Generate random numbers from an interval with holes

Question

Given a set $S$ of $k$ numbers in $[0, N)$. The task is to randomly generate numbers in the range $[0, N)$ such that none belongs to $S$.

Edit - Also given an API to generate random numbers between $[0, N)$. We have to use it to randomly generate numbers in the range $[0, N)$ such that none belongs to $S$.

I would also like a generic strategy for such questions. Another one I came across was to generate random numbers between [0,7] given a random number generator that generates numbers in range [0, 5].

Answer 1

0 1 2 3 4 5 6 7 8 9

0 1 2 3 - 5 - 7 8 -

"-" are in $S = \{4,6,9\}$

You can create mass distribution with $$\text{Prob}(x) = \begin{cases}\frac{1}{N - |S|} & \text{if } x \notin S\\ 0 & \text{if} \ x \in S \end{cases} $$

and then use this algorithm. Copied from here.

from collections import defaultdict
import random

def roll(massDist):
    randRoll = random.random() * sum(massDist) # in [0,1)
    s = 0
    result = 0
    for mass in massDist:
        s += mass
        if randRoll < s:
            return result
        result+=1

sampleMassDist = [1,1,1,1,0,1,0,1,1,0]

d = defaultdict(int)
for i in range(1000):
  d[roll(sampleMassDist)] += 1

print d

Answer 2

If $k$ is small (and smaller than $N$) you can generate a number $t$ in $[0,N)$ and test that it is not in $S$ before returning it. If $t$ is in $S$ (bad luck), you retry until you eventually succeeds. This is very simple and will do in many practical situations where $k$ is small and where is it easy to test for membership in $S$.

Another approach is to pick a number $t$ in $[0,N-k)$ and then compute and return the $t$-th number in $[0,N)\setminus S$. This is good when $S$ has a form simple or regular enough (e.g., the powers of 3) and it is easy to compute how many elements in $S$ are below some value. It is good also when $N$ is not too large, say $N\sim 100$, and you can precompute the $t$-th numbers.