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Say I were given a set of weights adding up to 1, and I lined them up one after another to make a series of bins with length proportional to their weight. I assign each bin an integer corresponding to its place in line.
Given any number in [0,1], I would like to be able to check which index corresponds to the bin this number lands in. Can I come up with an algorithm to do this in constant time?
A logarithmic time solution is straightforward, but I’m hoping for one better!
This can probably be accomplished using the alias method, which can be used to generate random samples from a discrete distribution in time O(1). I believe that you can adapt this algorithm to solve your problem by simply inverting the transform so that instead of mapping from a uniform random variable to a discrete bucket, you map from a discrete bucket to a uniform variable. From there, you can determine which bucket the value falls in.
Hope this helps!
EDIT: I recently wrote an extensive write-up of the alias method and other related tricks. Hopefully this makes the algorithm and its intuition clearer and easier!