  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 \emph{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.
