Suppose that you built a Bloom filter for a dictionary of...

Suppose that you built a Bloom filter for a dictionary of words with $m=2^{b}$ bits. A co-worker building an application wants to use your Bloom filter but has only $2^{b-1}$ bits available. Explain how your colleague can use your Bloom filter to avoid rebuilding a new Bloom filter using the original dictionary of words.

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Key Concepts

Bloom Filter

A Bloom filter is a probabilistic data structure used for set membership tests, which consists of a bit array and several independent hash functions. It allows fast insertion and querying while having a controlled false positive rate. The structure does not store the actual elements, only indications via bits set to 1 that some element hashed to that position.

Folding Technique

The folding technique is a method of reducing the size of a bit array by partitioning the array into segments and combining them, typically using bitwise OR operations. In this context, a Bloom filter built with 2^b bits can be folded into 2^(b-1) bits by dividing the original array into two equal halves and then taking the bitwise OR of corresponding pairs of bits. This operation preserves the membership information while fitting the smaller available storage.

Hash Functions Mapping

Hash functions in a Bloom filter generate indices into the bit array based on input elements. When folding the bit array, careful consideration must be given to how the hash outputs are interpreted so that they map consistently into the reduced-size array. This ensures that the probability of false positives is maintained as close as possible to the intended design even after the array size reduction.

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