Optimizing Data Popularity Conscious Bloom Filters

1
Optimizing Data Popularity Conscious Bloom Filters

• Ming Zhong Pin Lu Kai Shen Joel
  Seiferas
• University of Rochester

2
Problem Overview

• Bloom filters
• compact set representation in which each object
  is hashed into several bits in the filter
• allows possible false positives in membership
  queries
• useful in distributed applications communicating
  sets.
• Highly skewed data popularity distributions.
• Data popularity conscious Bloom filters
• use a large number of hashes for likely false
  positive candidates popular objects in queries
  unpopular objects in sets.
• Goal customize the hash number for each object
  to minimize the false positive prob.

3
Object Popularity Stability

• Stable object popularity is important for
  learning the object popularity and for low
  adjustment overhead.
• Illustration of stability across month-long trace
  segments

4
Problem Formulation and Result

• Problem formulation
• in a universe of N objects, an n-object set is
  represented by an m-bit filter
• object is membership pop. is pi, non-member
  query pop. is qi
• find object hash numbers k1, k2, , kN to
  minimize the false positive probability ?1iN
  qi pow(B,ki)
• B is the probability for an arbitrary filter bit
  to be 1, therefore ?1iN pi ki K ln(1-B) /
  (n ln(1-1/m)).
• Result (assume kis are unrestricted real
  numbers)
• Lagrangian function ?1iN qi pow(B,ki) ?
  (?1iN pi ki K)
• optimization is reached when the functions
  partial derivatives on kis and ? are all zero
• we find ki C log1/B(qi/pi), C is a
  constant
• also B 0.5.

5
Ranged Integer Problem

• Practical constraint
• object is hash number ki must be a positive
  integer, and often upper-bounded by kmax.
• Rounding real-number solutions to integers
• may increase the false positive rate
• no understanding on how much the increase may be.
• Overview of our approach
• introduce an importance score for each object
  (intuitively more important objects desire more
  hashes)
• the importance ranking helps produce fast
  approximation solutions.

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Object Importance Score

• Intuition
• revisit the optimal real-number solution ki C
  log2(qi/pi)
• Hint qi/pi provides a ranking on object hash
  numbers in a good solution.
• Results
• for the ranged real-number problem, an optimal
  solution k1, k2, , kN must follow the importance
  ranking
• k1, k2, ,kN is a 2-approximation solution
  to the ranged integer problem it also follows
  the importance ranking.

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Polynomial-Time 2-Approximation

• Our result indicates that at least one solution
  that follows the importance score ranking is
  provably 2-approximation.
• ? If we enumerate all importance-ranked
  solutions, the best is a 2-approximation.
• O(Nkmax) time 2-approximation
• no more than (N1)kmax-1 importance-ranked
  solutions in total
• it takes O(N) to check constraint and calculate
  the false positive rate for each solution.
• Practically expensive
• N can be huge
• the constant kmax may not be very small (e.g.,
  20).

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Faster Solutions

• (2e)-approximation
• the problem of identifying the best
  importance-ranked solution can be transformed
  into a knapsack problem
• dynamic programming produces (2e)-approximation
  solution in O(N2/e) time.
• Coarse-grained optimization
• partition large number of objects into a small
  number of groups (objects in each group have
  similar importance scores)
• optimize at the group granularity (then assign
  equal hash number to objects within one group) ?
  much smaller N.

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Evaluation on Synthetic Data

• Non-member query pop. qi follows Zipf-like
  distribution.
• Membership pop. pi follows a uniform
  distribution.
• Our integer approximation solution significantly
  outperforms the real-rounding solution,
  particularly at high popularity skewness.

10
Trace-driven Evaluation on Distributed Caching

• Distributed caches exchange their content (set of
  cached web objects) to cooperate.
• Evaluation driven by web access traces from
  IRCache.net.

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Trace-driven Evaluation on Distributed Keyword
Searching

• Distributed search engines pass keyword indexes
  to support distributed joins. False positives
  resolved by additional comm.
• Evaluation driven by web page listing at dmoz.com
  and keyword query traces at Ask.com.

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Related Work

• Compressed Bloom filters Mitzenmacher 2002.
• Bloom filters with additional functionalities
• deletion Fan et al. 2000
• frequency queries Cohen and Matias 2003
• associating objects with values Chazelle et al.
  2004.
• Alternative data structure Pagh et al. 2005.
• Weighted Bloom filters Bruck et al. 2006
• optimal real-number solution with integer
  rounding
• analytically, the rounding-induced error increase
  is unbounded
• practically, the error increase can be
  substantial.

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Conclusions

• Popularity conscious Bloom filters
• motivated by skewed, stable data popularity
  distributions
• customize each objects hash number according to
  its popularity in sets and queries.
• Unrestricted real-number problem
• optimal solution when object hash number is
  linear to log(query-pop/set-pop).
• Ranged integer problem
• query-pop/set-pop serves as an object importance
  indicator
• O(Nkmax) time 2-approximation
• O(N2/e) time (2e)-approximation.
• Quantitative evaluations driven by real
  distributed application traces.