When to Use Replacement vs. Non-Replacement in Weighted Random Selection?

Weighted Random Selection: Replacement vs. Non-Replacement

Weighted random selection is a fundamental technique used in various applications. It involves sampling elements from a given list with probability distributions determined by specified weights. When selecting elements with replacement, each item can be chosen multiple times, leading to a higher likelihood of picking items with higher weights. In contrast, selection without replacement restricts the selection of an item once it has been chosen.

Finding efficient algorithms for weighted random selection, especially with replacement, can be challenging. Existing methods, including modified reservoir algorithms, prove unsuitable for significant fraction selections from small list sizes.

An Efficient Approach: Alias Method

One approach that excels in this scenario is the alias method. This technique creates a structured set of bins, each representing a portion of the weighted list. By utilizing bit operations, the bins can be efficiently indexed, avoiding binary searches. Each bin contains two elements from the original list, enabling efficient representation of the distribution.

For instance, consider a list of five equally weighted choices: (a:1, b:1, c:1, d:1, e:1). The alias method creates a set of eight bins, each with a probability mass of 0.125.

1. Normalization: Adjust the weights to sum to 1.0. In this case, (a:0.2 b:0.2 c:0.2 d:0.2 e:0.2).
2. Partition: Allocate bins with weights lower than the partition probability (0.125), starting with the lowest weight. Here, (p1{a|null,1.0},p2,p3,p4,p5,p6,p7,p8).
3. Filling: Fill remaining space in the partition with the highest weight variable. For example, (p1{a|null,1.0},p2{a|b,0.6},p3,p4,p5,p6,p7,p8).

Runtime Selection:

At runtime, we generate a random number and use bit operations to efficiently determine the bin corresponding to the probability distribution. If the bin is split, we use the decimal portion of the random number to select between the two elements in the bin.

In summary, the alias method provides an efficient technique for weighted random selection with replacement. It utilizes bit operations for fast bin indexing and achieves accurate probability distributions by carefully dividing the weights into manageable bins.

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