Fast Hashing With Strong Concentration Bounds
Aamand Anders, Knudsen Jakob B. T., Knudsen Mathias B. T., Rasmussen Peter M. R., Thorup Mikkel. Arxiv 2019
Previous work on tabulation hashing by Patrascu and Thorup from STOC’11 on simple tabulation and from SODA’13 on twisted tabulation offered Chernoff-style concentration bounds on hash based sums, e.g., the number of balls/keys hashing to a given bin, but under some quite severe restrictions on the expected values of these sums. The basic idea in tabulation hashing is to view a key as consisting of \(c=O(1)\) characters, e.g., a 64-bit key as \(c=8\) characters of 8-bits. The character domain \(\Sigma\) should be small enough that character tables of size \(|\Sigma|\) fit in fast cache. The schemes then use \(O(1)\) tables of this size, so the space of tabulation hashing is \(O(|\Sigma|)\). However, the concentration bounds by Patrascu and Thorup only apply if the expected sums are \(\ll |\Sigma|\). To see the problem, consider the very simple case where we use tabulation hashing to throw \(n\) balls into \(m\) bins and want to analyse the number of balls in a given bin. With their concentration bounds, we are fine if \(n=m\), for then the expected value is \(1\). However, if \(m=2\), as when tossing \(n\) unbiased coins, the expected value \(n/2\) is \(\gg |\Sigma|\) for large data sets, e.g., data sets that do not fit in fast cache. To handle expectations that go beyond the limits of our small space, we need a much more advanced analysis of simple tabulation, plus a new tabulation technique that we call tabulation-permutation hashing which is at most twice as slow as simple tabulation. No other hashing scheme of comparable speed offers similar Chernoff-style concentration bounds.
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