Reservoir Sampling

Problem:

• Choose k entries from n numbers. Make sure each number is selected with the probability of k/n

Basic idea:

• Choose 1, 2, 3, ..., k first and put them into the reservoir.
• For k+1, pick it with a probability of k/(k+1), and randomly replace a number in the reservoir.
• For k+i, pick it with a probability of k/(k+i), and randomly replace a number in the reservoir.
• Repeat until k+i reaches n

Proof:

• For k+i, the probability that it is selected and will replace a number in the reservoir is k/(k+i)
• For a number in the reservoir before (let's say X), the probability that it keeps staying in the reservoir is
  • P(X was in the reservoir last time) × P(X is not replaced by k+i)
  • P(X was in the reservoir last time) × (1 - P(k+i is selected and replaces X))
  • = k/(k+i-1) × （1 - k/(k+i) × 1/k）
  • = k/(k+i)
• When k+i reaches n, the probability of each number staying in the reservoir is k/n

Example

• Choose 3 numbers from [111, 222, 333, 444]. Make sure each number is selected with a probability of 3/4
• First, choose [111, 222, 333] as the initial reservior
• Then choose 444 with a probability of 3/4
• For 111, it stays with a probability of
  • P(444 is not selected) + P(444 is selected but it replaces 222 or 333)
  • = 1/4 + 3/4*2/3
  • = 3/4
• The same case with 222 and 333
• Now all the numbers have the probability of 3/4 to be picked