reservoir sampling


Definition: Randomly select k items from a stream of items of unknown length. Save the first k items in an array of size k. For each item j, j > k, choose a random integer M from 1 to j (inclusive). If M ≤ k, replace item M of the array with item j.

Generalization (I am a kind of ...)
randomized algorithm.

Note: Each possible selection of k items has an equal probability of occurring.

This algorithm is an in-memory variant of Knuth's Algorithm 3.4.2R [Knuth97, 2:144]. He credits Alan G. Waterman. This algorithm is suggested in exercise 10.

Author: PEB



More information

Sahil (Thaker?)'s explanation of correctness. The extension to distributed reservoir sampling is flawed.

Vitter's algorithms X, Y, and Z use far fewer random numbers by choosing how many items to skip, rather than deciding whether or not to skip each item.
Jeffrey Scott Vitter, Random Sampling with a Reservoir, ACM Transactions on Mathematical Software (TOMS), 11(1):37-57, March 1985.
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Go to the Dictionary of Algorithms and Data Structures home page.

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Entry modified 21 January 2020.
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Cite this as:
Paul E. Black, "reservoir sampling", in Dictionary of Algorithms and Data Structures [online], Paul E. Black, ed. 21 January 2020. (accessed TODAY) Available from: