Bloom Filters

Benoit DonnetNovember 30th, 2006

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Context

• Introduced in 1970 ([bloom])

- Set membership problem- Trade-off between space and

computing complexity- Lossy summary technique

• Historical usage- Spell checking ([McIlroy])

- Database ([Bratbergsengen])

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Content

• Bloom filters• Extensions• Networking applications• Conclusion• References

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Bloom filters

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Construction

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Membership Query

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False Positive

• A Bloom filter can suffer of false positives

- The filter returns a positive answer for some elements that do not belong to A

• Can we evaluate a priori the impact of false positives on a Bloom filter?

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False Positive (2)

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False Positive (3)

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False Positive (4)

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Extensions

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Content

• Compressed Bloom filters ([mitzenmacher])

• Counting Bloom filters ([Fan et al.])

• Dynamic Bloom filters ([Guo et al.])

• Retouched Bloom filters ([Donnet et al.])

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Compressed BF

• A Bloom filter can be used a message exchanged by networked monitors

• New performance metric- Bandwidth

• Transmission size can be affected by compression

• Compressed Bloom filters ([mitzenmacher])

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Compressed BF (2)

• Positive aspects:- Quantity of bit exchanged reduced- False positive rate reduced- Amount of computation per query

reduced• Cost:

- Internal memory increased- Compression/decompression process

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Compressed BF (3)

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Counting BF

• The subset A is changing over time- Insertion- Deletion

• How to perform deletion?- Couting Bloom filters ([Fan et al.])

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Counting BF (2)

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Counting BF (3)

• Which size for the counter?- 4 bits per counter are OK for most

of the applications• What happens in case of an

overflow?

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Dynamic BF

• Statement:- During the execution of the

application, |A| can exceed its orignal size n

• Consequence:- The false positive rate is not

maintained anymore• Solution?

- Dynamic Bloom Filters ([Guo et al.])

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Dynamic BF (2)

• It uses a matrix of s Bloom filters- Each Bloom filter uses m bits and k

hash functions• It starts with s equals to 1.• A new Bloom filter (i.e., a new row in

the matrix) is created when needed.

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Dynamic BF (3)• How to insert an element?

- Check for an active Bloom filter- If there is no active Bloom filter,

create a new one- Add the element to the Bloom filter

• How to query an element?• If all s Bloom filters return false, the

element does not belong the DBF.• If, at least, one Bloom filter returns

true, the element probably belongs to the DBF.

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Dynamic BF (4)

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Retouched BF• Statement:

- Some false positives might be more troublesome than others

- Some applications might tolerate a small level of false negatives

• Question:- Can we trade-off the false positives

against false negatives?• Solution?

- Retouched Bloom filters ([Donnet et al. 06])

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Retouched BF (2)

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Retouched BF (3)

• Quid if we randomly reset s bits in the vector?

- Eliminates the same proportion of false positives as the proportion of false negatives generated

- Randomized bit clearing.• The process of removing selected false

positives is called selective clearing

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Retouched BF (4)

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Networking Applications

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Distributed Caching

•Proxies cooperate to exchange cache information

•Instead of sharing URLs list, proxies broadcast Bloom filters ([Fan et al.])

•A Bloom filter represents a proxy’s cache content

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Multicast

•A router maintains, for each multicast address, a list of associated interfaces/connections

•Replace the list by a Bloom filter ([Grönvall])

•Parallelization possible

•Deletion of an address can be achieved with a counting Bloom filter

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Measurement

• Topology discovery at the IP interface level

• Traceroute monitors exchange information about what was previously discovered

- Doubletree• This information shared can be encoded

as a Bloom filter- Communication cost reduction ([Donnet et al.

05])

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Conclusion

•A Bloom filter

•solves the set membership problem

•can generate false positives

•Extensions to standard Bloom filter were presented

•A few networking applications were discussed

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References

[Bloom]: Space/Time Trade-Offs in Hash Coding with Allowable Errors. In Communications of the ACM. vol. 13, n°7.

[McIlroy]: Development of a Spelling List. In Transactions on Communications. vol. 30, n° 1.

[Mitzenmacher]: Compressed Bloom Filters. In Transactions on Networking. vol. 10, n° 5.

[Fan et al.]: Summary Cache: a Scalable Wide-Area Web Cache Sharing Protocol. In Transactions on Networking. vol. 8, n°3.

[Guo et al.]: Theory and Network Applications of Dynamic Bloom Filters. In Proc. INFOCOM 2006.

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References (2)

[Bratbergsengen]: Hashing Methods and Relational Algebra Operations. In Proc. ICVLD 1984.

[Bruck et al.]: Weighted Bloom Filters. In Proc. ISIT 2006.

[Donnet et al. 06]: Retouched Bloom Filters: Allowing Networked Applications to Trade-Off Selected False Positives Against False Negatives. In Proc. CoNEXT 2006.

[Grönvall]: Scalable Multicast Forwarding. In Proc. ACM SIGCOMM 2001. Student Workshop.

[Donnet et al. 05]: Improved Algorithms for Network Topology Discovery. In Proc. PAM 2005.