Reservoir sampling

Reservoir sampling is a family of randomized algorithms for randomly choosing a sample of k items from a list S containing n items, where n is either a very large or unknown number. Typically n is large enough that the list doesn't fit into main memory.

Algorithm R

The most common example was labelled Algorithm R by Jeffrey Vitter in his paper on the subject.. This simple O(n) algorithm as described in the Dictionary of Algorithms and Data Structures consists of the following steps (assuming that the arrays are one-based, and that the number of items to select, k, is smaller than the size of the source array, S):

/*
  S has items to sample, R will contain the result
*/
ReservoirSample(S[1..n], R[1..k])
  // fill the reservoir array
  for i = 1 to k
      R[i] := S[i]

  // replace elements with gradually decreasing probability
  for i = k+1 to n
    j := random(1, i);   // important: inclusive range
    if j <= k
        R[j] := S[i]

The algorithm creates a "reservoir" array of size k and populates it with the first k items of S. It then iterates through the remaining elements of S until S is exhausted. At the i^th element of S, the algorithm generates a random number j between 1 and i. If j is less than or equal to k, the j^th element of the reservoir array is replaced with the i^th element of S. In effect, for all i, the i^th element of S is chosen to be included in the reservoir with probability k/i. Similarly, at each iteration the j^th element of the reservoir array is chosen to be replaced with probability 1/k * k/i, which simplifies to 1/i. It can be shown that when the algorithm has finished executing, each item in S has equal probability (i.e. k/length(S)) of being chosen for the reservoir. To see this, consider the following proof by induction. After the (i-1)^th round, let us assume, the probability of a number being in the reservoir array is k/(i-1). Since the probability of the number being replaced in the i^thround is 1/i, the probability that it survives the i^th round is (i-1)/i. Thus, the probability that a given number is in the reservoir after the i^th round is the product of these two probabilities, i.e. the probability of being in the reservoir after the (i-1)^th round, and surviving replacement in the i^th round. This is (k/(i-1)) * ((i-1)/i)=k/i. Hence, the result holds for i, and is therefore true by induction.

Reservoir with Random Sort

A simple reservoir-based algorithm can be designed using random sort and implemented using priority queue data structure. This algorithm assigns random number as keys to each item and maintain k items with minimum value for keys. In essence, this is equivalent to assigning a random number to each item as key, sorting items using these keys and taking top k items. The run time of the algoritm is $O(n\log n)$ which is worse then Algorithm R however this algorithm can easily be extended to weighted sampling. Note that both algorithms can operate on streams of unspecified lengths.

/*
  S is a stream of items to sample, R will contain the result
  S.Current returns current item in stream
  S.Next advances stream to next position
  max-priority-queue supports:
    Count -> number of items in priority queue
    Maximum() -> returns maximum key value of all items
    Extract-Max() -> Remove the item with max key
    Insert(key, Item) -> Adds item with specified key
*/
ReservoirSample(S[1..?], R[1..k])
  H = new max-priority-queue
  while S has data
    r = Random(0,1)  // important: inclusive range
    if H.Count < k
      H.Insert(r, S.Current)
    else
      if H.Maximum > r
        H.Extract-Max()
        H.Insert(r, S.Current)
    S.Next

Weighted Random Sampling using Reservoir

In many applications sampling is required to be according to the weights that are assigned to each items available in set. For example, it might be required to sample queries in a search engine with weight as number of times they were performed so that the sample can be analyzed for overall impact on user experience. There are two ways to interpret weights assigned to each item in the set:

1. Let the weight of each item be $w_{i}$ and sum of all weights be $W$. We can convert weight to probability of item getting selected in sample as $P_{i}=w_{i}/W$.
2. Let the weight of two items $i$ and $j$ be $w_{i}$ and $w_{j}$. Let the probability of item $i$ getting selected in sample be $p_{i}$, then we give $p_{j}=p_{i}\max(1,w_{j}/w_{i})$.

Algorithm A-Res

Following algorithm was given by Efraimidis and Spirakis that uses interpretation 1:

/*
  S is a stream of items to sample, R will contain the result
  S.Current returns current item in stream
  S.Weight  returns weight of current item in stream
  S.Next advances stream to next position
  The power operator is represented by ^
  max-priority-queue supports:
    Count -> number of items in priority queue
    Maximum() -> returns maximum key value of all items
    Extract-Max() -> Remove the item with max key
    Insert(key, Item) -> Adds item with specified key
*/
ReservoirSample(S[1..?], R[1..k])
  H = new max-priority-queue
  while S has data
    r = Random(0,1) ^ (1/S.Weight)  // important: inclusive range
    if H.Count < k
      H.Insert(r, S.Current)
    else
      if H.Maximum > r
        H.Extract-Max()
        H.Insert(r, S.Current)
    S.Next

This algorithm is identical to the algorithm given in Reservoir Sampling with Random Sort except for the line how we generate the key using random number generator. The algorithm is equivalent to assigning each item a key $r^{1/w_{i}}$ where $r$ is the random number and then sort items using these keys and finally select top k items for the sample.

Algorithm A-Chao

Following algorithm was given by M. T. Chao uses interpretation 2:

/*
  S has items to sample, R will contain the result
  S[i].Weight contains weight for each item
*/
WeightedReservoir-Chao(S[1..n], R[1..k])
  WSum = 0
  // fill the reservoir array
  for i = 1 to k
      R[i] := S[i]
      WSum = WSum + S[i].Weight
  for i = k+1 to n
    p = S[i].Weight / WSum // probability for this item
    j := random(0, 1);     // important: inclusive range
    if j <= p              // select item according to probability
        R[random(1,k)] := S[i]  //uniform selection in reservoir for replacement
    WSum = WSum + S[i].Weight

For each item, its relative weight is calculated and used to randomly decide if the item will be added into the reservoir. If the item is selected, then one of the existing items of the reservoir is uniformly selected and replaced with the new item. The trick here is that, if the probabilities of all items in the reservoir are already proportional to their weights, then by selecting uniformly which item to replace, the probabilities of all items remain proportional to their weight after the replacement.

Relation to Fisher-Yates shuffle

Suppose one wanted to draw k random cards from a deck of playing cards (i.e., n=52). A natural approach would be to shuffle the deck and then take the top k cards. In the general case, the shuffle also needs to work even if the number of cards in the deck is not known in advance, a condition which is satisfied by the inside-out version of the Fisher-Yates shuffle:

To initialize an array a of n elements to a randomly shuffled copy of S, both 0-based:
   a[0] ← S[0]
   for i from 1 to n - 1 do
       r ← random (0 .. i)
       a[i] ← a[r]
       a[r] ← S[i]

Note that although the rest of the cards are shuffled, only the top k are important in the present context. Therefore, the array a need only track the cards in the top k positions while performing the shuffle, reducing the amount of memory needed. Truncating a to length k, the algorithm is modified accordingly:

To initialize an array a to k random elements of S (which is of length n), both 0-based:
   a[0] ← S[0]
   for i from 1 to k - 1 do
       r ← random (0 .. i)
       a[i] ← a[r]

       a[r] ← S[i]

   for i from k to n - 1 do
       r ← random (0 .. i)
       if (r < k) then a[r] ← S[i]

Since the order of the first k cards is immaterial, the first loop can be removed and a can be initialized to be the first k items of S. This yields Algorithm R.

Example implementation

The following is a simple implementation of the algorithm in Python that samples the set of English Wikipedia page titles:

import random
SAMPLE_COUNT = 10

# Force the value of the seed so the results are repeatable
random.seed(12345)

sample_titles = []
for index, line in enumerate(open("enwiki-20091103-all-titles-in-ns0")):
        # Generate the reservoir
        if index < SAMPLE_COUNT:
                sample_titles.append(line)
        else:
                # Randomly replace elements in the reservoir
                # with a decreasing probability.
                # Choose an integer between 0 and index (inclusive)
                r = random.randint(0, index)
                if r < SAMPLE_COUNT:
                        sample_titles[r] = line
print sample_titles

Statistical properties

Probabilities of selection of the reservoir methods are discussed in Chao (1982) and Tillé (2006). While the first-order selection probabilities are equal to k/n (or, in case of Chao's procedure, to an arbitrary set of unequal probabilities), the second order selection probabilities depend on the order in which the records are sorted in the original reservoir. The problem is overcome by the cube sampling method of Deville and Tillé (2004).

Limitations

Reservoir sampling makes the assumption that the desired sample fits into main memory, often implying that k is a constant independent of n. In applications where we would like to select a large subset of the input list (say a third, i.e. k=n/3), other methods need to be adopted. Distributed implementations for this problem have been proposed.

See also

• Moving average
• Fisher-Yates shuffle