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This paper is a good survey of Bloom filter variants proposed in the history.

In Sect 2A, it introduces Bloom filters. Denotes number of bits as $$m$$, number of element as $$n$$ and number of hash function as $$k$$, when using $$m=\frac{1}{\ln 2}kn \approx 1.44kn$$, the false positive probability is $$\epsilon = (1-(1-\frac{1}{m})^{kn})^k \approx (1-e^{-kn/m})^k$$. However, the paper by Mitzenmacher in 2002 that proposed compressed Bloom filter pointed out that the standard size of $$1.44kn$$ bits actually maximises the information entropy of the Bloom filter. If we use more bits in the Bloom filter and apply compression, the resultant size can be smaller. In the extreme case of using infinite number of bits, the compressed size can reach the theoretical limit $$kn$$. This property is favorible to some applications such as P2P that the transferring cost is higher then the memory cost.

Section 2B mentioned the fingerprint hash table which proposed in the seminal paper ‘Networking Application of Bloom Filters: A Survey’ (Broder and Mitzenmacher, 2003). Its concept is that, if we have a perfect hash function that hashes the $$n$$ element into $$n$$ bits without collision, and we have certain fairly good hash function that return $$(\log 1/\epsilon)$$ random bits (fingerprint) from an element, we can implement a membership query structure similar to Bloom filter. The way to do is to put the $$(\log 1/\epsilon)$$ bits into the bin pointed by the perfect hash function. For an element being queried, it is a member if the bin’s content is exactly the same $$(\log 1/\epsilon)$$ bits. In this case, the total memory used is optimal. Although perfect hashing is hardly achievable.

In order to tolerate collision due to imperfect hash function, a linked list at the bins is proposed. However, storing pointers in a linked list is a huge overhead, compare to the $$(\log 1/\epsilon)$$ bits required for an element hash. Another approach is called the $$d$$-left hashing, which use multiple subtables. This is also using a lot of space.

This paper proposed rank-indexed hashing. Which is using linked list to store fingerprints but can avoid the storage of lengthy pointers. The idea assumes we have a popcount function, which returns the number of 1-bits in a bit vector. This popcount function is available as machine instruction in x86-64 architecture. Given we have this function, the linked list can be implemented as follows:

Assume we have $$L$$ buckets, each bucket may have a linked list or empty. Each element in a linked list can have a successor or not. So looking at all linked list as a whole, we can find some lists with 1st element, some lists with 2nd element, etc. We put all 1st elements in an array and all 2nd elements in another array, etc. We concatenate all these arrays into one fingerprint vector. Each $$n$$-th element array has a corresponding bit vector, so does the $$L$$ buckets.

To traverse a linked list, we do the following: The linked list at bucket $$j<L$$ starts at location $$k=\texttt{popcount}(I,j)$$ of the fingerprint vector, where $$I$$ is the bit vector of the $$L$$ buckets. The second element of such linked list is at $$\texttt{popcount}(I,L)+\texttt{popcount}(A,k)$$ where $$A$$ is the bit vector of 1st elements. So and so forth. A linked list is terminated if the $$k$$-th bit in $$A$$ is zero.

## Bibliographic data

@inproceedings{
title = "Rank-indexed Hashing: A Compact Construction of Bloom Filters and Variants",
author = "Nan Hua and Haiquan (Chuck) Zhao and Bill Lin and Jun (Jim) Xu",
booktitle = "Proc ICNP",
year = "2008",
}