Fast Pseudo-Random Fingerprints

We propose a method to exponentially speed up computation of various fingerprints, such as the ones used to compute similarity and rarity in massive data sets. Rather then maintaining the full stream of $b$ items of a universe $[u]$, such methods only maintain a concise fingerprint of the stream, and perform computations using the fingerprints. The computations are done approximately, and the required fingerprint size $k$ depends on the desired accuracy $\epsilon$ and confidence $\delta$. Our technique maintains a single bit per hash function, rather than a single integer, thus requiring a fingerprint of length $k = O(\frac{\ln \frac{1}{\delta}}{\epsilon^2})$ bits, rather than $O(\log u \cdot \frac{\ln \frac{1}{\delta}}{\epsilon^2})$ bits required by previous approaches. The main advantage of the fingerprints we propose is that rather than computing the fingerprint of a stream of $b$ items in time of $O(b \cdot k)$, we can compute it in time $O(b \log k)$. Thus this allows an exponential speedup for the fingerprint construction, or alternatively allows achieving a much higher accuracy while preserving computation time. Our methods rely on a specific family of pseudo-random hashes for which we can quickly locate hashes resulting in small values.