Weighted random selection from array

Question

I would like to randomly select one element from an array, but each element has a known probability of selection.

All chances together (within the array) sums to 1.<

Answer 1

This can be done in O(1) expected time per sample as follows.

Compute the CDF F(i) for each element i to be the sum of probabilities less than or equal to i.

Define the range r(i) of an element i to be the interval [F(i - 1), F(i)].

For each interval [(i - 1)/n, i/n], create a bucket consisting of the list of the elements whose range overlaps the interval. This takes O(n) time in total for the full array as long as you are reasonably careful.

When you randomly sample the array, you simply compute which bucket the random number is in, and compare with each element of the list until you find the interval that contains it.

The cost of a sample is O(the expected length of a randomly chosen list) <= 2.