1 // Copyright 2017 The Abseil Authors.
2 //
3 // Licensed under the Apache License, Version 2.0 (the "License");
4 // you may not use this file except in compliance with the License.
5 // You may obtain a copy of the License at
6 //
7 // https://www.apache.org/licenses/LICENSE-2.0
8 //
9 // Unless required by applicable law or agreed to in writing, software
10 // distributed under the License is distributed on an "AS IS" BASIS,
11 // WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
12 // See the License for the specific language governing permissions and
13 // limitations under the License.
14
15 #include "absl/random/zipf_distribution.h"
16
17 #include <algorithm>
18 #include <cstddef>
19 #include <cstdint>
20 #include <iterator>
21 #include <random>
22 #include <string>
23 #include <utility>
24 #include <vector>
25
26 #include "gmock/gmock.h"
27 #include "gtest/gtest.h"
28 #include "absl/base/internal/raw_logging.h"
29 #include "absl/random/internal/chi_square.h"
30 #include "absl/random/internal/pcg_engine.h"
31 #include "absl/random/internal/sequence_urbg.h"
32 #include "absl/random/random.h"
33 #include "absl/strings/str_cat.h"
34 #include "absl/strings/str_replace.h"
35 #include "absl/strings/strip.h"
36
37 namespace {
38
39 using ::absl::random_internal::kChiSquared;
40 using ::testing::ElementsAre;
41
42 template <typename IntType>
43 class ZipfDistributionTypedTest : public ::testing::Test {};
44
45 using IntTypes = ::testing::Types<int, int8_t, int16_t, int32_t, int64_t,
46 uint8_t, uint16_t, uint32_t, uint64_t>;
47 TYPED_TEST_SUITE(ZipfDistributionTypedTest, IntTypes);
48
TYPED_TEST(ZipfDistributionTypedTest,SerializeTest)49 TYPED_TEST(ZipfDistributionTypedTest, SerializeTest) {
50 using param_type = typename absl::zipf_distribution<TypeParam>::param_type;
51
52 constexpr int kCount = 1000;
53 absl::InsecureBitGen gen;
54 for (const auto& param : {
55 param_type(),
56 param_type(32),
57 param_type(100, 3, 2),
58 param_type(std::numeric_limits<TypeParam>::max(), 4, 3),
59 param_type(std::numeric_limits<TypeParam>::max() / 2),
60 }) {
61 // Validate parameters.
62 const auto k = param.k();
63 const auto q = param.q();
64 const auto v = param.v();
65
66 absl::zipf_distribution<TypeParam> before(k, q, v);
67 EXPECT_EQ(before.k(), param.k());
68 EXPECT_EQ(before.q(), param.q());
69 EXPECT_EQ(before.v(), param.v());
70
71 {
72 absl::zipf_distribution<TypeParam> via_param(param);
73 EXPECT_EQ(via_param, before);
74 }
75
76 // Validate stream serialization.
77 std::stringstream ss;
78 ss << before;
79 absl::zipf_distribution<TypeParam> after(4, 5.5, 4.4);
80
81 EXPECT_NE(before.k(), after.k());
82 EXPECT_NE(before.q(), after.q());
83 EXPECT_NE(before.v(), after.v());
84 EXPECT_NE(before.param(), after.param());
85 EXPECT_NE(before, after);
86
87 ss >> after;
88
89 EXPECT_EQ(before.k(), after.k());
90 EXPECT_EQ(before.q(), after.q());
91 EXPECT_EQ(before.v(), after.v());
92 EXPECT_EQ(before.param(), after.param());
93 EXPECT_EQ(before, after);
94
95 // Smoke test.
96 auto sample_min = after.max();
97 auto sample_max = after.min();
98 for (int i = 0; i < kCount; i++) {
99 auto sample = after(gen);
100 EXPECT_GE(sample, after.min());
101 EXPECT_LE(sample, after.max());
102 if (sample > sample_max) sample_max = sample;
103 if (sample < sample_min) sample_min = sample;
104 }
105 ABSL_INTERNAL_LOG(INFO,
106 absl::StrCat("Range: ", +sample_min, ", ", +sample_max));
107 }
108 }
109
110 class ZipfModel {
111 public:
ZipfModel(size_t k,double q,double v)112 ZipfModel(size_t k, double q, double v) : k_(k), q_(q), v_(v) {}
113
mean() const114 double mean() const { return mean_; }
115
116 // For the other moments of the Zipf distribution, see, for example,
117 // http://mathworld.wolfram.com/ZipfDistribution.html
118
119 // PMF(k) = (1 / k^s) / H(N,s)
120 // Returns the probability that any single invocation returns k.
PMF(size_t i)121 double PMF(size_t i) { return i >= hnq_.size() ? 0.0 : hnq_[i] / sum_hnq_; }
122
123 // CDF = H(k, s) / H(N,s)
CDF(size_t i)124 double CDF(size_t i) {
125 if (i >= hnq_.size()) {
126 return 1.0;
127 }
128 auto it = std::begin(hnq_);
129 double h = 0.0;
130 for (const auto end = it; it != end; it++) {
131 h += *it;
132 }
133 return h / sum_hnq_;
134 }
135
136 // The InverseCDF returns the k values which bound p on the upper and lower
137 // bound. Since there is no closed-form solution, this is implemented as a
138 // bisction of the cdf.
InverseCDF(double p)139 std::pair<size_t, size_t> InverseCDF(double p) {
140 size_t min = 0;
141 size_t max = hnq_.size();
142 while (max > min + 1) {
143 size_t target = (max + min) >> 1;
144 double x = CDF(target);
145 if (x > p) {
146 max = target;
147 } else {
148 min = target;
149 }
150 }
151 return {min, max};
152 }
153
154 // Compute the probability totals, which are based on the generalized harmonic
155 // number, H(N,s).
156 // H(N,s) == SUM(k=1..N, 1 / k^s)
157 //
158 // In the limit, H(N,s) == zetac(s) + 1.
159 //
160 // NOTE: The mean of a zipf distribution could be computed here as well.
161 // Mean := H(N, s-1) / H(N,s).
162 // Given the parameter v = 1, this gives the following function:
163 // (Hn(100, 1) - Hn(1,1)) / (Hn(100,2) - Hn(1,2)) = 6.5944
164 //
Init()165 void Init() {
166 if (!hnq_.empty()) {
167 return;
168 }
169 hnq_.clear();
170 hnq_.reserve(std::min(k_, size_t{1000}));
171
172 sum_hnq_ = 0;
173 double qm1 = q_ - 1.0;
174 double sum_hnq_m1 = 0;
175 for (size_t i = 0; i < k_; i++) {
176 // Partial n-th generalized harmonic number
177 const double x = v_ + i;
178
179 // H(n, q-1)
180 const double hnqm1 =
181 (q_ == 2.0) ? (1.0 / x)
182 : (q_ == 3.0) ? (1.0 / (x * x)) : std::pow(x, -qm1);
183 sum_hnq_m1 += hnqm1;
184
185 // H(n, q)
186 const double hnq =
187 (q_ == 2.0) ? (1.0 / (x * x))
188 : (q_ == 3.0) ? (1.0 / (x * x * x)) : std::pow(x, -q_);
189 sum_hnq_ += hnq;
190 hnq_.push_back(hnq);
191 if (i > 1000 && hnq <= 1e-10) {
192 // The harmonic number is too small.
193 break;
194 }
195 }
196 assert(sum_hnq_ > 0);
197 mean_ = sum_hnq_m1 / sum_hnq_;
198 }
199
200 private:
201 const size_t k_;
202 const double q_;
203 const double v_;
204
205 double mean_;
206 std::vector<double> hnq_;
207 double sum_hnq_;
208 };
209
210 using zipf_u64 = absl::zipf_distribution<uint64_t>;
211
212 class ZipfTest : public testing::TestWithParam<zipf_u64::param_type>,
213 public ZipfModel {
214 public:
ZipfTest()215 ZipfTest() : ZipfModel(GetParam().k(), GetParam().q(), GetParam().v()) {}
216
217 // We use a fixed bit generator for distribution accuracy tests. This allows
218 // these tests to be deterministic, while still testing the qualify of the
219 // implementation.
220 absl::random_internal::pcg64_2018_engine rng_{0x2B7E151628AED2A6};
221 };
222
TEST_P(ZipfTest,ChiSquaredTest)223 TEST_P(ZipfTest, ChiSquaredTest) {
224 const auto& param = GetParam();
225 Init();
226
227 size_t trials = 10000;
228
229 // Find the split-points for the buckets.
230 std::vector<size_t> points;
231 std::vector<double> expected;
232 {
233 double last_cdf = 0.0;
234 double min_p = 1.0;
235 for (double p = 0.01; p < 1.0; p += 0.01) {
236 auto x = InverseCDF(p);
237 if (points.empty() || points.back() < x.second) {
238 const double p = CDF(x.second);
239 points.push_back(x.second);
240 double q = p - last_cdf;
241 expected.push_back(q);
242 last_cdf = p;
243 if (q < min_p) {
244 min_p = q;
245 }
246 }
247 }
248 if (last_cdf < 0.999) {
249 points.push_back(std::numeric_limits<size_t>::max());
250 double q = 1.0 - last_cdf;
251 expected.push_back(q);
252 if (q < min_p) {
253 min_p = q;
254 }
255 } else {
256 points.back() = std::numeric_limits<size_t>::max();
257 expected.back() += (1.0 - last_cdf);
258 }
259 // The Chi-Squared score is not completely scale-invariant; it works best
260 // when the small values are in the small digits.
261 trials = static_cast<size_t>(8.0 / min_p);
262 }
263 ASSERT_GT(points.size(), 0);
264
265 // Generate n variates and fill the counts vector with the count of their
266 // occurrences.
267 std::vector<int64_t> buckets(points.size(), 0);
268 double avg = 0;
269 {
270 zipf_u64 dis(param);
271 for (size_t i = 0; i < trials; i++) {
272 uint64_t x = dis(rng_);
273 ASSERT_LE(x, dis.max());
274 ASSERT_GE(x, dis.min());
275 avg += static_cast<double>(x);
276 auto it = std::upper_bound(std::begin(points), std::end(points),
277 static_cast<size_t>(x));
278 buckets[std::distance(std::begin(points), it)]++;
279 }
280 avg = avg / static_cast<double>(trials);
281 }
282
283 // Validate the output using the Chi-Squared test.
284 for (auto& e : expected) {
285 e *= trials;
286 }
287
288 // The null-hypothesis is that the distribution is a poisson distribution with
289 // the provided mean (not estimated from the data).
290 const int dof = static_cast<int>(expected.size()) - 1;
291
292 // NOTE: This test runs about 15x per invocation, so a value of 0.9995 is
293 // approximately correct for a test suite failure rate of 1 in 100. In
294 // practice we see failures slightly higher than that.
295 const double threshold = absl::random_internal::ChiSquareValue(dof, 0.9999);
296
297 const double chi_square = absl::random_internal::ChiSquare(
298 std::begin(buckets), std::end(buckets), std::begin(expected),
299 std::end(expected));
300
301 const double p_actual =
302 absl::random_internal::ChiSquarePValue(chi_square, dof);
303
304 // Log if the chi_squared value is above the threshold.
305 if (chi_square > threshold) {
306 ABSL_INTERNAL_LOG(INFO, "values");
307 for (size_t i = 0; i < expected.size(); i++) {
308 ABSL_INTERNAL_LOG(INFO, absl::StrCat(points[i], ": ", buckets[i],
309 " vs. E=", expected[i]));
310 }
311 ABSL_INTERNAL_LOG(INFO, absl::StrCat("trials ", trials));
312 ABSL_INTERNAL_LOG(INFO,
313 absl::StrCat("mean ", avg, " vs. expected ", mean()));
314 ABSL_INTERNAL_LOG(INFO, absl::StrCat(kChiSquared, "(data, ", dof, ") = ",
315 chi_square, " (", p_actual, ")"));
316 ABSL_INTERNAL_LOG(INFO,
317 absl::StrCat(kChiSquared, " @ 0.9995 = ", threshold));
318 FAIL() << kChiSquared << " value of " << chi_square
319 << " is above the threshold.";
320 }
321 }
322
GenParams()323 std::vector<zipf_u64::param_type> GenParams() {
324 using param = zipf_u64::param_type;
325 const auto k = param().k();
326 const auto q = param().q();
327 const auto v = param().v();
328 const uint64_t k2 = 1 << 10;
329 return std::vector<zipf_u64::param_type>{
330 // Default
331 param(k, q, v),
332 // vary K
333 param(4, q, v), param(1 << 4, q, v), param(k2, q, v),
334 // vary V
335 param(k2, q, 0.5), param(k2, q, 1.5), param(k2, q, 2.5), param(k2, q, 10),
336 // vary Q
337 param(k2, 1.5, v), param(k2, 3, v), param(k2, 5, v), param(k2, 10, v),
338 // Vary V & Q
339 param(k2, 1.5, 0.5), param(k2, 3, 1.5), param(k, 10, 10)};
340 }
341
ParamName(const::testing::TestParamInfo<zipf_u64::param_type> & info)342 std::string ParamName(
343 const ::testing::TestParamInfo<zipf_u64::param_type>& info) {
344 const auto& p = info.param;
345 std::string name = absl::StrCat("k_", p.k(), "__q_", absl::SixDigits(p.q()),
346 "__v_", absl::SixDigits(p.v()));
347 return absl::StrReplaceAll(name, {{"+", "_"}, {"-", "_"}, {".", "_"}});
348 }
349
350 INSTANTIATE_TEST_SUITE_P(All, ZipfTest, ::testing::ValuesIn(GenParams()),
351 ParamName);
352
353 // NOTE: absl::zipf_distribution is not guaranteed to be stable.
TEST(ZipfDistributionTest,StabilityTest)354 TEST(ZipfDistributionTest, StabilityTest) {
355 // absl::zipf_distribution stability relies on
356 // absl::uniform_real_distribution, std::log, std::exp, std::log1p
357 absl::random_internal::sequence_urbg urbg(
358 {0x0003eb76f6f7f755ull, 0xFFCEA50FDB2F953Bull, 0xC332DDEFBE6C5AA5ull,
359 0x6558218568AB9702ull, 0x2AEF7DAD5B6E2F84ull, 0x1521B62829076170ull,
360 0xECDD4775619F1510ull, 0x13CCA830EB61BD96ull, 0x0334FE1EAA0363CFull,
361 0xB5735C904C70A239ull, 0xD59E9E0BCBAADE14ull, 0xEECC86BC60622CA7ull});
362
363 std::vector<int> output(10);
364
365 {
366 absl::zipf_distribution<int32_t> dist;
367 std::generate(std::begin(output), std::end(output),
368 [&] { return dist(urbg); });
369 EXPECT_THAT(output, ElementsAre(10031, 0, 0, 3, 6, 0, 7, 47, 0, 0));
370 }
371 urbg.reset();
372 {
373 absl::zipf_distribution<int32_t> dist(std::numeric_limits<int32_t>::max(),
374 3.3);
375 std::generate(std::begin(output), std::end(output),
376 [&] { return dist(urbg); });
377 EXPECT_THAT(output, ElementsAre(44, 0, 0, 0, 0, 1, 0, 1, 3, 0));
378 }
379 }
380
TEST(ZipfDistributionTest,AlgorithmBounds)381 TEST(ZipfDistributionTest, AlgorithmBounds) {
382 absl::zipf_distribution<int32_t> dist;
383
384 // Small values from absl::uniform_real_distribution map to larger Zipf
385 // distribution values.
386 const std::pair<uint64_t, int32_t> kInputs[] = {
387 {0xffffffffffffffff, 0x0}, {0x7fffffffffffffff, 0x0},
388 {0x3ffffffffffffffb, 0x1}, {0x1ffffffffffffffd, 0x4},
389 {0xffffffffffffffe, 0x9}, {0x7ffffffffffffff, 0x12},
390 {0x3ffffffffffffff, 0x25}, {0x1ffffffffffffff, 0x4c},
391 {0xffffffffffffff, 0x99}, {0x7fffffffffffff, 0x132},
392 {0x3fffffffffffff, 0x265}, {0x1fffffffffffff, 0x4cc},
393 {0xfffffffffffff, 0x999}, {0x7ffffffffffff, 0x1332},
394 {0x3ffffffffffff, 0x2665}, {0x1ffffffffffff, 0x4ccc},
395 {0xffffffffffff, 0x9998}, {0x7fffffffffff, 0x1332f},
396 {0x3fffffffffff, 0x2665a}, {0x1fffffffffff, 0x4cc9e},
397 {0xfffffffffff, 0x998e0}, {0x7ffffffffff, 0x133051},
398 {0x3ffffffffff, 0x265ae4}, {0x1ffffffffff, 0x4c9ed3},
399 {0xffffffffff, 0x98e223}, {0x7fffffffff, 0x13058c4},
400 {0x3fffffffff, 0x25b178e}, {0x1fffffffff, 0x4a062b2},
401 {0xfffffffff, 0x8ee23b8}, {0x7ffffffff, 0x10b21642},
402 {0x3ffffffff, 0x1d89d89d}, {0x1ffffffff, 0x2fffffff},
403 {0xffffffff, 0x45d1745d}, {0x7fffffff, 0x5a5a5a5a},
404 {0x3fffffff, 0x69ee5846}, {0x1fffffff, 0x73ecade3},
405 {0xfffffff, 0x79a9d260}, {0x7ffffff, 0x7cc0532b},
406 {0x3ffffff, 0x7e5ad146}, {0x1ffffff, 0x7f2c0bec},
407 {0xffffff, 0x7f95adef}, {0x7fffff, 0x7fcac0da},
408 {0x3fffff, 0x7fe55ae2}, {0x1fffff, 0x7ff2ac0e},
409 {0xfffff, 0x7ff955ae}, {0x7ffff, 0x7ffcaac1},
410 {0x3ffff, 0x7ffe555b}, {0x1ffff, 0x7fff2aac},
411 {0xffff, 0x7fff9556}, {0x7fff, 0x7fffcaab},
412 {0x3fff, 0x7fffe555}, {0x1fff, 0x7ffff2ab},
413 {0xfff, 0x7ffff955}, {0x7ff, 0x7ffffcab},
414 {0x3ff, 0x7ffffe55}, {0x1ff, 0x7fffff2b},
415 {0xff, 0x7fffff95}, {0x7f, 0x7fffffcb},
416 {0x3f, 0x7fffffe5}, {0x1f, 0x7ffffff3},
417 {0xf, 0x7ffffff9}, {0x7, 0x7ffffffd},
418 {0x3, 0x7ffffffe}, {0x1, 0x7fffffff},
419 };
420
421 for (const auto& instance : kInputs) {
422 absl::random_internal::sequence_urbg urbg({instance.first});
423 EXPECT_EQ(instance.second, dist(urbg));
424 }
425 }
426
427 } // namespace
428