absl/random/zipf_distribution_test.cc - third_party/abseil-cpp - Git at Google

 // 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/base/internal/raw_logging.h"
 #include "absl/random/internal/chi_square.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_CASE(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;
     }
     ABSL_INTERNAL_LOG(INFO,
                       absl::StrCat("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()) {}

   absl::InsecureBitGen rng_;
 };

 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) {
     ABSL_INTERNAL_LOG(INFO, "values");
     for (size_t i = 0; i < expected.size(); i++) {
       ABSL_INTERNAL_LOG(INFO, absl::StrCat(points[i], ": ", buckets[i],
                                            " vs. E=", expected[i]));
     }
     ABSL_INTERNAL_LOG(INFO, absl::StrCat("trials ", trials));
     ABSL_INTERNAL_LOG(INFO,
                       absl::StrCat("mean ", avg, " vs. expected ", mean()));
     ABSL_INTERNAL_LOG(INFO, absl::StrCat(kChiSquared, "(data, ", dof, ") = ",
                                          chi_square, " (", p_actual, ")"));
     ABSL_INTERNAL_LOG(INFO,
                       absl::StrCat(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
	// 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/base/internal/raw_logging.h"
	#include "absl/random/internal/chi_square.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_CASE(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;
	}
	ABSL_INTERNAL_LOG(INFO,
	absl::StrCat("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()) {}

	absl::InsecureBitGen rng_;
	};

	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) {
	ABSL_INTERNAL_LOG(INFO, "values");
	for (size_t i = 0; i < expected.size(); i++) {
	ABSL_INTERNAL_LOG(INFO, absl::StrCat(points[i], ": ", buckets[i],
	" vs. E=", expected[i]));
	}
	ABSL_INTERNAL_LOG(INFO, absl::StrCat("trials ", trials));
	ABSL_INTERNAL_LOG(INFO,
	absl::StrCat("mean ", avg, " vs. expected ", mean()));
	ABSL_INTERNAL_LOG(INFO, absl::StrCat(kChiSquared, "(data, ", dof, ") = ",
	chi_square, " (", p_actual, ")"));
	ABSL_INTERNAL_LOG(INFO,
	absl::StrCat(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