Rainer Gemulla, Wolfgang Lehner and Peter J. Haas

1
A Dip in the Reservoir Maintaining Sample
Synopses of Evolving Datasets

• Rainer Gemulla, Wolfgang Lehner and Peter J. Haas
• VLDB 2006

2
Outline

• Introduction
• Bernoulli Sampling
• Reservoir Sampling
• Random Pairing
• Conclusion

3
Introduction

• Random sampling
• approximate query answering
• data mining
• data stream processing
• query optimization
• data integration
• For example
• It is often infeasible to process or store the
  entire data stream
• Random sampling is an appealing approach to build
  synopses of large data streams

4
Uniform Sampling

• Uniform sampling
• all samples of the same size are equally likely
• many statistical procedures assume uniformity
• flexibility
• Example
• a data set (also called population)
• possible samples of size 2

5
Bernoulli Sampling

• Each inserted item is included in the sample with
  probability q and excluded with probability 1-q
• For a dataset R, the sample size follows the
  binomial distribution BINOM(R,q), so that
• The main disadvantage is the uncontrollable
  variability of the sample size.

6
Reservoir Sampling

• Reservoir sampling
• Maintains a random sample of fixed size M
• building block for many sophisticated sampling
  schemes
• single-scan algorithm
• add the first M elements
• afterwards, flip a coin
• ignore the element (reject)
• replace a random element in the sample (accept)
• accept probability of the ith element

7
Reservoir Sampling (Example)

• Example
• sample size M 2

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Problems with Reservoir Sampling

• Problems with reservoir sampling
• lacks support for deletions (stable data sets)

?
9
An Incorrect Approach

• Idea
• use arriving insertions to refill the sample

Not uniform!
10
Random Pairing

• Random pairing
• compensates deletions with arriving insertions
• corrects inclusion probabilies
• General idea (insertion)
• no uncompensated deletions ? reservoir sampling
• otherwise,
• randomly select an uncompensated deletion
  (partner)
• compensate it Was it in the sample?
• yes ? add arriving element to sample
• no ? ignore arriving element

11
(Cont.)

• The RP algorithm maintains two counters
• c1 records the number of uncompensated deletions
  in which the deleted item was in the sample
• c 2 records the number of uncompensated deletions
  in which the deleted item was not in the sample
• d c1 c 2 the total number of uncompensated
  deletions

12
Random Pairing

• Example

13
Conclusion

• Reservoir Sampling
• lacks support for deletions
• Random Pairing
• uses arriving insertions to compensate for
  deletions
• Can this sampling schemes be applied to sliding
  windows??
• It may be difficult, because that the number of
  items in the window is unknown in advance.