Show Lecture.RandomNumbers as a slide show.
CS253 Random Numbers
Inclusion
To use C++ random numbers, you need to:
#include <random>
To use old C random numbers (don’t ), you need to:
#include <cstdlib>
Philosophy
“Computers can’t do anything truly random. Only a person can do that.”
- Stop trying to prove your superiority.
- If you believe that you have something special that distinguishes you
from machines, you’re talking religion, not CS.
- My dog is pretty random.
- You’re somewhat predictable.
- An online rock-paper-scissors
program beats people 60% of the time over more than a million games,
because people are lousy at being random.
Old Stuff
Patron Saint of Randomness
- There are several C random number generators,
of varying degrees of standardization:
- They still work ok, but avoid them for new C++ code.
- They mix up generation and distribution something terrible.
- Also, each family has a separate seeding function.
- Also also, there’s no way to save/restore state!
Traditional Method
Traditional random number generators work like this:
unsigned long n = 1;
for (int i=0; i<5; i++) {
n = n * 16807 % 2147483647;
cout << n << '\n';
}
16807
282475249
1622650073
984943658
1144108930
- It’s fast, simple, and good enough for many tasks. However …
- What happens if
n
is zero?
- What number always follows 16807?
- How many possible states does this RNG
(Random Number Generator) have?
Overview
- In C++, random numbers have:
- Generators:
Generate uniformly-distributed random integers,
typically zero or one 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
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()
.
🤨 That sequence looks familiar …
#include <random>
#include <iostream>
using namespace std;
int main() {
default_random_engine gen;
for (int i=0; i<5; i++)
cout << gen() << '\n';
}
16807
282475249
1622650073
984943658
1144108930
I won’t bother with the #includes in subsequent examples.
Mersenne Twister
- Here’s a different, 64-bit generator.
- Use
.min()
and .max()
to find out the range of a given generator.
mt19937_64 gen;
cout << "range is " << gen.min() << "…" << gen.max() << "\n\n";
for (int i=0; i<3; i++)
cout << gen() << '\n';
range is 0…18446744073709551615
14514284786278117030
4620546740167642908
13109570281517897720
Ranges
Generators have varying ranges:
ranlux24 rl;
minstd_rand mr;
random_device rd;
mt19937_64 mt;
cout << "ranlux24: " << rl.min() << "…" << rl.max() << '\n'
<< "minstd_rand: " << mr.min() << "…" << mr.max() << '\n'
<< "random_device: " << rd.min() << "…" << rd.max() << '\n'
<< "mt19937_64: " << mt.min() << "…" << mt.max() << '\n';
ranlux24: 0…16777215
minstd_rand: 1…2147483646
random_device: 0…4294967295
mt19937_64: 0…18446744073709551615
Hey, look! Zero is not a possible return value for minstd_rand.
Save/Restore
A generator can save & restore state to an I/O stream:
ranlux24 gen;
cout << gen() << ' ';
cout << gen() << endl;
ofstream("state") << gen;
system("wc -c state");
cout << gen() << ' ';
cout << gen() << '\n';
ifstream("state") >> gen;
cout << gen() << ' ';
cout << gen() << '\n';
15039276 16323925
209 state
14283486 7150092
14283486 7150092
endl! Isn’t that a sin? 😈 🔥
Needed to flush output before wc ran.
True randomness
random_device a, b, c;
cout << a() << '\n'
<< b() << '\n'
<< c() << '\n';
2672176592
413119524
2629878844
- random_device is, ideally, truly random, and not pseudo-random.
- Intel computers have an RDRAND instruction.
- It might depend on random things like human typing intervals,
network packets arrival times, or radioactive decay.
- If true randomness isn’t available, it resorts to pseudo-random numbers.
- It could pause waiting for randomness to become available.
- Use it sparingly.
Cloudflare
The hosting service Cloudflare uses a unique source of randomness.
Seeding
minstd_rand a, b, c(123);
cout << a() << ' ' << a() << '\n';
cout << b() << ' ' << b() << '\n';
cout << c() << ' ' << c() << '\n';
48271 182605794
48271 182605794
5937333 985676192
- Great—we can “seed” the random number generator with a value.
- This way, we can reproduce our pseudo-random sequences.
- Consider random testing: we want to be able to reproduce the sequence
if we find an error.
- How to choose the random seed?
- It should probably be … random.
Seed with process ID
auto seed = getpid();
minstd_rand a(seed);
for (int i=0; i<5; i++)
cout << a() << '\n';
528006589
1070135023
916050295
1975498215
262991230
- You can seed with your process id.
- OK for casual use, but the seed is easily guessed.
- Process IDs are usually 15- or 16-bit quantities, so there are
generally only 32768 or 65536 of them.
Somebody could easily try them all.
Seed with time
// seconds since start of 1970
auto seed = time(nullptr);
minstd_rand a(seed);
for (int i=0; i<5; i++)
cout << a() << '\n';
1377731695
1213069049
619461530
465213802
98939663
- You can seed with a time-related value.
- Two runs may occur within the same second,
and so produce identical random sequences.
- OK for casual use, but the seed is easily guessed.
- There are only 86,400 seconds in a day.
Somebody could easily try them all.
Seed with more accurate time
Nanoseconds make more possibilities:
auto seed = chrono::high_resolution_clock::now()
.time_since_epoch().count();
cout << "Seed: " << seed << '\n';
minstd_rand a(seed);
for (int i=0; i<5; i++)
cout << a() << '\n';
Seed: 1716244834790610198
520896016
1433049260
2125075943
689478304
105651178
- There are 86,400,000,000,000 nanoseconds in a day.
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
🂺 🂻 🂽 🂾 🂱
- encryption of military communications
⚔️ 🔫 💣 🥆 ☢️
- encrypted email re: extra-marital affairs 💔
- That’s beyond the scope of this discussion.
Seed with random_device
random_device rd;
auto seed = rd();
minstd_rand0 a(seed);
for (int i=0; i<5; i++)
cout << a() << '\n';
1525741784
39934861
1170310963
613632268
1101055382
You can seed with random_device, if you know that
it’s truly random.
Not good enough.
- Great, so we know how to generate a number 1…2,147,483,646
or perhaps 0…18,446,744,073,709,551,615
- How often do we want to do that?
- Sometimes, we want integers with different ranges.
- Or, perhaps we want floating-point numbers.
- Maybe spread out linearly, or a bell-shaped curve, Poisson, etc.
- This is a job for a distribution.
Caution
Resist the urge to hack your own distribution—it’s hard. Just use the standard distributions.
minstd_rand r;
int first_half = 0;
for (int i=0; i<100'000'000; i++)
if (r() % 1'000'000'000 < 500'000'000)
first_half++;
cout << first_half << '\n';
53435616
Shouldn’t the result be close to 50 million?
minstd_rand, on this computer, produces a number 1…2,147,483,646.
If you take that mod a billion, the range 1…147,473,646 appears three times,
whereas 147,473,647…999,999,999 only appears twice,
so 1…147,473,646 is overrepresented. Tricky to get right!
Distributions
- Uniform:
- Bernoulli (yes/no) trials:
- Piecewise distributions:
|
- Related to Normal distribution:
- Rate-based distributions:
|
uniform_int_distribution
auto seed = random_device()(); //❓❓❓
mt19937 gen(seed);
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';
}
2 3 6 2 2 4 2 1 1 4 5 5 4 3 6 4 6 1 1 5 5 6 3 1 2 2 5 2 6 4 1 6 6 4 4 1 3 6 4 4
1 5 1 1 6 4 3 6 3 2 6 1 2 1 4 2 4 3 1 5 5 3 1 2 3 1 1 1 3 3 6 4 5 5 5 5 5 1 6 6
6 6 5 2 1 1 2 4 5 3 6 4 1 1 2 5 2 1 4 6 4 4 6 4 2 4 1 4 4 5 4 6 4 2 2 2 4 6 6 2
1 6 3 6 5 5 3 4 5 5 4 3 3 4 4 2 3 1 3 4 5 3 2 1 3 3 2 3 2 3 6 4 2 2 4 1 2 6 5 4
5 2 3 4 3 3 5 1 4 4 2 4 5 6 1 5 3 2 1 5 5 3 4 3 1 6 2 2 2 4 4 2 3 6 2 1 1 5 5 6
1 4 1 2 6 2 3 5 4 4 6 3 5 2 1 2 5 1 3 5 6 6 3 2 6 2 5 1 2 3 2 3 2 6 3 5 4 5 6 5
5 6 5 1 5 3 3 5 2 2 4 6 6 4 3 2 1 1 6 4 1 4 2 6 4 3 1 6 5 2 1 5 4 5 1 4 1 5 5 6
4 5 6 1 6 2 5 6 4 6 6 4 5 6 6 3 6 1 4 4 2 4 3 6 2 2 6 2 3 6 6 2 2 2 3 5 5 5 3 6
3 3 5 3 1 3 4 4 2 5 5 3 1 6 4 6 3 6 2 1 3 5 2 5 2 6 5 4 2 5 3 5 6 6 6 5 6 5 6 3
4 3 3 6 1 6 1 2 4 2 1 4 6 3 1 4 6 4 5 2 2 1 5 6 1 2 6 5 2 2 5 2 3 3 1 6 4 2 3 1
uniform_real_distribution
auto seed = random_device()();
ranlux48 gen(seed);
uniform_real_distribution<> dist(18.0, 25.0);
for (int y=0; y<5; y++) {
for (int x=0; x<10; x++)
cout << fixed << setprecision(3) << dist(gen) << ' ';
cout << '\n';
}
24.455 23.931 19.791 22.218 22.428 18.769 19.596 22.935 18.155 22.458
18.630 20.955 24.109 23.674 21.836 22.961 21.552 23.473 21.299 18.928
24.693 24.686 22.658 24.349 20.535 19.012 19.814 21.684 19.152 23.332
22.062 22.694 21.806 21.930 19.112 23.059 24.871 21.152 22.697 21.237
24.137 18.919 20.865 21.411 20.608 23.143 22.969 18.074 20.768 21.665
OMG—what’s that <>
doing there?
uniform_real_distribution’s template argument defaults to double,
because … real
.
Boolean Values
Yield true 42% of time:
random_device rd;
knuth_b gen(rd());
bernoulli_distribution dist(0.42);
constexpr int nrolls = 1'000'000;
int count=0;
for (int i=0; i<nrolls; i++)
if (dist(gen))
count++;
cout << "true: " << count*100.0/nrolls << "%\n";
true: 42.0281%
Histogram
random_device rd;
mt19937_64 gen(rd());
normal_distribution<> dist(21.5, 1.5);
map<int,int> tally;
for (int i=0; i<10000; i++)
tally[dist(gen)]++;
for (auto p : tally)
cout << p.first << ": " << string(p.second/100,'#') << '\n';
16:
17:
18: ###
19: ###########
20: ####################
21: #########################
22: ####################
23: ###########
24: ####
25:
26:
27:
Passwords
random_device rd;
auto seed = rd();
ranlux24 gen(seed);
uniform_int_distribution<char> dist('A','~');
for (int y=0; y<8; y++) {
string pw;
for (int x=0; x<32; x++)
pw += dist(gen);
cout << "Password: " << pw << '\n';
}
Password: Bg`usRhAjS\bwna~gQgjpFEzfNM\gh]e
Password: RUOecjAhndbcxgHJsGutYRSMntP]Jbq\
Password: vhpFolSW]xh]PROxhfEgSCxQlPUdgp~F
Password: }HOQ~EkE_DF`Kmgw]ybVGSgQpOYWdzqR
Password: |CuC}r{HoOA[\HZemSy~ZEucEzLdxtKt
Password: dYeNVecvjYy~IRp`r`Qy_yUv_]ebibD~
Password: hOF[WGuE`XGosqVgHo~gL]qbjDkxcY`A
Password: OS|mD`agIzg~soLBVamsYHJIRFvJgHIz
Even though we’re using uniform_int_distribution, which has int
right there in its name, it’s
uniform_int_distribution<char>
, so we get characters.
Think of them as 8-bit integers that display differently.