Weighted random numbers

Solution 1:

There is a straightforward algorithm for picking an item at random, where items have individual weights:

1) calculate the sum of all the weights

2) pick a random number that is 0 or greater and is less than the sum of the weights

3) go through the items one at a time, subtracting their weight from your random number, until you get the item where the random number is less than that item's weight

Pseudo-code illustrating this:

int sum_of_weight = 0;
for(int i=0; i<num_choices; i++) {
   sum_of_weight += choice_weight[i];
}
int rnd = random(sum_of_weight);
for(int i=0; i<num_choices; i++) {
  if(rnd < choice_weight[i])
    return i;
  rnd -= choice_weight[i];
}
assert(!"should never get here");

This should be straightforward to adapt to your boost containers and such.

If your weights are rarely changed but you often pick one at random, and as long as your container is storing pointers to the objects or is more than a few dozen items long (basically, you have to profile to know if this helps or hinders), then there is an optimisation:

By storing the cumulative weight sum in each item you can use a binary search to pick the item corresponding to the pick weight.

If you do not know the number of items in the list, then there's a very neat algorithm called reservoir sampling that can be adapted to be weighted.

Solution 2:

Updated answer to an old question. You can easily do this in C++11 with just the std::lib:

#include <iostream>
#include <random>
#include <iterator>
#include <ctime>
#include <type_traits>
#include <cassert>

int main()
{
    // Set up distribution
    double interval[] = {1,   2,   3,   4};
    double weights[] =  {  .90, .56, .04};
    std::piecewise_constant_distribution<> dist(std::begin(interval),
                                                std::end(interval),
                                                std::begin(weights));
    // Choose generator
    std::mt19937 gen(std::time(0));  // seed as wanted
    // Demonstrate with N randomly generated numbers
    const unsigned N = 1000000;
    // Collect number of times each random number is generated
    double avg[std::extent<decltype(weights)>::value] = {0};
    for (unsigned i = 0; i < N; ++i)
    {
        // Generate random number using gen, distributed according to dist
        unsigned r = static_cast<unsigned>(dist(gen));
        // Sanity check
        assert(interval[0] <= r && r <= *(std::end(interval)-2));
        // Save r for statistical test of distribution
        avg[r - 1]++;
    }
    // Compute averages for distribution
    for (double* i = std::begin(avg); i < std::end(avg); ++i)
        *i /= N;
    // Display distribution
    for (unsigned i = 1; i <= std::extent<decltype(avg)>::value; ++i)
        std::cout << "avg[" << i << "] = " << avg[i-1] << '\n';
}

Output on my system:

avg[1] = 0.600115
avg[2] = 0.373341
avg[3] = 0.026544

Note that most of the code above is devoted to just displaying and analyzing the output. The actual generation is just a few lines of code. The output demonstrates that the requested "probabilities" have been obtained. You have to divide the requested output by 1.5 since that is what the requests add up to.