Reservoir sampling is a family of randomized online algorithms for choosing a simple random sample, without replacement, of items from a population of unknown size in a single pass over the items. The size of the population is not known to the algorithm and is typically too large for all items to fit into main memory. The population is revealed to the algorithm over time, and the algorithm cannot look back at previous items. At any point, the current state of the algorithm must permit extraction of a simple random sample without replacement of size over the part of the population seen so far.
Example Consider the case when we want to maintain a sample of one item. A simple algorithm is to store the first item with probability and the -th item with probability . We want to show that the sample is uniformly distributed over all of the items that we have seen at each step since.
The proof is by induction on the number of items . If there is 1 item we always choose it and we are done. If there are already items, then the probability of one of the items already in the store remaining the store becomes so the probability of any of the items remaining in the sample is now The proof of correctness for the general algorithm where we have an arbitrary number of slots is an extension of this argument.
History and Algorithm R
Reservoir sampling was introduced as early as 1962 by Fan, Muller, and Rezucha, who described a sampling algorithm which did not need to know the population size in advance. A simpler algorithm, known as Algorithm R, was discovered independently by Alan G. Waterman and McLeod & Bellhouse; it can also be viewed as a variant of the Fisher–Yates shuffle. Algorithm R works as follows.
Initialize an array indexed from to , containing the first items of the input . This is the reservoir. For each new input , generate a random number uniformly in . If , then set Otherwise, discard . Return after all inputs are processed.
Optimizing running time by skipping elements
The Algorithm R described in the previous section has linear runtime, since it generates a random number for every element in the input stream to determine whether it should be included in the reservoir. It was observed by Jeffrey Vitter that it is possible to obtain faster algorithms with sublinear runtime when the algorithm is allowed to skip past sections of the input stream without examining them. Vitter gave a reservoir sampling algorithm which runs in time , and proved that this is optimal. Kim-Hung Li described a simple algorithm, Algorithm L, achieving the same asymptotic runtime, and which is described in this section.
The starting point for Algorithm L is the observation that if random numbers are generated uniformly and independently, then the indices of the smallest of them form a uniform sample of the -subsets of .
The process can be done without knowing : Keep the smallest of that has been seen so far, as well as , the index of the largest among them.
For each new , compare it with . If , then discard , store , and set to be the index of the largest among them. Otherwise, discard , and set .Now couple this with the stream of inputs . Every time some is accepted, store the corresponding . Every time some is discarded, discard the corresponding .
This algorithm still needs random numbers, thus taking time. However, it can be improved by observing that after each step, the number of following items which will be skipped by the algorithm follows a geometric distribution with parameter . Thus, the sampling algorithm can simply generate a sample from this distribution to determine how many items to skip over, rather than examining them one at a time. A further simplification is to store only the maximum value at each point, and avoid storing . This results in Algorithm L, which is implemented as follows:
// S has items to sample, R will contain the result ReservoirSample(S[1..n], R[1..k]) // fill the reservoir array with the first k items for i = 1 to k R[i] := S[i] end
// random() generates a uniform (0,1) random number W := exp(log(random())/k)
while i <= n // skip items according to a geometric distribution i := i + floor(log(random())/log(1-W)) + 1 if i <= n // replace a random item of the reservoir with item i R[randomInteger(1,k)] := S[i] W := W * exp(log(random())/k) end end end
This algorithm computes three random numbers for each item that becomes part of the reservoir, and does not spend any time on items that do not. Its expected running time is , which is optimal.
Sampling with replacement and the single-item case
Reservoir sampling can also be performed with replacement, meaning that the same input item may appear more than once in the final sample. In sampling without replacement, each input item can appear at most once. For a sample of size one, however, the resulting sample distribution is the same.
Park, Ostrouchov, Samatova, and Geist (2004) derived a skip-based algorithm for reservoir sampling with replacement.
For a reservoir of size one, suppose the currently selected item is at 1-based position in the stream. The position of the next replacement can be sampled directly from a single uniform random value :
This gives a particularly simple algorithm:
if stream is empty return no sample
sample := first item n := 1
loop q := random() // uniform in (0,1) n := floor(n / q) + 1
advance stream to 1-based item n if stream is exhausted break
sample := item n
return sample
Here denotes a uniform random value strictly between 0 and 1. An implementation whose random-number generator can return an endpoint must exclude or resample such values. The variable is the absolute 1-based position in the stream, rather than a count of items to skip. The algorithm therefore generates replacement positions directly instead of making a random replacement decision for every input item.
For a reservoir of size one, each new replacement position requires only one random variate. By comparison, Algorithm L normally uses three. When Algorithm L is specialized to , choosing the reservoir slot requires no randomness, leaving two nontrivial random variates per replacement.
A weighted generalization was later given by Meligrana and Fazzone.
For a single-item weighted reservoir with current cumulative weight , the cumulative-weight threshold for the next replacement is
where is uniform on . The stream is scanned until its cumulative weight reaches that threshold.
With random sort
If we associate with each item of the input a uniformly generated random number, the items with the largest (or, equivalently, smallest) associated values form a simple random sample. A simple reservoir-sampling thus maintains the items with the currently largest associated values in a priority queue.
(* S is a stream of items to sample S.Current returns current item in stream S.Next advances stream to next position min-priority-queue supports: Count -> number of items in priority queue Minimum -> returns minimum key value of all items Extract-Min() -> Remove the item with minimum key Insert(key, Item) -> Adds item with specified key *) ReservoirSample(S[1..?]) H := new min-priority-queue while S has data r := random() // uniformly random between 0 and 1, exclusive if H.Count < k H.Insert(r, S.Current) else // keep k items with largest associated keys if r > H.Minimum H.Extract-Min() H.Insert(r, S.Current) end S.Next end return items in H end
The expected running time of this algorithm is and it is relevant mainly because it can easily be extended to items with weights.
Weighted random sampling The methods presented in the previous sections do not allow to obtain a priori fixed inclusion probabilities. Some applications require items' sampling probabilities to be according to weights associated with each item. 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. Let the weight of item be , and the sum of all weights be . There are two ways to interpret weights assigned to each item in the set: In each round, the probability of every unselected item to be selected in that round is proportional to its weight relative to the weights of all unselected items. If is the current sample, then the probability of an item to be selected in the current round is . The probability of each item to be included in the random sample is proportional to its relative weight, i.e., . It is not achievable when , since each item is then included with probability 1.
Algorithm A-Res The following algorithm was given by Efraimidis and Spirakis that uses interpretation 1:(* S is a stream of items to sample 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 ^ min-priority-queue supports: Count -> number of items in priority queue Minimum() -> returns minimum key value of all items Extract-Min() -> Remove the item with minimum key Insert(key, Item) -> Adds item with specified key *) ReservoirSample(S[1..?]) H := new min-priority-queue while S has data r := random() ^ (1/S.Weight) // random() produces a uniformly random number in (0,1) if H.Count < k H.Insert(r, S.Current) else // keep k items with largest associated keys if r > H.Minimum H.Extract-Min() H.Insert(r, S.Current) end end S.Next end return items in H endThis algorithm is identical to the algorithm given in Reservoir Sampling with Random Sort except for the generation of the items' keys. The algorithm is equivalent to assigning each item a key where is the random number and then selecting the items with the largest keys. Equivalently, a more numerically stable formulation of this algorithm computes the keys as and select the items with the smallest keys.
Algorithm A-ExpJ The following algorithm is a more efficient version of A-Res, also given by Efraimidis and Spirakis: (* S is a stream of items to sample 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 ^ min-priority-queue supports: Count -> number of items in the priority queue Minimum -> minimum key of any item in the priority queue Extract-Min() -> Remove the item with minimum key Insert(Key, Item) -> Adds item with specified key *) ReservoirSampleWithJumps(S[1..?]) H := new min-priority-queue while S has data and H.Count < k r := random() ^ (1/S.Weight) // random() produces a uniformly random number in (0,1) H.Insert(r, S.Current) S.Next end X := log(random()) / log(H.Minimum) // this is the amount of weight that needs to be jumped over while S has data X := X - S.Weight if X <= 0 t := H.Minimum ^ S.Weight r := random(t, 1) ^ (1/S.Weight) // random(x, y) produces a uniformly random number in (x, y)
H.Extract-Min() H.Insert(r, S.Current)
X := log(random()) / log(H.Minimum) end S.Next end return items in H end
This algorithm follows the same mathematical properties that are used in A-Res, but instead of calculating the key for each item and checking whether that item should be inserted or not, it calculates an exponential jump to the next item which will be inserted. This avoids having to create random variates for each item, which may be expensive. The number of random variates required is reduced from to in expectation, where is the reservoir size, and is the number of items in the stream.
Algorithm A-Chao Warning: the following description is wrong, see Chao's original paper and the discussion here. Following algorithm was given by M. T. Chao uses interpretation 2: and Tillé (2006).(* S has items to sample, R will contain the result S[i].Weight contains weight for each item *)