No Repetition: Fast Streaming with Highly Concentrated Hashing

TL;DR

This paper demonstrates that using hash functions with strong concentration bounds allows for high-probability estimators in streaming algorithms without repetitions, leading to faster and simpler algorithms.

Contribution

It introduces a method to achieve high-probability bounds in streaming estimators using a single hash function with strong concentration, eliminating the need for multiple independent repetitions.

Findings

Single hash function achieves exponential error reduction

Algorithms become faster and simpler with strong concentration hashing

Suitable for online processing of high-volume data streams

Abstract

To get estimators that work within a certain error bound with high probability, a common strategy is to design one that works with constant probability, and then boost the probability using independent repetitions. Important examples of this approach are small space algorithms for estimating the number of distinct elements in a stream, or estimating the set similarity between large sets. Using standard strongly universal hashing to process each element, we get a sketch based estimator where the probability of a too large error is, say, 1/4. By performing $r$ independent repetitions and taking the median of the estimators, the error probability falls exponentially in $r$. However, running $r$ independent experiments increases the processing time by a factor $r$. Here we make the point that if we have a hash function with strong concentration bounds, then we get the same high probability bounds without any need for repetitions. Instead of $r$ independent sketches, we have a single sketch that is $r$ times bigger, so the total space is the same. However, we only apply a single hash function, so we save a factor $r$ in time, and the overall algorithms just get simpler. Fast practical hash functions with strong concentration bounds were recently proposed by Aamand em et al. (to appear in STOC 2020). Using their hashing schemes, the algorithms thus become very fast and practical, suitable for online processing of high volume data streams.