Bloom Filters and Locality - Sensitive Hashing

This section follows 5.5.3 in Mitzenmacher/Upfal. Bloom Filters are a compact data structure for approximate membership queries. The data structure uses only linear space. It makes mistakes in the sense that non-members may be declared members with small probability. Let S = {e1, . . . , em} be the elements to be stored; S is a subset of some universe U . We use a boolean array T of length n. We assume that we have k hash functions that map U to the integers from 1 to n. The hash functions are assumed to be independent and random. We initialize T to the all-zero array and then set T [i] to one iff there is an element ej ∈ S and a hash function h` such that h`(ej) = i. Note that an array entry may be set to one by several elements. In order to answer a membership query for e, we compute the hash values h`(e), 1 ≤ ` ≤ k, and return “YES” if and only if T [h`(e)] = 1 for all `. Clearly, if e ∈ S, the answer will be YES. Also, if the answer is NO, then e 6∈ S. However, there is the possibility of false positives, i.e., the data structure might answer YES for elements e 6∈ S. What is the probability that e is a false positive? Let us first compute the probability that a particular T [i] stays zero. This is ( 1− 1 n )km = ( 1− 1 n )n·km/n ≈ e−km/n.