.
//
// The goals of this implementation are to provide good performance while still
// beig thread-safe: This limits the implementation to not using lgamma provided
// by <math.h>.
//
template <typename IntType = int>
class poisson_distribution {
public:
using result_type = IntType;
class param_type {
public:
using distribution_type = poisson_distribution;
explicit param_type(double mean = 1.0);
double mean() const { return mean_; }
friend bool operator==(const param_type& a, const param_type& b) {
return a.mean_ == b.mean_;
}
friend bool operator!=(const param_type& a, const param_type& b) {
return !(a == b);
}
private:
friend class poisson_distribution;
double mean_;
double emu_; // e ^ -mean_
double lmu_; // ln(mean_)
double s_;
double log_k_;
int split_;
static_assert(std::is_integral<IntType>::value,
"Class-template absl::poisson_distribution<> must be "
"parameterized using an integral type.");
};
poisson_distribution() : poisson_distribution(1.0) {}
explicit poisson_distribution(double mean) : param_(mean) {}
explicit poisson_distribution(const param_type& p) : param_(p) {}
void reset() {}
// generating functions
template <typename URBG>
result_type operator()(URBG& g) { // NOLINT(runtime/references)
return (*this)(g, param_);
}
template <typename URBG>
result_type operator()(URBG& g, // NOLINT(runtime/references)
const param_type& p);
param_type param() const { return param_; }
void param(const param_type& p) { param_ = p; }
result_type(min)() const { return 0; }
result_type(max)() const { return (std::numeric_limits<result_type>::max)(); }
double mean() const { return param_.mean(); }
friend bool operator==(const poisson_distribution& a,
const poisson_distribution& b) {
return a.param_ == b.param_;
}
friend bool operator!=(const poisson_distribution& a,
const poisson_distribution& b) {
return a.param_ != b.param_;
}
private:
param_type param_;
random_internal::FastUniformBits<uint64_t> fast_u64_;
};
// -----------------------------------------------------------------------------
// Implementation details follow
// -----------------------------------------------------------------------------
template <typename IntType>
poisson_distribution<IntType>::param_type::param_type(double mean)
: mean_(mean), split_(0) {
assert(mean >= 0);
assert(mean <= (std::numeric_limits<result_type>::max)());
// As a defensive measure, avoid large values of the mean. The rejection
// algorithm used does not support very large values well. It my be worth
// changing algorithms to better deal with these cases.
assert(mean <= 1e10);
if (mean_ < 10) {
// For small lambda, use the knuth method.
split_ = 1;
emu_ = std::exp(-mean_);
} else if (mean_ <= 50) {
// Use split-knuth method.
split_ = 1 + static_cast<int>(mean_ / 10.0);
emu_ = std::exp(-mean_ / static_cast<double>(split_));
} else {
// Use ratio of uniforms method.
constexpr double k2E = 0.7357588823428846;
constexpr double kSA = 0.4494580810294493;
lmu_ = std::log(mean_);
double a = mean_ + 0.5;
s_ = kSA + std::sqrt(k2E * a);
const double mode = std::ceil(mean_) - 1;
log_k_ = lmu_ * mode - absl::random_internal::StirlingLogFactorial(mode);
}
}
template <typename IntType>
template <typename URBG>
typename poisson_distribution<IntType>::result_type
poisson_distribution<IntType>::operator()(
URBG& g, // NOLINT(runtime/references)
const param_type& p) {
using random_internal::GeneratePositiveTag;
using random_internal::GenerateRealFromBits;
using random_internal::GenerateSignedTag;
if (p.split_ != 0) {
// Use Knuth's algorithm with range splitting to avoid floating-point
// errors. Knuth's algorithm is: Ui is a sequence of uniform variates on
// (0,1); return the number of variates required for product(Ui) <
// exp(-lambda).
//
// The expected number of variates required for Knuth's method can be
// computed as follows:
// The expected value of U is 0.5, so solving for 0.5^n < exp(-lambda) gives
// the expected number of uniform variates
// required for a given lambda, which is:
// lambda = [2, 5, 9, 10, 11, 12, 13, 14, 15, 16, 17]
// n = [3, 8, 13, 15, 16, 18, 19, 21, 22, 24, 25]
//
result_type n = 0;
for (int split = p.split_; split > 0; --split) {
double r = 1.0;
do {
r *= GenerateRealFromBits<double, GeneratePositiveTag, true>(
fast_u64_(g)); // U(-1, 0)
++n;
} while (r > p.emu_);
--n;
}
return n;
}
// Use ratio of uniforms method.
//
// Let u ~ Uniform(0, 1), v ~ Uniform(-1, 1),
// a = lambda + 1/2,
// s = 1.5 - sqrt(3/e) + sqrt(2(lambda + 1/2)/e),
// x = s * v/u + a.
// P(floor(x) = k | u^2 < f(floor(x))/k), where
// f(m) = lambda^m exp(-lambda)/ m!, for 0 <= m, and f(m) = 0 otherwise,
// and k = max(f).
const double a = p.mean_ + 0.5;
for (;;) {
const double u = GenerateRealFromBits<double, GeneratePositiveTag, false>(
fast_u64_(g)); // U(0, 1)
const double v = GenerateRealFromBits<double, GenerateSignedTag, false>(
fast_u64_(g)); // U(-1, 1)
const double x = std::floor(p.s_ * v / u + a);
if (x < 0) continue; // f(negative) = 0
const double rhs = x * p.lmu_;
// clang-format off
double s = (x <= 1.0) ? 0.0
: (x == 2.0) ? 0.693147180559945
: absl::random_internal::StirlingLogFactorial(x);
// clang-format on
const double lhs = 2.0 * std::log(u) + p.log_k_ + s;
if (lhs < rhs) {
return x > (max)() ? (max)()
: static_cast<result_type>(x); // f(x)/k >= u^2
}
}
}
template <typename CharT, typename Traits, typename IntType>
std::basic_ostream<CharT, Traits>& operator<<(
std::basic_ostream<CharT, Traits>& os, // NOLINT(runtime/references)
const poisson_distribution<IntType>& x) {
auto saver = random_internal::make_ostream_state_saver(os);
os.precision(random_internal::stream_precision_helper<double>::kPrecision);
os << x.mean();
return os;
}
template <typename CharT, typename Traits, typename IntType>
std::basic_istream<CharT, Traits>& operator>>(
std::basic_istream<CharT, Traits>& is, // NOLINT(runtime/references)
poisson_distribution<IntType>& x) { // NOLINT(runtime/references)
using param_type = typename poisson_distribution<IntType>::param_type;
auto saver = random_internal::make_istream_state_saver(is);
double mean = random_internal::read_floating_point<double>(is);
if (!is.fail()) {
x.param(param_type(mean));
}
return is;
}
ABSL_NAMESPACE_END
} // namespace absl
#endif // ABSL_RANDOM_POISSON_DISTRIBUTION_H_
