Weighted Random Selection with and Without Replacement

In response to a programming challenge, we seek efficient algorithms for weighted random selection from a list, both with and without replacement.

Weighted Selection With Replacement

One effective method for weighted selection with replacement is the Alias Method. This technique creates a set of equal-sized bins for each weighted element. By utilizing bit operations, we can index these bins efficiently without resorting to binary search. Each bin stores a single percentage representing the boundary between the original weighted elements.

Consider the example of five elements with equal weights: (a, b, c, d, e).

Alias Method Implementation

1. Normalize weights: Divide each weight by the total to sum to 1.0.
2. Determine the smallest power of 2 greater than or equal to the number of elements (here, 8).
3. Assign an empty partition to each element.

4. Repeat the following steps until all weights are distributed:

   • Place as much of the element with the least remaining weight into an empty partition.
   • If a partition is not filled, add the heaviest remaining element.

For our example, after several iterations, we have the following partition:

• p1: {a, 1.0}
• p2: {a, b, 0.6}

Runtime Selection

1. Generate a random number between 0 and 1.
2. Bitshift the random number to find the partition index.
3. If the partition is split, use the decimal portion of the bitshifted number to decide which element to return.

Weighted Selection Without Replacement

While algorithms like the weighted reservoir method exist for unweighted selection without replacement, this problem remains unsolved.