Fast hashing with Strong Concentration Bounds
share
Previous work on tabulation hashing of Pǎtraşcu and Thorup from STOC'11 on simple tabulation and from SODA'13 on twisted tabulation offered Chernoff-style concentration bounds on hash based sums, but under some quite severe restrictions on the expected values of these sums. More precisely, 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 Σ should be small enough that character tables of size |Σ| fit in fast cache. The schemes then use O(1) tables of this size, so the space of tabulation hashing is O(|Σ|). However the above concentration bounds only apply if the expected sums are ≪ |Σ|. To see the problem, consider the very simple case where we use tabulation hashing to throw n balls into m bins and apply Chernoff bounds to the number of balls that land in a given bin. We are fine if n=m, for then the expected value is 1. However, if m=2 bins as when tossing n unbiased coins, then the expectancy n/2 is ≫ |Σ| for large data sets, e.g., data sets that don't 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.
READ FULL TEXT