On 06/05/2012 12:30 AM, Robert Bradshaw wrote:
I just found http://cmph.sourceforge.net/ which looks quite
interesting. Though the resulting hash functions are supposedly cheap,
I have the feeling that branching is considered cheap in this context.
Actually, this lead was *very* promising. I believe the very first
reference I actually read through and didn't eliminate after the
abstract totally swept away our home-grown solutions!
"Hash & Displace" by Pagh (1999) is actually very simple, easy to
understand, and fast both for generation and (the branch-free) lookup:
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.69.3753&rep=rep1&type=pdf
The idea is:
- Find a hash `g(x)` to partition the keys into `b` groups (the paper
requires b > 2n, though I think in practice you can often get away with
less)
- Find a hash `f(x)` such that f is 1:1 within each group (which is
easily achieved since groups only has a few elements)
- For each group, from largest to smallest: Find a displacement
`d[group]` so that `f(x) ^ d` doesn't cause collisions.
It requires extra storage for the displacement table. However, I think 8
bits per element might suffice even for vtables of 512 or 1024 in size.
Even with 16 bits it's rather negligible compared to the minimum-128-bit
entries of the table.
I benchmarked these hash functions:
displace1: ((h >> r1) ^ d[h & 63]) & m1
displace2: ((h >> r1) ^ d[h & m2]) & m1
displace3: ((h >> r1) ^ d[(h >> r2) & m2]) & m1
Only the third one is truly in the spirit of the algorithm, but I think
the first two should work well too (and when h is known compile-time,
looking up d[h & 63] isn't harder than looking up r1 or m1).
My computer is acting up and all my numbers today are slower than the
earlier ones (yes, I've disabled turbo-mode in the BIOS for a year ago,
and yes, I've pinned the CPU speed). But here's today's numbers,
compiled with -DIMHASH:
direct: min=5.37e-09 mean=5.39e-09 std=1.96e-11
val=2400000000.000000
index: min=6.45e-09 mean=6.46e-09 std=1.15e-11
val=1800000000.000000
twoshift: min=6.99e-09 mean=7.00e-09 std=1.35e-11
val=1800000000.000000
threeshift: min=7.53e-09 mean=7.54e-09 std=1.63e-11
val=1800000000.000000
displace1: min=6.99e-09 mean=7.00e-09 std=1.66e-11
val=1800000000.000000
displace2: min=6.99e-09 mean=7.02e-09 std=2.77e-11
val=1800000000.000000
displace3: min=7.52e-09 mean=7.54e-09 std=1.19e-11
val=1800000000.000000
I did a dirty prototype of the table-finder as well and it works:
https://github.com/dagss/hashvtable/blob/master/pagh99.py
Dag
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