<s>
In	O
computer	B-General_Concept
science	I-General_Concept
and	O
data	B-Application
mining	I-Application
,	O
MinHash	B-Algorithm
(	O
or	O
the	O
min-wise	O
independent	O
permutations	B-Algorithm
locality	B-Algorithm
sensitive	I-Algorithm
hashing	I-Algorithm
scheme	O
)	O
is	O
a	O
technique	O
for	O
quickly	O
estimating	O
how	O
similar	O
two	O
sets	O
are	O
.	O
</s>
<s>
It	O
has	O
also	O
been	O
applied	O
in	O
large-scale	O
clustering	B-Algorithm
problems	O
,	O
such	O
as	O
clustering	B-Algorithm
documents	O
by	O
the	O
similarity	O
of	O
their	O
sets	O
of	O
words	O
.	O
</s>
<s>
This	O
value	O
is	O
0	O
when	O
the	O
two	O
sets	O
are	O
disjoint	B-Algorithm
,	O
1	O
when	O
they	O
are	O
equal	O
,	O
and	O
strictly	O
between	O
0	O
and	O
1	O
otherwise	O
.	O
</s>
<s>
The	O
goal	O
of	O
MinHash	B-Algorithm
is	O
to	O
estimate	O
quickly	O
,	O
without	O
explicitly	O
computing	O
the	O
intersection	O
and	O
union	O
.	O
</s>
<s>
Let	O
be	O
a	O
hash	B-Error_Name
function	I-Error_Name
that	O
maps	O
the	O
members	O
of	O
to	O
distinct	O
integers	O
,	O
let	O
be	O
a	O
random	B-Algorithm
permutation	I-Algorithm
of	O
the	O
elements	O
of	O
the	O
set	O
,	O
and	O
for	O
any	O
subset	O
of	O
define	O
to	O
be	O
the	O
minimal	O
member	O
of	O
with	O
respect	O
to	O
—	O
that	O
is	O
,	O
the	O
member	O
of	O
with	O
the	O
minimum	O
value	O
of	O
.	O
</s>
<s>
(	O
In	O
cases	O
where	O
the	O
hash	B-Error_Name
function	I-Error_Name
used	O
is	O
assumed	O
to	O
have	O
pseudo-random	O
properties	O
,	O
the	O
random	B-Algorithm
permutation	I-Algorithm
would	O
not	O
be	O
used	O
.	O
)	O
</s>
<s>
Now	O
,	O
applying	O
to	O
both	O
and	O
,	O
and	O
assuming	O
no	O
hash	B-Error_Name
collisions	O
,	O
we	O
see	O
that	O
the	O
values	O
are	O
equal	O
(	O
)	O
if	O
and	O
only	O
if	O
among	O
all	O
elements	O
of	O
,	O
the	O
element	O
with	O
the	O
minimum	O
hash	B-Error_Name
value	I-Error_Name
lies	O
in	O
the	O
intersection	O
.	O
</s>
<s>
The	O
idea	O
of	O
the	O
MinHash	B-Algorithm
scheme	O
is	O
to	O
reduce	O
this	O
variance	O
by	O
averaging	O
together	O
several	O
variables	O
constructed	O
in	O
the	O
same	O
way	O
.	O
</s>
<s>
The	O
simplest	O
version	O
of	O
the	O
minhash	B-Algorithm
scheme	O
uses	O
different	O
hash	B-Error_Name
functions	I-Error_Name
,	O
where	O
is	O
a	O
fixed	O
integer	O
parameter	O
,	O
and	O
represents	O
each	O
set	O
by	O
the	O
values	O
of	O
for	O
these	O
functions	O
.	O
</s>
<s>
To	O
estimate	O
using	O
this	O
version	O
of	O
the	O
scheme	O
,	O
let	O
be	O
the	O
number	O
of	O
hash	B-Error_Name
functions	I-Error_Name
for	O
which	O
,	O
and	O
use	O
as	O
the	O
estimate	O
.	O
</s>
<s>
For	O
example	O
,	O
400	O
hashes	B-Error_Name
would	O
be	O
required	O
to	O
estimate	O
with	O
an	O
expected	O
error	O
less	O
than	O
or	O
equal	O
to	O
.05	O
.	O
</s>
<s>
It	O
may	O
be	O
computationally	O
expensive	O
to	O
compute	O
multiple	O
hash	B-Error_Name
functions	I-Error_Name
,	O
but	O
a	O
related	O
version	O
of	O
MinHash	B-Algorithm
scheme	O
avoids	O
this	O
penalty	O
by	O
using	O
only	O
a	O
single	O
hash	B-Error_Name
function	I-Error_Name
and	O
uses	O
it	O
to	O
select	O
multiple	O
values	O
from	O
each	O
set	O
rather	O
than	O
selecting	O
only	O
a	O
single	O
minimum	O
value	O
per	O
hash	B-Error_Name
function	I-Error_Name
.	O
</s>
<s>
Let	O
be	O
a	O
hash	B-Error_Name
function	I-Error_Name
,	O
and	O
let	O
be	O
a	O
fixed	O
integer	O
.	O
</s>
<s>
The	O
difference	O
between	O
this	O
estimator	O
and	O
the	O
estimator	O
produced	O
by	O
multiple	O
hash	B-Error_Name
functions	I-Error_Name
is	O
that	O
always	O
has	O
exactly	O
members	O
,	O
whereas	O
the	O
multiple	O
hash	B-Error_Name
functions	I-Error_Name
may	O
lead	O
to	O
a	O
smaller	O
number	O
of	O
sampled	O
elements	O
due	O
to	O
the	O
possibility	O
that	O
two	O
different	O
hash	B-Error_Name
functions	I-Error_Name
may	O
have	O
the	O
same	O
minima	O
.	O
</s>
<s>
By	O
standard	O
Chernoff	O
bounds	O
for	O
sampling	O
without	O
replacement	O
,	O
this	O
estimator	O
has	O
expected	O
error	O
,	O
matching	O
the	O
performance	O
of	O
the	O
multiple-hash-function	O
scheme	O
.	O
</s>
<s>
Specifically	O
,	O
for	O
set	O
size	O
the	O
many	O
hash	B-Error_Name
variant	O
takes	O
time	O
.	O
</s>
<s>
The	O
single	O
hash	B-Error_Name
variant	O
is	O
generally	O
faster	O
,	O
requiring	O
time	O
to	O
maintain	O
the	O
queue	O
of	O
minimum	O
hash	B-Error_Name
values	I-Error_Name
assuming	O
.	O
</s>
<s>
A	O
variety	O
of	O
techniques	O
to	O
introduce	O
weights	O
into	O
the	O
computation	O
of	O
MinHashes	B-Algorithm
have	O
been	O
developed	O
.	O
</s>
<s>
Extend	O
our	O
hash	B-Error_Name
function	I-Error_Name
to	O
accept	O
both	O
a	O
set	O
member	O
and	O
an	O
integer	O
,	O
then	O
generate	O
multiple	O
hashes	B-Error_Name
for	O
each	O
item	O
,	O
according	O
to	O
its	O
weight	O
.	O
</s>
<s>
If	O
item	O
occurs	O
times	O
,	O
generate	O
hashes	B-Error_Name
.	O
</s>
<s>
Run	O
the	O
original	O
algorithm	O
on	O
this	O
expanded	O
set	O
of	O
hashes	B-Error_Name
.	O
</s>
<s>
Another	O
family	O
of	O
extensions	O
use	O
exponentially	O
distributed	O
hashes	B-Error_Name
.	O
</s>
<s>
A	O
uniformly	O
random	O
hash	B-Error_Name
between	O
0	O
and	O
1	O
can	O
be	O
converted	O
to	O
follow	O
an	O
exponential	O
distribution	O
by	O
CDF	B-Algorithm
inversion	I-Algorithm
.	O
</s>
<s>
In	O
order	O
to	O
implement	O
the	O
MinHash	B-Algorithm
scheme	O
as	O
described	O
above	O
,	O
one	O
needs	O
the	O
hash	B-Error_Name
function	I-Error_Name
to	O
define	O
a	O
random	B-Algorithm
permutation	I-Algorithm
on	O
elements	O
,	O
where	O
is	O
the	O
total	O
number	O
of	O
distinct	O
elements	O
in	O
the	O
union	O
of	O
all	O
of	O
the	O
sets	O
to	O
be	O
compared	O
.	O
</s>
<s>
But	O
because	O
there	O
are	O
different	O
permutations	B-Algorithm
,	O
it	O
would	O
require	O
bits	O
just	O
to	O
specify	O
a	O
truly	O
random	B-Algorithm
permutation	I-Algorithm
,	O
an	O
infeasibly	O
large	O
number	O
for	O
even	O
moderate	O
values	O
of	O
.	O
</s>
<s>
Because	O
of	O
this	O
fact	O
,	O
by	O
analogy	O
to	O
the	O
theory	O
of	O
universal	B-Algorithm
hashing	I-Algorithm
,	O
there	O
has	O
been	O
significant	O
work	O
on	O
finding	O
a	O
family	O
of	O
permutations	B-Algorithm
that	O
is	O
"	O
min-wise	O
independent	O
"	O
,	O
meaning	O
that	O
for	O
any	O
subset	O
of	O
the	O
domain	O
,	O
any	O
element	O
is	O
equally	O
likely	O
to	O
be	O
the	O
minimum	O
.	O
</s>
<s>
different	O
permutations	B-Algorithm
,	O
and	O
therefore	O
that	O
it	O
needs	O
bits	O
to	O
specify	O
a	O
single	O
permutation	B-Algorithm
,	O
still	O
infeasibly	O
large	O
.	O
</s>
<s>
Because	O
of	O
the	O
above	O
impracticality	O
,	O
two	O
variant	O
notions	O
of	O
min-wise	O
independence	O
have	O
been	O
introduced	O
:	O
restricted	O
min-wise	O
independent	O
permutations	B-Algorithm
families	O
,	O
and	O
approximate	O
min-wise	O
independent	O
families	O
.	O
</s>
<s>
In	O
1999	O
Piotr	O
Indyk	O
proved	O
that	O
any	O
k-wise	O
independent	O
family	O
of	O
hash	B-Error_Name
functions	I-Error_Name
is	O
also	O
approximately	O
min-wise	O
independent	O
for	O
large	O
enough	O
.	O
</s>
<s>
This	O
guarantee	O
is	O
,	O
among	O
other	O
things	O
,	O
sufficient	O
to	O
give	O
the	O
Jaccard	O
bound	O
required	O
by	O
the	O
MinHash	B-Algorithm
algorithm	O
.	O
</s>
<s>
Since	O
k-wise	O
independent	O
hash	B-Error_Name
functions	I-Error_Name
can	O
be	O
specified	O
using	O
just	O
bits	O
,	O
this	O
approach	O
is	O
much	O
more	O
practical	O
than	O
using	O
completely	O
min-wise	O
independent	O
permutations	B-Algorithm
.	O
</s>
<s>
Another	O
practical	O
family	O
of	O
hash	B-Error_Name
functions	I-Error_Name
that	O
give	O
approximate	O
min-wise	O
independence	O
is	O
Tabulation	B-Algorithm
hashing	I-Algorithm
.	O
</s>
<s>
The	O
original	O
applications	O
for	O
MinHash	B-Algorithm
involved	O
clustering	B-Algorithm
and	O
eliminating	O
near-duplicates	O
among	O
web	O
documents	O
,	O
represented	O
as	O
sets	O
of	O
the	O
words	O
occurring	O
in	O
those	O
documents	O
.	O
</s>
<s>
Similar	O
techniques	O
have	O
also	O
been	O
used	O
for	O
clustering	B-Algorithm
and	O
near-duplicate	O
elimination	O
for	O
other	O
types	O
of	O
data	O
,	O
such	O
as	O
images	O
:	O
in	O
the	O
case	O
of	O
image	O
data	O
,	O
an	O
image	O
can	O
be	O
represented	O
as	O
a	O
set	O
of	O
smaller	O
subimages	O
cropped	O
from	O
it	O
,	O
or	O
as	O
sets	O
of	O
more	O
complex	O
image	O
feature	O
descriptions	O
.	O
</s>
<s>
In	O
data	B-Application
mining	I-Application
,	O
use	O
MinHash	B-Algorithm
as	O
a	O
tool	O
for	O
association	B-Algorithm
rule	I-Algorithm
learning	I-Algorithm
.	O
</s>
<s>
Given	O
a	O
database	O
in	O
which	O
each	O
entry	O
has	O
multiple	O
attributes	O
(	O
viewed	O
as	O
a	O
0	B-Algorithm
–	I-Algorithm
1	I-Algorithm
matrix	I-Algorithm
with	O
a	O
row	O
per	O
database	O
entry	O
and	O
a	O
column	O
per	O
attribute	O
)	O
they	O
use	O
MinHash-based	O
approximations	O
to	O
the	O
Jaccard	O
index	O
to	O
identify	O
candidate	O
pairs	O
of	O
attributes	O
that	O
frequently	O
co-occur	O
,	O
and	O
then	O
compute	O
the	O
exact	O
value	O
of	O
the	O
index	O
for	O
only	O
those	O
pairs	O
to	O
determine	O
the	O
ones	O
whose	O
frequencies	O
of	O
co-occurrence	O
are	O
below	O
a	O
given	O
strict	O
threshold	O
.	O
</s>
<s>
The	O
MinHash	B-Algorithm
algorithm	O
has	O
been	O
adapted	O
for	O
bioinformatics	O
,	O
where	O
the	O
problem	O
of	O
comparing	O
genome	O
sequences	O
has	O
a	O
similar	O
theoretical	O
underpinning	O
to	O
that	O
of	O
comparing	O
documents	O
on	O
the	O
web	O
.	O
</s>
<s>
MinHash-based	O
tools	O
allow	O
rapid	O
comparison	O
of	O
whole	O
genome	O
sequencing	O
data	O
with	O
reference	O
genomes	O
(	O
around	O
3	O
minutes	O
to	O
compare	O
one	O
genome	O
with	O
the	O
90000	O
reference	O
genomes	O
in	O
RefSeq	O
)	O
,	O
and	O
are	O
suitable	O
for	O
speciation	O
and	O
maybe	O
a	O
limited	O
degree	O
of	O
microbial	O
sub-typing	O
.	O
</s>
<s>
There	O
are	O
also	O
applications	O
for	O
metagenomics	O
and	O
the	O
use	O
of	O
MinHash	B-Algorithm
derived	O
algorithms	O
for	O
genome	O
alignment	O
and	O
genome	O
assembly	O
.	O
</s>
<s>
Accurate	O
average	O
nucleotide	O
identity	O
(	O
ANI	O
)	O
values	O
can	O
be	O
generated	O
very	O
efficiently	O
with	O
MinHash-based	O
algorithms	O
.	O
</s>
<s>
The	O
MinHash	B-Algorithm
scheme	O
may	O
be	O
seen	O
as	O
an	O
instance	O
of	O
locality-sensitive	B-Algorithm
hashing	I-Algorithm
,	O
a	O
collection	O
of	O
techniques	O
for	O
using	O
hash	B-Error_Name
functions	I-Error_Name
to	O
map	O
large	O
sets	O
of	O
objects	O
down	O
to	O
smaller	O
hash	B-Error_Name
values	I-Error_Name
in	O
such	O
a	O
way	O
that	O
,	O
when	O
two	O
objects	O
have	O
a	O
small	O
distance	O
from	O
each	O
other	O
,	O
their	O
hash	B-Error_Name
values	I-Error_Name
are	O
likely	O
to	O
be	O
the	O
same	O
.	O
</s>
<s>
In	O
this	O
instance	O
,	O
the	O
signature	O
of	O
a	O
set	O
may	O
be	O
seen	O
as	O
its	O
hash	B-Error_Name
value	I-Error_Name
.	O
</s>
<s>
Other	O
locality	B-Algorithm
sensitive	I-Algorithm
hashing	I-Algorithm
techniques	O
exist	O
for	O
Hamming	O
distance	O
between	O
sets	O
and	O
cosine	O
distance	O
between	O
vectors	O
;	O
locality	B-Algorithm
sensitive	I-Algorithm
hashing	I-Algorithm
has	O
important	O
applications	O
in	O
nearest	B-Algorithm
neighbor	I-Algorithm
search	I-Algorithm
algorithms	O
.	O
</s>
<s>
For	O
large	O
distributed	O
systems	O
,	O
and	O
in	O
particular	O
MapReduce	B-Operating_System
,	O
there	O
exist	O
modified	O
versions	O
of	O
MinHash	B-Algorithm
to	O
help	O
compute	O
similarities	O
with	O
no	O
dependence	O
on	O
the	O
point	O
dimension	O
.	O
</s>
<s>
A	O
large	O
scale	O
evaluation	O
was	O
conducted	O
by	O
Google	B-Application
in	O
2006	O
to	O
compare	O
the	O
performance	O
of	O
Minhash	B-Algorithm
and	O
SimHash	B-Algorithm
algorithms	O
.	O
</s>
<s>
In	O
2007	O
Google	B-Application
reported	O
using	O
Simhash	B-Algorithm
for	O
duplicate	O
detection	O
for	O
web	O
crawling	O
and	O
using	O
Minhash	B-Algorithm
and	O
LSH	B-Algorithm
for	O
Google	B-Application
News	I-Application
personalization	O
.	O
</s>
