Generating random integer from a range

Question

The simplest (and hence best) C++ (using the 2011 standard) answer is

#include <random>

std::random_device rd;     // only used once to initialise (seed) engine
std::mt19937 rng(rd());    // random-number engine used (Mersenne-Twister in this case)
std::uniform_int_distribution<int> uni(min,max); // guaranteed unbiased

auto random_integer = uni(rng);

No need to re-invent the wheel. No need to worry about bias. No need to worry about using time as random seed.

A fast, somewhat better than yours, but still not properly uniform distributed solution is

output = min + (rand() % static_cast<int>(max - min + 1))

Except when the size of the range is a power of 2, this method produces biased non-uniform distributed numbers regardless the quality of rand(). For a comprehensive test of the quality of this method, please read this.

If your compiler supports C++0x and using it is an option for you, then the new standard <random> header is likely to meet your needs. It has a high quality uniform_int_distribution which will accept minimum and maximum bounds (inclusive as you need), and you can choose among various random number generators to plug into that distribution.

Here is code that generates a million random ints uniformly distributed in [-57, 365]. I've used the new std <chrono> facilities to time it as you mentioned performance is a major concern for you.

#include <iostream>
#include <random>
#include <chrono>

int main()
{
    typedef std::chrono::high_resolution_clock Clock;
    typedef std::chrono::duration<double> sec;
    Clock::time_point t0 = Clock::now();
    const int N = 10000000;
    typedef std::minstd_rand G;                // Select the engine
    G g;                                       // Construct the engine
    typedef std::uniform_int_distribution<> D; // Select the distribution
    D d(-57, 365);                             // Construct the distribution
    int c = 0;
    for (int i = 0; i < N; ++i) 
        c += d(g);                             // Generate a random number
    Clock::time_point t1 = Clock::now();
    std::cout << N/sec(t1-t0).count() << " random numbers per second.\n";
    return c;
}

For me (2.8 GHz Intel Core i5) this prints out:

2.10268e+07 random numbers per second.

You can seed the generator by passing in an int to its constructor:

    G g(seed);

If you later find that int doesn't cover the range you need for your distribution, this can be remedied by changing the uniform_int_distribution like so (e.g. to long long):

    typedef std::uniform_int_distribution<long long> D;

If you later find that the minstd_rand isn't a high enough quality generator, that can also easily be swapped out. E.g.:

    typedef std::mt19937 G;  // Now using mersenne_twister_engine

Having separate control over the random number generator, and the random distribution can be quite liberating.

I've also computed (not shown) the first 4 "moments" of this distribution (using minstd_rand) and compared them to the theoretical values in an attempt to quantify the quality of the distribution:

min = -57
max = 365
mean = 154.131
x_mean = 154
var = 14931.9
x_var = 14910.7
skew = -0.00197375
x_skew = 0
kurtosis = -1.20129
x_kurtosis = -1.20001

(The x_ prefix refers to "expected")

Let's split the problem into two parts:

• Generate a random number n in the range 0 through (max-min).
• Add min to that number

The first part is obviously the hardest. Let's assume that the return value of rand() is perfectly uniform. Using modulo will add bias to the first (RAND_MAX + 1) % (max-min+1) numbers. So if we could magically change RAND_MAX to RAND_MAX - (RAND_MAX + 1) % (max-min+1), there would no longer be any bias.

It turns out that we can use this intuition if we are willing to allow pseudo-nondeterminism into the running time of our algorithm. Whenever rand() returns a number which is too large, we simply ask for another random number until we get one which is small enough.

The running time is now geometrically distributed, with expected value 1/p where p is the probability of getting a small enough number on the first try. Since RAND_MAX - (RAND_MAX + 1) % (max-min+1) is always less than (RAND_MAX + 1) / 2, we know that p > 1/2, so the expected number of iterations will always be less than two for any range. It should be possible to generate tens of millions of random numbers in less than a second on a standard CPU with this technique.

EDIT:

Although the above is technically correct, DSimon's answer is probably more useful in practice. You shouldn't implement this stuff yourself. I have seen a lot of implementations of rejection sampling and it is often very difficult to see if it's correct or not.

How about the Mersenne Twister? The boost implementation is rather easy to use and is well tested in many real-world applications. I've used it myself in several academic projects such as artificial intelligence and evolutionary algorithms.

Here's their example where they make a simple function to roll a six-sided die:

#include <boost/random/mersenne_twister.hpp>
#include <boost/random/uniform_int.hpp>
#include <boost/random/variate_generator.hpp>

boost::mt19937 gen;

int roll_die() {
    boost::uniform_int<> dist(1, 6);
    boost::variate_generator<boost::mt19937&, boost::uniform_int<> > die(gen, dist);
    return die();
}

Oh, and here's some more pimping of this generator just in case you aren't convinced you should use it over the vastly inferior rand():

The Mersenne Twister is a "random number" generator invented by Makoto Matsumoto and Takuji Nishimura; their website includes numerous implementations of the algorithm.

Essentially, the Mersenne Twister is a very large linear-feedback shift register. The algorithm operates on a 19,937 bit seed, stored in an 624-element array of 32-bit unsigned integers. The value 2^19937-1 is a Mersenne prime; the technique for manipulating the seed is based on an older "twisting" algorithm -- hence the name "Mersenne Twister".

An appealing aspect of the Mersenne Twister is its use of binary operations -- as opposed to time-consuming multiplication -- for generating numbers. The algorithm also has a very long period, and good granularity. It is both fast and effective for non-cryptographic applications.