Bloom filters provide a space-efficient probabilistic data structure for representing a set in order to support membership queries. They allow false positives but no false negatives. The structure uses k hash functions to map elements to bit positions in a bit array. Querying whether an element is in the set checks if the corresponding bit positions are all set to 1. Modern applications include distributed caching, peer-to-peer networks, routing, and measurement infrastructure where Bloom filters trade off exact representation for speed and space efficiency.

1. Bloom Filters: A History and Modern Applications Michael Mitzenmacher

4. Bloom Filters Start with an m bit array, filled with 0s. Hash each item x j in S k times. If H i ( x j ) = a , set B [ a ] = 1 . To check if y is in S , check B at H i ( y ) . All k values must be 1 . Possible to have a false positive; all k values are 1 , but y is not in S . n items m = cn bits k hash functions 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 B 0 1 0 0 1 0 1 0 0 1 1 1 0 1 1 0 B 0 1 0 0 1 0 1 0 0 1 1 1 0 1 1 0 B 0 1 0 0 1 0 1 0 0 1 1 1 0 1 1 0 B

6. Example m / n = 8 Opt k = 8 ln 2 = 5.45... n items m = cn bits k hash functions

9. Perfect Hashing Approach Element 1 Element 2 Element 3 Element 4 Element 5 Fingerprint(4) Fingerprint(5) Fingerprint(2) Fingerprint(1) Fingerprint(3)

11. Bloom filters and some related schemes are covered in other texts

17. A Modern Application: Distributed Web Caches Web Cache 1 Web Cache 2 Web Cache 3 The Web

21. Counting Bloom Filters Start with an m bit array, filled with 0s. Hash each item x j in S k times. If H i ( x j ) = a , add 1 to B [ a ] . To delete x j decrement the corresponding counters. Can obtain a corresponding Bloom filter by reducing to 0/1. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 B 0 3 0 0 1 0 2 0 0 3 2 1 0 2 1 0 B 0 2 0 0 0 0 2 0 0 3 2 1 0 1 1 0 B 0 1 0 0 0 0 1 0 0 1 1 1 0 1 1 0 B

26. Compressed Bloom Filters n=10,000 0.0108 0.0177 0.0216 f False positive rate 1 2 6 k Hash functions 7.923 7.923 8 z/n Transmit bits per/el 92 14 8 m/n Array bits per/el

30. 2. Resource Location: Framework Queries sent to root. Each node keeps a list of resources reachable through it, through children. List = Bloom filter.

37. Conservative Update 0 3 4 1 8 1 1 0 3 2 5 4 2 0 y Increment +2 The flow associated with y can only have been responsible for 3 packets; counters should be updated to 5. 0 3 4 1 8 1 1 0 5 2 5 5 2 0