Maintaining Stream Statistics Over Sliding Windows

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Maintaining Stream Statistics Over Sliding Windows

• Paper by Mayur Datar, Aristides Gionis, Piotr
  Indyk, Rajeev Motwani

Presentation by Adam Morrison.
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Sliding Window Intro

• Infinite stream.
• Only last N elements relevant.
• Packet streams.
• N is huge.
• Stronger model

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Model

• Count memory bits.
• Online algorithm.

Arrival
5
6
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Timestamp
3 2 1
3 2 1
3 2 1
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Plan

• Basic Counting
• Given a bit stream, maintain at every time
  instant the count of 1s in the last N elements.
• Sum
• Given an integer stream, maintain the sum of the
  last N elements.
• Everything else

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Basic Counting

• Exact Solution? (Counter?)

Exact solution requires ?(N) bits.
1
1
0
2
2
1
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Approximate Basic Counting

• Solution Approximate the answer and bound the
  relative error

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The idea
Bucket sizes? Policy for creating new
buckets? What is it good for?

• Dynamic histogram of active 1s.

• New 1s go into right most bucket.

• For each bucket keep the timestamp of the most
  recent 1 and the buckets size.

• When timestamp expires, free bucket.

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Example (N4)
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(Timestamps are easy)
Cyclic counter mod N.
N15
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What does the histogram buy us?

• Active bucket ? Contains an active 1.
• Only the last bucket might contain expired 1s.

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Estimating number of 1s

• Conclusion
• T sum of all bucket sizes but last.
• So there are at least T 1s.
• C size of last bucket.
• Actual of 1s can be anything from 1 to C.

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Absolute ?Relative
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Bounding the error

• Goal Relative error at most ?1/k.

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How can we do that?(With as few buckets as
possible?)

• Non-decreasing bucket sizes.
• Bucket sizes constrained to
• At most buckets of each size.
• For all sizes but that of last bucket, at least
  buckets of each size.

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New 1 create bucket
Check if invariant violated.
Too many buckets merge
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Why it works (correctness)

• If there are at least
• buckets of sizes

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Why it works (space)

• Can account for all 1s with just
• 
  

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Space usage
of buckets
Bucket size
T counter for estimation
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Operations

• Estimation O(1)

• Insertion Cascading makes it
• worst case.

• But only O(1) amortized!

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Plan

• Basic Counting
• Given a bit stream, maintain at every time
  instant the count of 1s in the last N elements.
• Sum
• Given an integer stream, maintain the sum of the
  last N elements.
• Everything else

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Extending to Sum

• Integers in range 0, R.
• On value V, insert V 1s.
• Timestamps
• Bucket counter
• of buckets
• Total space

Insertion takes ?(R)!
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Reducing insertion time

• If we had a way to rebuild the entire histogram
• We could buffer new values
• And rebuild histogram when buffer reaches size B.
• If it takes ,
  amortized is

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k/2 canonical representation
Would it really? Is this representation unique?

• The k/2 canonical representation of S

If S is the total size of the buckets, computing
its k/2 canonical representation would help us
rebuild the histogram.
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Total time required is O(log S).
01
j2
5
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If a value gets unindexed, it will never be
indexed in the future.
8 6 4 3 2 1
9 7 5 4 3 2
10 8 6 5 4 3
Calculate S1S2 representation
10 6 2 1 1 1 1
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Plan

• Lower Bounds
• More about timestamps.
• Applications.
• More problems

• Basic Counting
• Given a bit stream, maintain at every time
  instant the count of 1s in the last N elements.
• Sum
• Given an integer stream, maintain the sum of the
  last N elements.
• Everything else

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Lower bounds

• Lower Bounds
• More about timestamps.
• Applications.
• More problems

• Basic Counting and Sum algorithms are optimal.
• Similar techniques will show that lots of other
  problems are intractable. (Later.)

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Basic Counting bound
N
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Big block d
Same idea works for Sum.
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Randomized bound
Lower bound applies to randomized algorithms.

• Yao minimax principle
• Expected space complexity of optimal algorithm
  for an input distribution is a lower bound on
  expected space complexity of randomized algorithm.

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Timestamps

• Lower Bounds
• More about timestamps.
• Applications.
• More problems

If much less than N items can arrive during the
window, memory usage is reduced.

• Define window based on real time equate
  timestamp with clock.
• No work needs to be done when items dont arrive,
  so deletions can be deferred.

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Applications

• Lower Bounds
• More about timestamps.
• Applications.
• More problems

• Adapting algorithms to the sliding window model
  using EH to replace counters.
• Counters require bits, EH takes
  .
• Also factor loss in accuracy.

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More Problems

• Lower Bounds
• More about timestamps.
• Applications.
• More problems

• Min/Max
• Storing subsequence of (say) mins is optimal.
• Distinct values
• Basic Counting reduces to it.

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Other Problems

• Distinct values with deletions.
• Factor 2 estimation requires ?(N) space.
• Map 1s in a bit string to distinct values. Pad
  with zeros to infer value of last bit, then use
  deletion to cancel that bit.
• Repeat.

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Other Problems

• Sum with negative integers.
• Factor 2 estimation requires ?(N) space.
• Maps 1s in bit string to (-1,1) and 0s to (1,-1).
• Pad with 0s and query at odd time instants.

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END