On the analysis of Bloom filters

The Bloom filter is a simple random binary data structure which can be efficiently used for approximate set membership testing. When testing for membership of an object, the Bloom filter may give a false positive, whose probability is the main performance figure of the structure. We complete and extend the analysis of the Bloom filter available in the literature by means of the gamma-transform approach. Known results are confirmed and new results are provided, including the variance of the number of bits set to 1 in the filter. We consider the choice of bits to be set to 1 when an object is inserted both with and without replacement, in what we call standard and classic Bloom filter, respectively. Simple iterative schemes for the computation of the false positive probability and a new non-iterative approximation, taking into account the variance of bits set to 1, are also provided.