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Random Numbers

Philosophy

“Computers can’t do anything truly random. Only a person can do that.”

• Stop trying to prove your superiority.
• My dog is pretty random.
• You’re somewhat predictable.
  • An online rock-paper-scissors program beats people 60% of the time over nearly a million games, because people are lousy at being random.

Traditional Method

Traditionally, random number generators work like this:

unsigned n = 1;
for (int i=0; i<5; i++) {
    n = n * 16807 % 2147483647;
    cout << n << '\n';
}

16807
282475249
1622647863
947787490
1578127215

• Good enough for many tasks. However …
• How many possible states does this RNG have?
• What number always follows 16807?
• What happens if it goes to zero?

Overview

• In C++, random numbers have:
  • Generators
    Generate uniformly-distributed random integers, typically zero to a big number.
  • Distributions
    Take uniformly-distributed random integers, and transform them into other distributions with different ranges.
• Examples:
  • Picking a card (uniform, but discrete)
  • Rolling 3d6 (bell-shaped, but discrete)
  • Human height (bell-shaped, continuous)

Generators

Engine	Description
default_random_engine	Default random engine
minstd_rand	Minimal Standard minstd_rand generator
minstd_rand0	Minimal Standard minstd_rand0 generator
mt19937	Mersenne Twister 19937 generator
mt19937_64	Mersenne Twister 19937 generator (64 bit)
ranlux24_base	Ranlux 24 base generator
ranlux48_base	Ranlux 48 base generator
ranlux24	Ranlux 24 generator
ranlux48	Ranlux 48 generator
knuth_b	Knuth-B generator
random_device	True random number generator

Default Engine

Define a random-number generator, and use () to generate a number. This is not a function call, because gen is an object, not a function. It’s operator().

#include <random>
#include <iostream>

using namespace std;

int main() {
    default_random_engine gen;
    for (int i=0; i<3; i++)
        cout << gen() << '\n';
}

16807
282475249
1622650073

I won’t bother with the #includes in subsequent examples.

Mersenne Twister

mt19937_64 gen;
cout << "min=" << gen.min() << '\n'
     << "max=" << gen.max() << "\n\n";
for (int i=0; i<3; i++)
    cout << gen() << '\n';

min=0
max=18446744073709551615

14514284786278117030
4620546740167642908
13109570281517897720

Here’s a different, 64-bit generator. You can use .min() and .max() to find out the range of a given generator.

True randomness

random_device gen;
for (int i=0; i<3; i++)
    cout << gen() << '\n';

1865252563
1461172128
52382921

• random_device is, ideally, truly random, and not pseudo-random.
• It might depend on random things like human typing intervals, internet packets, or radioactive decay.
• If no source of true randomness is available, it resorts to pseudo-random numbers.
• It could pause waiting for randomness to become available.
• Use it sparingly.

Seeding

minstd_rand gen, a(1), b(123), c(8675309);
cout << gen() << '\n';
cout << a() << '\n';
cout << b() << '\n';
cout << c() << '\n';

48271
48271
5937333
6529574

• Great! However, most of these are pseudo-random sequences.
• They can be seeded, though they do have a default seed.
• How to choose the random seed? It should probably be … random.

Seed with random_device

random_device gen;
const auto seed = gen();
minstd_rand a, b(seed);
for (int i=0; i<3; i++)
    cout << a() << '\n';
for (int i=0; i<3; i++)
    cout << b() << '\n';

48271
182605794
1291394886
1871231674
914459187
333051592

You can seed with random_device, if you know that it’s truly random.

Seed with time

// seconds since start of 1970
const auto seed = time(nullptr);
minstd_rand a, b(seed);
for (int i=0; i<3; i++)
    cout << a() << '\n';
for (int i=0; i<3; i++)
    cout << b() << '\n';

48271
182605794
1291394886
1679553393
1919191959
1018004956

• You can seed with the time of day.
• OK for casual use, but the seed is easily guessed.
• There are only 86,400 seconds in a day.

Seed with more accurate time

C++ provides a more accurate time—nanoseconds make more possibilities:

auto seed = chrono::high_resolution_clock::now()
            .time_since_epoch().count();
minstd_rand a, b(seed);
for (int i=0; i<3; i++)
    cout << a() << '\n';
for (int i=0; i<3; i++)
    cout << b() << '\n';

48271
182605794
1291394886
1889508170
523418686
798284951

Better Seeding

• Many generators have more than 32 or 64 bits of state.
• Therefore, you can seed them with more than 32 or 64 bits.
  • If you’re doing something very important, and somebody guessing your seed, and hence predicting your sequence, would be catastrophic:
    • on-line poker
    • extra-marital affairs
    • encryption of military communications.
• That’s beyond the scope of this discussion.

Distribution

• Enough about generators.
• How do we get different distributions?

Distributions

Uniform: uniform_int_distribution uniform_real_distribution Related to Bernoulli (yes/no) trials: bernoulli_distribution binomial_distribution geometric_distribution negative_binomial_distribution Rate-based distributions: poisson_distribution exponential_distribution gamma_distribution weibull_distribution extreme_value_distribution

Related to Normal distribution: normal_distribution lognormal_distribution chi_squared_distribution cauchy_distribution fisher_f_distribution student_t_distribution Piecewise distributions: discrete_distribution piecewise_constant_distribution piecewise_linear_distribution

uniform_int_distribution

mt19937 gen;
uniform_int_distribution<int> dist(1,6);
for (int y=0; y<10; y++) {
    for (int x=0; x<40; x++)
        cout << dist(gen) << ' ';
    cout << '\n';
}

5 1 6 6 1 6 6 2 4 2 1 4 2 2 4 6 6 6 6 6 1 5 6 6 6 1 3 5 5 2 1 1 3 1 6 4 5 6 6 4 
4 5 1 3 6 2 6 5 5 3 5 5 5 3 3 3 4 2 2 2 5 5 1 2 2 6 1 1 1 6 5 5 5 1 2 5 6 3 1 4 
3 1 3 2 5 1 5 1 2 3 3 3 3 3 4 5 5 6 5 5 2 5 5 1 4 5 1 4 1 3 3 5 6 4 3 6 4 5 2 1 
5 5 2 5 4 6 5 3 6 3 6 4 4 1 1 5 1 2 2 5 6 6 2 2 5 2 2 1 6 2 3 6 2 6 2 6 4 5 3 6 
3 1 5 5 4 4 4 5 6 5 2 5 5 3 5 4 3 2 4 4 1 3 1 3 4 4 5 3 6 6 1 1 4 3 3 3 1 5 3 3 
1 6 5 3 2 6 4 4 1 1 4 3 2 3 4 4 5 5 5 6 3 5 1 6 2 4 6 4 1 4 5 3 4 4 6 3 1 6 3 4 
1 3 6 6 1 1 5 6 5 3 6 5 1 3 3 6 2 1 5 4 3 5 6 1 2 6 2 1 1 2 1 5 6 1 4 4 4 1 1 5 
6 4 4 3 3 2 4 1 3 3 1 5 2 1 1 2 2 4 2 5 3 6 1 5 6 2 6 1 3 2 3 4 3 6 6 3 3 1 1 1 
5 6 3 2 2 2 3 3 1 2 1 3 6 6 6 4 4 5 1 3 2 4 3 1 5 2 1 3 1 6 2 5 4 2 5 1 4 3 3 3 
4 5 2 2 5 4 2 5 5 2 2 3 3 3 4 5 5 5 1 3 6 3 5 2 3 6 3 3 3 5 2 1 4 3 4 6 5 3 5 6 

uniform_real_distribution

mt19937 gen;
uniform_real_distribution<float> dist(18.0, 25.0);
for (int y=0; y<10; y++) {
    for (int x=0; x<12; x++)
        cout << fixed << setprecision(2) << dist(gen) << ' ';
    cout << '\n';
}

23.70 18.95 24.34 23.85 18.89 24.78 24.39 19.55 22.43 20.16 18.68 21.83 
19.95 19.32 21.83 24.95 24.70 24.98 24.75 24.77 19.10 23.08 24.79 24.87 
24.70 18.77 21.40 23.59 23.60 20.08 18.99 18.03 20.95 18.79 24.41 22.48 
23.55 24.15 24.72 21.53 22.59 23.59 18.25 20.53 23.94 19.48 24.54 22.77 
22.75 20.79 23.30 23.18 23.20 21.32 20.75 20.95 22.59 19.22 19.20 20.11 
22.94 23.58 18.22 20.22 19.94 24.11 18.32 19.04 18.68 24.96 23.76 23.75 
22.86 18.88 20.22 23.35 24.65 21.43 18.24 22.65 21.07 18.88 20.67 19.47 
23.36 18.36 23.57 18.26 19.31 20.86 21.43 21.21 21.12 21.41 22.52 23.56 
22.97 24.45 23.28 23.65 19.93 22.94 22.76 18.02 22.59 22.97 19.14 22.51 
18.83 21.19 21.49 23.42 24.72 22.02 20.38 24.14 22.10 23.66 19.57 18.12 

Binding

mt19937 gen;
uniform_real_distribution<float> dist(18.0, 25.0);
auto rng = bind(dist, gen);
for (int y=0; y<10; y++) {
    for (int x=0; x<12; x++)
        cout << fixed << setprecision(2) << rng() << ' ';
    cout << '\n';
}

23.70 18.95 24.34 23.85 18.89 24.78 24.39 19.55 22.43 20.16 18.68 21.83 
19.95 19.32 21.83 24.95 24.70 24.98 24.75 24.77 19.10 23.08 24.79 24.87 
24.70 18.77 21.40 23.59 23.60 20.08 18.99 18.03 20.95 18.79 24.41 22.48 
23.55 24.15 24.72 21.53 22.59 23.59 18.25 20.53 23.94 19.48 24.54 22.77 
22.75 20.79 23.30 23.18 23.20 21.32 20.75 20.95 22.59 19.22 19.20 20.11 
22.94 23.58 18.22 20.22 19.94 24.11 18.32 19.04 18.68 24.96 23.76 23.75 
22.86 18.88 20.22 23.35 24.65 21.43 18.24 22.65 21.07 18.88 20.67 19.47 
23.36 18.36 23.57 18.26 19.31 20.86 21.43 21.21 21.12 21.41 22.52 23.56 
22.97 24.45 23.28 23.65 19.93 22.94 22.76 18.02 22.59 22.97 19.14 22.51 
18.83 21.19 21.49 23.42 24.72 22.02 20.38 24.14 22.10 23.66 19.57 18.12 

Histogram

mt19937 gen;
normal_distribution<double> dist(21.5, 1.5);
auto rng = bind(dist, gen);
map<int,int> tally;
for (int i=0; i<10000; i++)
    tally[rng()]++;
for (auto p : tally)
    cout << p.first << ": " << string(p.second/100,'#') << '\n';

15: 
16: 
17: #
18: ###
19: ###########
20: #####################
21: #########################
22: ####################
23: ###########
24: ###
25: 
26: