Bloom filters

Bloom filters are used for answering queries on set membership. In this data structure, the whole element is not stored at the hashed address. Only a few bits are set in an array. Given a set S of cardinality n, we store it in an array of m bits using k hash functions h1(), . . . , hk(). Initially, all the cells in the array are set to 0. Then, for each element in the set, x ∈ S, for each 1 ≤ i ≤ k, we set hi(x) = 1. If an element is already 1, then it is not modified. To lookup an element z, the k locations h1(z), h2(z), . . . hk(z) are checked. If atleast one of the locations has a 0, then z does not belong to the set and false is returned. If all the locations contain a 1, then true is returned. But the element may still not belong to the set S. This situation is known as false positive. What about the universality of the hash functions? The hash functions need to be (n+ 1)k-universal. The hash addresses of the elements in the set and one element that is looked up should be independent. Given a constant bits per element ratio mn = c, we would like to reduce the probability of a false positive. It can be shown that the minimum probability of false positive occurs when the fraction of 0’s in the array is 12 and the number of hash functions k = mn ln 2. Then, the probability of a false positive is (0.6185) m n . The analysis for this can be found in the book by Mitzenmacher [MU05, Section 5.5.3]. This allows for a tradeoff between the space efficiency and the probability of a false positive.