// 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.
#include "absl/random/zipf_distribution.h"
#include <algorithm>
#include <cstddef>
#include <cstdint>
#include <iterator>
#include <random>
#include <string>
#include <utility>
#include <vector>
#include "gmock/gmock.h"
#include "gtest/gtest.h"
#include "absl/log/log.h"
#include "absl/random/internal/chi_square.h"
#include "absl/random/internal/pcg_engine.h"
#include "absl/random/internal/sequence_urbg.h"
#include "absl/random/random.h"
#include "absl/strings/str_cat.h"
#include "absl/strings/str_replace.h"
#include "absl/strings/strip.h"
namespace {
using ::absl::random_internal::kChiSquared;
using ::testing::ElementsAre;
template <typename IntType>
class ZipfDistributionTypedTest : public ::testing::Test {};
using IntTypes = ::testing::Types<int, int8_t, int16_t, int32_t, int64_t,
uint8_t, uint16_t, uint32_t, uint64_t>;
TYPED_TEST_SUITE(ZipfDistributionTypedTest, IntTypes);
TYPED_TEST(ZipfDistributionTypedTest, SerializeTest) {
using param_type = typename absl::zipf_distribution<TypeParam>::param_type;
constexpr int kCount = 1000;
absl::InsecureBitGen gen;
for (const auto& param : {
param_type(),
param_type(32),
param_type(100, 3, 2),
param_type(std::numeric_limits<TypeParam>::max(), 4, 3),
param_type(std::numeric_limits<TypeParam>::max() / 2),
}) {
// Validate parameters.
const auto k = param.k();
const auto q = param.q();
const auto v = param.v();
absl::zipf_distribution<TypeParam> before(k, q, v);
EXPECT_EQ(before.k(), param.k());
EXPECT_EQ(before.q(), param.q());
EXPECT_EQ(before.v(), param.v());
{
absl::zipf_distribution<TypeParam> via_param(param);
EXPECT_EQ(via_param, before);
}
// Validate stream serialization.
std::stringstream ss;
ss << before;
absl::zipf_distribution<TypeParam> after(4, 5.5, 4.4);
EXPECT_NE(before.k(), after.k());
EXPECT_NE(before.q(), after.q());
EXPECT_NE(before.v(), after.v());
EXPECT_NE(before.param(), after.param());
EXPECT_NE(before, after);
ss >> after;
EXPECT_EQ(before.k(), after.k());
EXPECT_EQ(before.q(), after.q());
EXPECT_EQ(before.v(), after.v());
EXPECT_EQ(before.param(), after.param());
EXPECT_EQ(before, after);
// Smoke test.
auto sample_min = after.max();
auto sample_max = after.min();
for (int i = 0; i < kCount; i++) {
auto sample = after(gen);
EXPECT_GE(sample, after.min());
EXPECT_LE(sample, after.max());
if (sample > sample_max) sample_max = sample;
if (sample < sample_min) sample_min = sample;
}
LOG(INFO) << "Range: " << sample_min << ", " << sample_max;
}
}
class ZipfModel {
public:
ZipfModel(size_t k, double q, double v) : k_(k), q_(q), v_(v) {}
double mean() const { return mean_; }
// For the other moments of the Zipf distribution, see, for example,
// http://mathworld.wolfram.com/ZipfDistribution.html
// PMF(k) = (1 / k^s) / H(N,s)
// Returns the probability that any single invocation returns k.
double PMF(size_t i) { return i >= hnq_.size() ? 0.0 : hnq_[i] / sum_hnq_; }
// CDF = H(k, s) / H(N,s)
double CDF(size_t i) {
if (i >= hnq_.size()) {
return 1.0;
}
auto it = std::begin(hnq_);
double h = 0.0;
for (const auto end = it; it != end; it++) {
h += *it;
}
return h / sum_hnq_;
}
// The InverseCDF returns the k values which bound p on the upper and lower
// bound. Since there is no closed-form solution, this is implemented as a
// bisction of the cdf.
std::pair<size_t, size_t> InverseCDF(double p) {
size_t min = 0;
size_t max = hnq_.size();
while (max > min + 1) {
size_t target = (max + min) >> 1;
double x = CDF(target);
if (x > p) {
max = target;
} else {
min = target;
}
}
return {min, max};
}
// Compute the probability totals, which are based on the generalized harmonic
// number, H(N,s).
// H(N,s) == SUM(k=1..N, 1 / k^s)
//
// In the limit, H(N,s) == zetac(s) + 1.
//
// NOTE: The mean of a zipf distribution could be computed here as well.
// Mean := H(N, s-1) / H(N,s).
// Given the parameter v = 1, this gives the following function:
// (Hn(100, 1) - Hn(1,1)) / (Hn(100,2) - Hn(1,2)) = 6.5944
//
void Init() {
if (!hnq_.empty()) {
return;
}
hnq_.clear();
hnq_.reserve(std::min(k_, size_t{1000}));
sum_hnq_ = 0;
double qm1 = q_ - 1.0;
double sum_hnq_m1 = 0;
for (size_t i = 0; i < k_; i++) {
// Partial n-th generalized harmonic number
const double x = v_ + i;
// H(n, q-1)
const double hnqm1 =
(q_ == 2.0) ? (1.0 / x)
: (q_ == 3.0) ? (1.0 / (x * x)) : std::pow(x, -qm1);
sum_hnq_m1 += hnqm1;
// H(n, q)
const double hnq =
(q_ == 2.0) ? (1.0 / (x * x))
: (q_ == 3.0) ? (1.0 / (x * x * x)) : std::pow(x, -q_);
sum_hnq_ += hnq;
hnq_.push_back(hnq);
if (i > 1000 && hnq <= 1e-10) {
// The harmonic number is too small.
break;
}
}
assert(sum_hnq_ > 0);
mean_ = sum_hnq_m1 / sum_hnq_;
}
private:
const size_t k_;
const double q_;
const double v_;
double mean_;
std::vector<double> hnq_;
double sum_hnq_;
};
using zipf_u64 = absl::zipf_distribution<uint64_t>;
class ZipfTest : public testing::TestWithParam<zipf_u64::param_type>,
public ZipfModel {
public:
ZipfTest() : ZipfModel(GetParam().k(), GetParam().q(), GetParam().v()) {}
// We use a fixed bit generator for distribution accuracy tests. This allows
// these tests to be deterministic, while still testing the qualify of the
// implementation.
absl::random_internal::pcg64_2018_engine rng_{0x2B7E151628AED2A6};
};
TEST_P(ZipfTest, ChiSquaredTest) {
const auto& param = GetParam();
Init();
size_t trials = 10000;
// Find the split-points for the buckets.
std::vector<size_t> points;
std::vector<double> expected;
{
double last_cdf = 0.0;
double min_p = 1.0;
for (double p = 0.01; p < 1.0; p += 0.01) {
auto x = InverseCDF(p);
if (points.empty() || points.back() < x.second) {
const double p = CDF(x.second);
points.push_back(x.second);
double q = p - last_cdf;
expected.push_back(q);
last_cdf = p;
if (q < min_p) {
min_p = q;
}
}
}
if (last_cdf < 0.999) {
points.push_back(std::numeric_limits<size_t>::max());
double q = 1.0 - last_cdf;
expected.push_back(q);
if (q < min_p) {
min_p = q;
}
} else {
points.back() = std::numeric_limits<size_t>::max();
expected.back() += (1.0 - last_cdf);
}
// The Chi-Squared score is not completely scale-invariant; it works best
// when the small values are in the small digits.
trials = static_cast<size_t>(8.0 / min_p);
}
ASSERT_GT(points.size(), 0);
// Generate n variates and fill the counts vector with the count of their
// occurrences.
std::vector<int64_t> buckets(points.size(), 0);
double avg = 0;
{
zipf_u64 dis(param);
for (size_t i = 0; i < trials; i++) {
uint64_t x = dis(rng_);
ASSERT_LE(x, dis.max());
ASSERT_GE(x, dis.min());
avg += static_cast<double>(x);
auto it = std::upper_bound(std::begin(points), std::end(points),
static_cast<size_t>(x));
buckets[std::distance(std::begin(points), it)]++;
}
avg = avg / static_cast<double>(trials);
}
// Validate the output using the Chi-Squared test.
for (auto& e : expected) {
e *= trials;
}
// The null-hypothesis is that the distribution is a poisson distribution with
// the provided mean (not estimated from the data).
const int dof = static_cast<int>(expected.size()) - 1;
// NOTE: This test runs about 15x per invocation, so a value of 0.9995 is
// approximately correct for a test suite failure rate of 1 in 100. In
// practice we see failures slightly higher than that.
const double threshold = absl::random_internal::ChiSquareValue(dof, 0.9999);
const double chi_square = absl::random_internal::ChiSquare(
std::begin(buckets), std::end(buckets), std::begin(expected),
std::end(expected));
const double p_actual =
absl::random_internal::ChiSquarePValue(chi_square, dof);
// Log if the chi_squared value is above the threshold.
if (chi_square > threshold) {
LOG(INFO) << "values";
for (size_t i = 0; i < expected.size(); i++) {
LOG(INFO) << points[i] << ": " << buckets[i] << " vs. E=" << expected[i];
}
LOG(INFO) << "trials " << trials;
LOG(INFO) << "mean " << avg << " vs. expected " << mean();
LOG(INFO) << kChiSquared << "(data, " << dof << ") = " << chi_square << " ("
<< p_actual << ")";
LOG(INFO) << kChiSquared << " @ 0.9995 = " << threshold;
FAIL() << kChiSquared << " value of " << chi_square
<< " is above the threshold.";
}
}
std::vector<zipf_u64::param_type> GenParams() {
using param = zipf_u64::param_type;
const auto k = param().k();
const auto q = param().q();
const auto v = param().v();
const uint64_t k2 = 1 << 10;
return std::vector<zipf_u64::param_type>{
// Default
param(k, q, v),
// vary K
param(4, q, v), param(1 << 4, q, v), param(k2, q, v),
// vary V
param(k2, q, 0.5), param(k2, q, 1.5), param(k2, q, 2.5), param(k2, q, 10),
// vary Q
param(k2, 1.5, v), param(k2, 3, v), param(k2, 5, v), param(k2, 10, v),
// Vary V & Q
param(k2, 1.5, 0.5), param(k2, 3, 1.5), param(k, 10, 10)};
}
std::string ParamName(
const ::testing::TestParamInfo<zipf_u64::param_type>& info) {
const auto& p = info.param;
std::string name = absl::StrCat("k_", p.k(), "__q_", absl::SixDigits(p.q()),
"__v_", absl::SixDigits(p.v()));
return absl::StrReplaceAll(name, {{"+", "_"}, {"-", "_"}, {".", "_"}});
}
INSTANTIATE_TEST_SUITE_P(All, ZipfTest, ::testing::ValuesIn(GenParams()),
ParamName);
// NOTE: absl::zipf_distribution is not guaranteed to be stable.
TEST(ZipfDistributionTest, StabilityTest) {
// absl::zipf_distribution stability relies on
// absl::uniform_real_distribution, std::log, std::exp, std::log1p
absl::random_internal::sequence_urbg urbg(
{0x0003eb76f6f7f755ull, 0xFFCEA50FDB2F953Bull, 0xC332DDEFBE6C5AA5ull,
0x6558218568AB9702ull, 0x2AEF7DAD5B6E2F84ull, 0x1521B62829076170ull,
0xECDD4775619F1510ull, 0x13CCA830EB61BD96ull, 0x0334FE1EAA0363CFull,
0xB5735C904C70A239ull, 0xD59E9E0BCBAADE14ull, 0xEECC86BC60622CA7ull});
std::vector<int> output(10);
{
absl::zipf_distribution<int32_t> dist;
std::generate(std::begin(output), std::end(output),
[&] { return dist(urbg); });
EXPECT_THAT(output, ElementsAre(10031, 0, 0, 3, 6, 0, 7, 47, 0, 0));
}
urbg.reset();
{
absl::zipf_distribution<int32_t> dist(std::numeric_limits<int32_t>::max(),
3.3);
std::generate(std::begin(output), std::end(output),
[&] { return dist(urbg); });
EXPECT_THAT(output, ElementsAre(44, 0, 0, 0, 0, 1, 0, 1, 3, 0));
}
}
TEST(ZipfDistributionTest, AlgorithmBounds) {
absl::zipf_distribution<int32_t> dist;
// Small values from absl::uniform_real_distribution map to larger Zipf
// distribution values.
const std::pair<uint64_t, int32_t> kInputs[] = {
{0xffffffffffffffff, 0x0}, {0x7fffffffffffffff, 0x0},
{0x3ffffffffffffffb, 0x1}, {0x1ffffffffffffffd, 0x4},
{0xffffffffffffffe, 0x9}, {0x7ffffffffffffff, 0x12},
{0x3ffffffffffffff, 0x25}, {0x1ffffffffffffff, 0x4c},
{0xffffffffffffff, 0x99}, {0x7fffffffffffff, 0x132},
{0x3fffffffffffff, 0x265}, {0x1fffffffffffff, 0x4cc},
{0xfffffffffffff, 0x999}, {0x7ffffffffffff, 0x1332},
{0x3ffffffffffff, 0x2665}, {0x1ffffffffffff, 0x4ccc},
{0xffffffffffff, 0x9998}, {0x7fffffffffff, 0x1332f},
{0x3fffffffffff, 0x2665a}, {0x1fffffffffff, 0x4cc9e},
{0xfffffffffff, 0x998e0}, {0x7ffffffffff, 0x133051},
{0x3ffffffffff, 0x265ae4}, {0x1ffffffffff, 0x4c9ed3},
{0xffffffffff, 0x98e223}, {0x7fffffffff, 0x13058c4},
{0x3fffffffff, 0x25b178e}, {0x1fffffffff, 0x4a062b2},
{0xfffffffff, 0x8ee23b8}, {0x7ffffffff, 0x10b21642},
{0x3ffffffff, 0x1d89d89d}, {0x1ffffffff, 0x2fffffff},
{0xffffffff, 0x45d1745d}, {0x7fffffff, 0x5a5a5a5a},
{0x3fffffff, 0x69ee5846}, {0x1fffffff, 0x73ecade3},
{0xfffffff, 0x79a9d260}, {0x7ffffff, 0x7cc0532b},
{0x3ffffff, 0x7e5ad146}, {0x1ffffff, 0x7f2c0bec},
{0xffffff, 0x7f95adef}, {0x7fffff, 0x7fcac0da},
{0x3fffff, 0x7fe55ae2}, {0x1fffff, 0x7ff2ac0e},
{0xfffff, 0x7ff955ae}, {0x7ffff, 0x7ffcaac1},
{0x3ffff, 0x7ffe555b}, {0x1ffff, 0x7fff2aac},
{0xffff, 0x7fff9556}, {0x7fff, 0x7fffcaab},
{0x3fff, 0x7fffe555}, {0x1fff, 0x7ffff2ab},
{0xfff, 0x7ffff955}, {0x7ff, 0x7ffffcab},
{0x3ff, 0x7ffffe55}, {0x1ff, 0x7fffff2b},
{0xff, 0x7fffff95}, {0x7f, 0x7fffffcb},
{0x3f, 0x7fffffe5}, {0x1f, 0x7ffffff3},
{0xf, 0x7ffffff9}, {0x7, 0x7ffffffd},
{0x3, 0x7ffffffe}, {0x1, 0x7fffffff},
};
for (const auto& instance : kInputs) {
absl::random_internal::sequence_urbg urbg({instance.first});
EXPECT_EQ(instance.second, dist(urbg));
}
}
} // namespace
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