<s>
In	O
computer	B-General_Concept
science	I-General_Concept
,	O
tabulation	B-Algorithm
hashing	I-Algorithm
is	O
a	O
method	O
for	O
constructing	O
universal	B-Algorithm
families	I-Algorithm
of	I-Algorithm
hash	I-Algorithm
functions	I-Algorithm
by	O
combining	O
table	B-Data_Structure
lookup	I-Data_Structure
with	O
exclusive	O
or	O
operations	O
.	O
</s>
<s>
It	O
was	O
first	O
studied	O
in	O
the	O
form	O
of	O
Zobrist	B-Application
hashing	I-Application
for	O
computer	O
games	O
;	O
later	O
work	O
by	O
Carter	O
and	O
Wegman	O
extended	O
this	O
method	O
to	O
arbitrary	O
fixed-length	O
keys	O
.	O
</s>
<s>
Generalizations	O
of	O
tabulation	B-Algorithm
hashing	I-Algorithm
have	O
also	O
been	O
developed	O
that	O
can	O
handle	O
variable-length	O
keys	O
such	O
as	O
text	O
strings	O
.	O
</s>
<s>
Despite	O
its	O
simplicity	O
,	O
tabulation	B-Algorithm
hashing	I-Algorithm
has	O
strong	O
theoretical	O
properties	O
that	O
distinguish	O
it	O
from	O
some	O
other	O
hash	O
functions	O
.	O
</s>
<s>
In	O
particular	O
,	O
it	O
is	O
3-independent	B-Error_Name
:	O
every	O
3-tuple	O
of	O
keys	O
is	O
equally	O
likely	O
to	O
be	O
mapped	O
to	O
any	O
3-tuple	O
of	O
hash	O
values	O
.	O
</s>
<s>
More	O
sophisticated	O
but	O
slower	O
variants	O
of	O
tabulation	B-Algorithm
hashing	I-Algorithm
extend	O
the	O
method	O
to	O
higher	O
degrees	O
of	O
independence	O
.	O
</s>
<s>
Because	O
of	O
its	O
high	O
degree	O
of	O
independence	O
,	O
tabulation	B-Algorithm
hashing	I-Algorithm
is	O
usable	O
with	O
hashing	O
methods	O
that	O
require	O
a	O
high-quality	O
hash	O
function	O
,	O
including	O
hopscotch	B-Algorithm
hashing	I-Algorithm
,	O
cuckoo	B-Algorithm
hashing	I-Algorithm
,	O
and	O
the	O
MinHash	B-Algorithm
technique	O
for	O
estimating	O
the	O
size	O
of	O
set	O
intersections	O
.	O
</s>
<s>
Then	O
,	O
create	O
a	O
set	O
of	O
lookup	B-Data_Structure
tables	I-Data_Structure
,	O
one	O
for	O
each	O
block	O
,	O
and	O
fill	O
them	O
with	O
random	O
values	O
.	O
</s>
<s>
Choose	O
a	O
block	O
size	O
r	O
≤	O
p	O
;	O
the	O
choice	O
of	O
block	O
size	O
controls	O
the	O
tradeoff	O
between	O
time	O
and	O
memory	O
usage	O
,	O
so	O
it	O
should	O
be	O
made	O
so	O
that	O
the	O
tables	O
are	O
not	O
too	O
large	O
,	O
e.g.	O
,	O
so	O
that	O
the	O
tables	O
fit	O
into	O
the	O
computer	O
's	O
cache	B-General_Concept
memory	I-General_Concept
.	O
</s>
<s>
The	O
first	O
instance	O
of	O
tabulation	B-Algorithm
hashing	I-Algorithm
is	O
Zobrist	B-Application
hashing	I-Application
,	O
a	O
method	O
for	O
hashing	O
positions	O
in	O
abstract	O
board	O
games	O
such	O
as	O
chess	B-Application
named	O
after	O
Albert	O
Lindsey	O
Zobrist	O
,	O
who	O
published	O
it	O
in	O
1970	O
.	O
</s>
<s>
In	O
this	O
method	O
,	O
a	O
random	O
bitstring	O
is	O
generated	O
for	O
each	O
game	O
feature	O
such	O
as	O
a	O
combination	O
of	O
a	O
chess	B-Application
piece	O
and	O
a	O
square	O
of	O
the	O
chessboard	O
.	O
</s>
<s>
The	O
resulting	O
hash	O
value	O
can	O
then	O
be	O
used	O
as	O
an	O
index	O
into	O
a	O
transposition	B-General_Concept
table	I-General_Concept
.	O
</s>
<s>
Tabulation	B-Algorithm
hashing	I-Algorithm
in	O
greater	O
generality	O
,	O
for	O
arbitrary	O
binary	O
values	O
,	O
was	O
later	O
rediscovered	O
by	O
and	O
studied	O
in	O
more	O
detail	O
by	O
.	O
</s>
<s>
define	O
a	O
randomized	O
scheme	O
for	O
generating	O
hash	O
functions	O
to	O
be	O
universal	B-Algorithm
if	O
,	O
for	O
any	O
two	O
keys	O
,	O
the	O
probability	O
that	O
they	O
collide	B-Algorithm
(	O
that	O
is	O
,	O
they	O
are	O
mapped	O
to	O
the	O
same	O
value	O
as	O
each	O
other	O
)	O
is	O
1/m	O
,	O
where	O
m	O
is	O
the	O
number	O
of	O
values	O
that	O
the	O
keys	O
can	O
take	O
on	O
.	O
</s>
<s>
2-independent	O
hashing	O
schemes	O
are	O
automatically	O
universal	B-Algorithm
,	O
and	O
any	O
universal	B-Algorithm
hashing	I-Algorithm
scheme	O
can	O
be	O
converted	O
into	O
a	O
2-independent	O
scheme	O
by	O
storing	O
a	O
random	O
number	O
x	O
as	O
part	O
of	O
the	O
initialization	O
phase	O
of	O
the	O
algorithm	O
and	O
adding	O
x	O
to	O
each	O
hash	O
value	O
.	O
</s>
<s>
As	O
observe	O
,	O
tabulation	B-Algorithm
hashing	I-Algorithm
is	O
3-independent	B-Error_Name
but	O
not	O
4-independent	O
.	O
</s>
<s>
For	O
these	O
four	O
keys	O
,	O
the	O
hash	O
values	O
computed	O
by	O
tabulation	B-Algorithm
hashing	I-Algorithm
will	O
always	O
satisfy	O
the	O
equation	O
,	O
whereas	O
for	O
a	O
4-independent	O
hashing	O
scheme	O
the	O
same	O
equation	O
would	O
only	O
be	O
satisfied	O
with	O
probability	O
1/m	O
.	O
</s>
<s>
Therefore	O
,	O
tabulation	B-Algorithm
hashing	I-Algorithm
is	O
not	O
4-independent	O
.	O
</s>
<s>
Because	O
tabulation	B-Algorithm
hashing	I-Algorithm
is	O
a	O
universal	B-Algorithm
hashing	I-Algorithm
scheme	O
,	O
it	O
can	O
be	O
used	O
in	O
any	O
hashing-based	O
algorithm	O
in	O
which	O
universality	O
is	O
sufficient	O
.	O
</s>
<s>
For	O
instance	O
,	O
in	O
hash	O
chaining	O
,	O
the	O
expected	O
time	O
per	O
operation	O
is	O
proportional	O
to	O
the	O
sum	O
of	O
collision	O
probabilities	O
,	O
which	O
is	O
the	O
same	O
for	O
any	O
universal	B-Algorithm
scheme	O
as	O
it	O
would	O
be	O
for	O
truly	O
random	O
hash	O
functions	O
,	O
and	O
is	O
constant	O
whenever	O
the	O
load	O
factor	O
of	O
the	O
hash	B-Algorithm
table	I-Algorithm
is	O
constant	O
.	O
</s>
<s>
Therefore	O
,	O
tabulation	B-Algorithm
hashing	I-Algorithm
can	O
be	O
used	O
to	O
compute	O
hash	O
functions	O
for	O
hash	O
chaining	O
with	O
a	O
theoretical	O
guarantee	O
of	O
constant	O
expected	O
time	O
per	O
operation	O
.	O
</s>
<s>
However	O
,	O
universal	B-Algorithm
hashing	I-Algorithm
is	O
not	O
strong	O
enough	O
to	O
guarantee	O
the	O
performance	O
of	O
some	O
other	O
hashing	O
algorithms	O
.	O
</s>
<s>
For	O
instance	O
,	O
for	O
linear	B-Algorithm
probing	I-Algorithm
,	O
5-independent	O
hash	O
functions	O
are	O
strong	O
enough	O
to	O
guarantee	O
constant	O
time	O
operation	O
,	O
but	O
there	O
are	O
4-independent	O
hash	O
functions	O
that	O
fail.For	O
the	O
sufficiency	O
of	O
5-independent	O
hashing	O
for	O
linear	B-Algorithm
probing	I-Algorithm
,	O
see	O
.	O
</s>
<s>
Nevertheless	O
,	O
despite	O
only	O
being	O
3-independent	B-Error_Name
,	O
tabulation	B-Algorithm
hashing	I-Algorithm
provides	O
the	O
same	O
constant-time	O
guarantee	O
for	O
linear	B-Algorithm
probing	I-Algorithm
.	O
</s>
<s>
Cuckoo	B-Algorithm
hashing	I-Algorithm
,	O
another	O
technique	O
for	O
implementing	O
hash	B-Algorithm
tables	I-Algorithm
,	O
guarantees	O
constant	O
time	O
per	O
lookup	B-Data_Structure
(	O
regardless	O
of	O
the	O
hash	O
function	O
)	O
.	O
</s>
<s>
Insertions	O
into	O
a	O
cuckoo	B-Algorithm
hash	I-Algorithm
table	I-Algorithm
may	O
fail	O
,	O
causing	O
the	O
entire	O
table	O
to	O
be	O
rebuilt	O
,	O
but	O
such	O
failures	O
are	O
sufficiently	O
unlikely	O
that	O
the	O
expected	O
time	O
per	O
insertion	O
(	O
using	O
either	O
a	O
truly	O
random	O
hash	O
function	O
or	O
a	O
hash	O
function	O
with	O
logarithmic	O
independence	O
)	O
is	O
constant	O
.	O
</s>
<s>
With	O
tabulation	B-Algorithm
hashing	I-Algorithm
,	O
on	O
the	O
other	O
hand	O
,	O
the	O
best	O
bound	O
known	O
on	O
the	O
failure	O
probability	O
is	O
higher	O
,	O
high	O
enough	O
that	O
insertions	O
cannot	O
be	O
guaranteed	O
to	O
take	O
constant	O
expected	O
time	O
.	O
</s>
<s>
Nevertheless	O
,	O
tabulation	B-Algorithm
hashing	I-Algorithm
is	O
adequate	O
to	O
ensure	O
the	O
linear-expected-time	O
construction	O
of	O
a	O
cuckoo	B-Algorithm
hash	I-Algorithm
table	I-Algorithm
for	O
a	O
static	O
set	O
of	O
keys	O
that	O
does	O
not	O
change	O
as	O
the	O
table	O
is	O
used	O
.	O
</s>
<s>
Although	O
tabulation	B-Algorithm
hashing	I-Algorithm
as	O
described	O
above	O
(	O
"	O
simple	O
tabulation	B-Algorithm
hashing	I-Algorithm
"	O
)	O
is	O
only	O
3-independent	B-Error_Name
,	O
variations	O
of	O
this	O
method	O
can	O
be	O
used	O
to	O
obtain	O
hash	O
functions	O
with	O
much	O
higher	O
degrees	O
of	O
independence	O
.	O
</s>
<s>
uses	O
the	O
same	O
idea	O
of	O
using	O
exclusive	O
or	O
operations	O
to	O
combine	O
random	O
values	O
from	O
a	O
table	O
,	O
with	O
a	O
more	O
complicated	O
algorithm	O
based	O
on	O
expander	O
graphs	O
for	O
transforming	O
the	O
key	O
bits	O
into	O
table	O
indices	O
,	O
to	O
define	O
hashing	O
schemes	O
that	O
are	O
k-independent	O
for	O
any	O
constant	O
or	O
even	O
logarithmic	O
value	O
of	O
k	O
.	O
However	O
,	O
the	O
number	O
of	O
table	B-Data_Structure
lookups	I-Data_Structure
needed	O
to	O
compute	O
each	O
hash	O
value	O
using	O
Siegel	O
's	O
variation	O
of	O
tabulation	B-Algorithm
hashing	I-Algorithm
,	O
while	O
constant	O
,	O
is	O
still	O
too	O
large	O
to	O
be	O
practical	O
,	O
and	O
the	O
use	O
of	O
expanders	O
in	O
Siegel	O
's	O
technique	O
also	O
makes	O
it	O
not	O
fully	O
constructive	O
.	O
</s>
<s>
provides	O
a	O
scheme	O
based	O
on	O
tabulation	B-Algorithm
hashing	I-Algorithm
that	O
reaches	O
high	O
degrees	O
of	O
independence	O
more	O
quickly	O
,	O
in	O
a	O
more	O
constructive	O
way	O
.	O
</s>
<s>
He	O
observes	O
that	O
using	O
one	O
round	O
of	O
simple	O
tabulation	B-Algorithm
hashing	I-Algorithm
to	O
expand	O
the	O
input	O
keys	O
to	O
six	O
times	O
their	O
original	O
length	O
,	O
and	O
then	O
a	O
second	O
round	O
of	O
simple	O
tabulation	B-Algorithm
hashing	I-Algorithm
on	O
the	O
expanded	O
keys	O
,	O
results	O
in	O
a	O
hashing	O
scheme	O
whose	O
independence	O
number	O
is	O
exponential	O
in	O
the	O
parameter	O
r	O
,	O
the	O
number	O
of	O
bits	O
per	O
block	O
in	O
the	O
partition	O
of	O
the	O
keys	O
into	O
blocks	O
.	O
</s>
<s>
studies	O
variations	O
of	O
tabulation	B-Algorithm
hashing	I-Algorithm
suitable	O
for	O
variable-length	O
keys	O
such	O
as	O
character	O
strings	O
.	O
</s>
<s>
Lemire	O
shows	O
that	O
no	O
scheme	O
of	O
this	O
type	O
can	O
be	O
3-independent	B-Error_Name
.	O
</s>
<s>
Mixed	O
Tabulation	B-Algorithm
hashing	I-Algorithm
(	O
and	O
the	O
less	O
general	O
Twisted	O
Tabulation	O
)	O
were	O
introduced	O
by	O
Dahlgaard	O
and	O
ThorupDahlgaard	O
,	O
Søren	O
,	O
and	O
Mikkel	O
Thorup	O
.	O
</s>
<s>
Springer	O
,	O
Cham	O
,	O
2014	O
.	O
as	O
a	O
way	O
to	O
strengthen	O
the	O
properties	O
of	O
Tabulation	B-Algorithm
hashing	I-Algorithm
while	O
keeping	O
nearly	O
the	O
same	O
performance	O
.	O
</s>
<s>
2015	O
IEEE	O
56th	O
Annual	O
Symposium	O
on	O
Foundations	O
of	O
Computer	B-General_Concept
Science	I-General_Concept
.	O
</s>
<s>
IEEE	O
,	O
2015	O
.	O
to	O
have	O
strong	O
concentration	O
with	O
regards	O
to	O
k''-partitions	O
,	O
which	O
are	O
useful	O
in	O
algorithms	O
for	O
counting	O
distinct	O
elements	O
,	O
such	O
as	O
the	O
classical	O
method	O
by	O
Flajolet	B-Algorithm
and	I-Algorithm
Martin	I-Algorithm
.	O
</s>
