<s>
Linear	B-Algorithm
probing	I-Algorithm
is	O
a	O
scheme	O
in	O
computer	B-General_Concept
programming	I-General_Concept
for	O
resolving	O
collisions	B-Algorithm
in	O
hash	B-Algorithm
tables	I-Algorithm
,	O
data	B-General_Concept
structures	I-General_Concept
for	O
maintaining	O
a	O
collection	O
of	O
key	B-Application
–	I-Application
value	I-Application
pairs	I-Application
and	O
looking	O
up	O
the	O
value	O
associated	O
with	O
a	O
given	O
key	O
.	O
</s>
<s>
Along	O
with	O
quadratic	B-Algorithm
probing	I-Algorithm
and	O
double	B-Algorithm
hashing	I-Algorithm
,	O
linear	B-Algorithm
probing	I-Algorithm
is	O
a	O
form	O
of	O
open	B-Algorithm
addressing	I-Algorithm
.	O
</s>
<s>
In	O
these	O
schemes	O
,	O
each	O
cell	O
of	O
a	O
hash	B-Algorithm
table	I-Algorithm
stores	O
a	O
single	O
key	B-Application
–	I-Application
value	I-Application
pair	I-Application
.	O
</s>
<s>
When	O
the	O
hash	B-Algorithm
function	I-Algorithm
causes	O
a	O
collision	O
by	O
mapping	O
a	O
new	O
key	O
to	O
a	O
cell	O
of	O
the	O
hash	B-Algorithm
table	I-Algorithm
that	O
is	O
already	O
occupied	O
by	O
another	O
key	O
,	O
linear	B-Algorithm
probing	I-Algorithm
searches	O
the	O
table	O
for	O
the	O
closest	O
following	O
free	O
location	O
and	O
inserts	O
the	O
new	O
key	O
there	O
.	O
</s>
<s>
Lookups	O
are	O
performed	O
in	O
the	O
same	O
way	O
,	O
by	O
searching	O
the	O
table	O
sequentially	O
starting	O
at	O
the	O
position	O
given	O
by	O
the	O
hash	B-Algorithm
function	I-Algorithm
,	O
until	O
finding	O
a	O
cell	O
with	O
a	O
matching	O
key	O
or	O
an	O
empty	O
cell	O
.	O
</s>
<s>
As	O
write	O
,	O
"	O
Hash	B-Algorithm
tables	I-Algorithm
are	O
the	O
most	O
commonly	O
used	O
nontrivial	O
data	B-General_Concept
structures	I-General_Concept
,	O
and	O
the	O
most	O
popular	O
implementation	O
on	O
standard	O
hardware	O
uses	O
linear	B-Algorithm
probing	I-Algorithm
,	O
which	O
is	O
both	O
fast	O
and	O
simple.	O
"	O
</s>
<s>
Linear	B-Algorithm
probing	I-Algorithm
can	O
provide	O
high	O
performance	O
because	O
of	O
its	O
good	O
locality	B-General_Concept
of	I-General_Concept
reference	I-General_Concept
,	O
but	O
is	O
more	O
sensitive	O
to	O
the	O
quality	O
of	O
its	O
hash	B-Algorithm
function	I-Algorithm
than	O
some	O
other	O
collision	O
resolution	O
schemes	O
.	O
</s>
<s>
It	O
takes	O
constant	O
expected	O
time	O
per	O
search	O
,	O
insertion	O
,	O
or	O
deletion	O
when	O
implemented	O
using	O
a	O
random	O
hash	B-Algorithm
function	I-Algorithm
,	O
a	O
5-independent	B-Error_Name
hash	I-Error_Name
function	I-Error_Name
,	O
or	O
tabulation	B-Algorithm
hashing	I-Algorithm
.	O
</s>
<s>
Good	O
results	O
can	O
also	O
be	O
achieved	O
in	O
practice	O
with	O
other	O
hash	B-Algorithm
functions	I-Algorithm
such	O
as	O
MurmurHash	B-Error_Name
.	O
</s>
<s>
Linear	B-Algorithm
probing	I-Algorithm
is	O
a	O
component	O
of	O
open	B-Algorithm
addressing	I-Algorithm
schemes	O
for	O
using	O
a	O
hash	B-Algorithm
table	I-Algorithm
to	O
solve	O
the	O
dictionary	B-Application
problem	I-Application
.	O
</s>
<s>
In	O
the	O
dictionary	B-Application
problem	I-Application
,	O
a	O
data	B-General_Concept
structure	I-General_Concept
should	O
maintain	O
a	O
collection	O
of	O
key	B-Application
–	I-Application
value	I-Application
pairs	I-Application
subject	O
to	O
operations	O
that	O
insert	O
or	O
delete	O
pairs	O
from	O
the	O
collection	O
or	O
that	O
search	O
for	O
the	O
value	O
associated	O
with	O
a	O
given	O
key	O
.	O
</s>
<s>
In	O
open	B-Algorithm
addressing	I-Algorithm
solutions	O
to	O
this	O
problem	O
,	O
the	O
data	B-General_Concept
structure	I-General_Concept
is	O
an	O
array	B-Data_Structure
(	O
the	O
hash	B-Algorithm
table	I-Algorithm
)	O
whose	O
cells	O
(	O
when	O
nonempty	O
)	O
each	O
store	O
a	O
single	O
key	B-Application
–	I-Application
value	I-Application
pair	I-Application
.	O
</s>
<s>
A	O
hash	B-Algorithm
function	I-Algorithm
is	O
used	O
to	O
map	O
each	O
key	O
into	O
the	O
cell	O
of	O
where	O
that	O
key	O
should	O
be	O
stored	O
,	O
typically	O
scrambling	O
the	O
keys	O
so	O
that	O
keys	O
with	O
similar	O
values	O
are	O
not	O
placed	O
near	O
each	O
other	O
in	O
the	O
table	O
.	O
</s>
<s>
A	O
hash	B-Algorithm
collision	I-Algorithm
occurs	O
when	O
the	O
hash	B-Algorithm
function	I-Algorithm
maps	O
a	O
key	O
into	O
a	O
cell	O
that	O
is	O
already	O
occupied	O
by	O
a	O
different	O
key	O
.	O
</s>
<s>
Linear	B-Algorithm
probing	I-Algorithm
is	O
a	O
strategy	O
for	O
resolving	O
collisions	B-Algorithm
,	O
by	O
placing	O
the	O
new	O
key	O
into	O
the	O
closest	O
following	O
empty	O
cell	O
.	O
</s>
<s>
To	O
search	O
for	O
a	O
given	O
key	O
,	O
the	O
cells	O
of	O
are	O
examined	O
,	O
beginning	O
with	O
the	O
cell	O
at	O
index	O
(	O
where	O
is	O
the	O
hash	B-Algorithm
function	I-Algorithm
)	O
and	O
continuing	O
to	O
the	O
adjacent	O
cells	O
,	O
,	O
...	O
,	O
until	O
finding	O
either	O
an	O
empty	O
cell	O
or	O
a	O
cell	O
whose	O
stored	O
key	O
is	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
the	O
search	O
returns	O
as	O
its	O
result	O
that	O
the	O
key	O
is	O
not	O
present	O
in	O
the	O
dictionary	B-Application
.	O
</s>
<s>
To	O
insert	O
a	O
key	B-Application
–	I-Application
value	I-Application
pair	I-Application
into	O
the	O
table	O
(	O
possibly	O
replacing	O
any	O
existing	O
pair	O
with	O
the	O
same	O
key	O
)	O
,	O
the	O
insertion	O
algorithm	O
follows	O
the	O
same	O
sequence	O
of	O
cells	O
that	O
would	O
be	O
followed	O
for	O
a	O
search	O
,	O
until	O
finding	O
either	O
an	O
empty	O
cell	O
or	O
a	O
cell	O
whose	O
stored	O
key	O
is	O
.	O
</s>
<s>
The	O
new	O
key	B-Application
–	I-Application
value	I-Application
pair	I-Application
is	O
then	O
placed	O
into	O
that	O
cell	O
.	O
</s>
<s>
If	O
the	O
insertion	O
would	O
cause	O
the	O
load	O
factor	O
of	O
the	O
table	O
(	O
its	O
fraction	O
of	O
occupied	O
cells	O
)	O
to	O
grow	O
above	O
some	O
preset	O
threshold	O
,	O
the	O
whole	O
table	O
may	O
be	O
replaced	O
by	O
a	O
new	O
table	O
,	O
larger	O
by	O
a	O
constant	O
factor	O
,	O
with	O
a	O
new	O
hash	B-Algorithm
function	I-Algorithm
,	O
as	O
in	O
a	O
dynamic	B-Data_Structure
array	I-Data_Structure
.	O
</s>
<s>
Setting	O
this	O
threshold	O
close	O
to	O
zero	O
and	O
using	O
a	O
high	O
growth	O
rate	O
for	O
the	O
table	O
size	O
leads	O
to	O
faster	O
hash	B-Algorithm
table	I-Algorithm
operations	O
but	O
greater	O
memory	O
usage	O
than	O
threshold	O
values	O
close	O
to	O
one	O
and	O
low	O
growth	O
rates	O
.	O
</s>
<s>
It	O
is	O
also	O
possible	O
to	O
remove	O
a	O
key	B-Application
–	I-Application
value	I-Application
pair	I-Application
from	O
the	O
dictionary	B-Application
.	O
</s>
<s>
This	O
would	O
affect	O
searches	O
for	O
other	O
keys	O
that	O
have	O
a	O
hash	B-Error_Name
value	I-Error_Name
earlier	O
than	O
the	O
emptied	O
cell	O
,	O
but	O
that	O
are	O
stored	O
in	O
a	O
position	O
later	O
than	O
the	O
emptied	O
cell	O
.	O
</s>
<s>
Instead	O
,	O
when	O
a	O
cell	O
is	O
emptied	O
,	O
it	O
is	O
necessary	O
to	O
search	O
forward	O
through	O
the	O
following	O
cells	O
of	O
the	O
table	O
until	O
finding	O
either	O
another	O
empty	O
cell	O
or	O
a	O
key	O
that	O
can	O
be	O
moved	O
to	O
cell	O
(	O
that	O
is	O
,	O
a	O
key	O
whose	O
hash	B-Error_Name
value	I-Error_Name
is	O
equal	O
to	O
or	O
earlier	O
than	O
)	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
time	O
to	O
complete	O
the	O
whole	O
process	O
is	O
proportional	O
to	O
the	O
length	O
of	O
the	O
block	O
of	O
occupied	O
cells	O
containing	O
the	O
deleted	O
key	O
,	O
matching	O
the	O
running	O
time	O
of	O
the	O
other	O
hash	B-Algorithm
table	I-Algorithm
operations	O
.	O
</s>
<s>
Alternatively	O
,	O
it	O
is	O
possible	O
to	O
use	O
a	O
lazy	B-Algorithm
deletion	I-Algorithm
strategy	O
in	O
which	O
a	O
key	B-Application
–	I-Application
value	I-Application
pair	I-Application
is	O
removed	O
by	O
replacing	O
the	O
value	O
by	O
a	O
special	O
flag	B-Data_Structure
value	I-Data_Structure
indicating	O
a	O
deleted	O
key	O
.	O
</s>
<s>
However	O
,	O
these	O
flag	B-Data_Structure
values	I-Data_Structure
will	O
contribute	O
to	O
the	O
load	O
factor	O
of	O
the	O
hash	B-Algorithm
table	I-Algorithm
.	O
</s>
<s>
With	O
this	O
strategy	O
,	O
it	O
may	O
become	O
necessary	O
to	O
clean	O
the	O
flag	B-Data_Structure
values	I-Data_Structure
out	O
of	O
the	O
array	B-Data_Structure
and	O
rehash	B-Algorithm
all	O
the	O
remaining	O
key	B-Application
–	I-Application
value	I-Application
pairs	I-Application
once	O
too	O
large	O
a	O
fraction	O
of	O
the	O
array	B-Data_Structure
becomes	O
occupied	O
by	O
deleted	O
keys	O
.	O
</s>
<s>
Linear	B-Algorithm
probing	I-Algorithm
provides	O
good	O
locality	B-General_Concept
of	I-General_Concept
reference	I-General_Concept
,	O
which	O
causes	O
it	O
to	O
require	O
few	O
uncached	O
memory	O
accesses	O
per	O
operation	O
.	O
</s>
<s>
However	O
,	O
compared	O
to	O
some	O
other	O
open	B-Algorithm
addressing	I-Algorithm
strategies	O
,	O
its	O
performance	O
degrades	O
more	O
quickly	O
at	O
high	O
load	O
factors	O
because	O
of	O
primary	B-Algorithm
clustering	I-Algorithm
,	O
a	O
tendency	O
for	O
one	O
collision	O
to	O
cause	O
more	O
nearby	O
collisions	B-Algorithm
.	O
</s>
<s>
Additionally	O
,	O
achieving	O
good	O
performance	O
with	O
this	O
method	O
requires	O
a	O
higher-quality	O
hash	B-Algorithm
function	I-Algorithm
than	O
for	O
some	O
other	O
collision	O
resolution	O
schemes	O
.	O
</s>
<s>
When	O
used	O
with	O
low-quality	O
hash	B-Algorithm
functions	I-Algorithm
that	O
fail	O
to	O
eliminate	O
nonuniformities	O
in	O
the	O
input	O
distribution	O
,	O
linear	B-Algorithm
probing	I-Algorithm
can	O
be	O
slower	O
than	O
other	O
open-addressing	O
strategies	O
such	O
as	O
double	B-Algorithm
hashing	I-Algorithm
,	O
which	O
probes	O
a	O
sequence	O
of	O
cells	O
whose	O
separation	O
is	O
determined	O
by	O
a	O
second	O
hash	B-Algorithm
function	I-Algorithm
,	O
or	O
quadratic	B-Algorithm
probing	I-Algorithm
,	O
where	O
the	O
size	O
of	O
each	O
step	O
varies	O
depending	O
on	O
its	O
position	O
within	O
the	O
probe	O
sequence	O
.	O
</s>
<s>
Using	O
linear	B-Algorithm
probing	I-Algorithm
,	O
dictionary	B-Application
operations	O
can	O
be	O
implemented	O
in	O
constant	O
expected	O
time	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
insert	O
,	O
remove	O
and	O
search	O
operations	O
can	O
be	O
implemented	O
in	O
O(1 )	O
,	O
as	O
long	O
as	O
the	O
load	O
factor	O
of	O
the	O
hash	B-Algorithm
table	I-Algorithm
is	O
a	O
constant	O
strictly	O
less	O
than	O
one	O
.	O
</s>
<s>
If	O
all	O
starting	O
cells	O
are	O
equally	O
likely	O
,	O
in	O
a	O
hash	B-Algorithm
table	I-Algorithm
with	O
cells	O
,	O
then	O
a	O
maximal	O
block	O
of	O
occupied	O
cells	O
will	O
have	O
probability	O
of	O
containing	O
the	O
starting	O
location	O
of	O
a	O
search	O
,	O
and	O
will	O
take	O
time	O
whenever	O
it	O
is	O
the	O
starting	O
location	O
.	O
</s>
<s>
A	O
similar	O
sum	O
of	O
squared	O
block	O
lengths	O
gives	O
the	O
expected	O
time	O
bound	O
for	O
a	O
random	O
hash	B-Algorithm
function	I-Algorithm
(	O
rather	O
than	O
for	O
a	O
random	O
starting	O
location	O
into	O
a	O
specific	O
state	O
of	O
the	O
hash	B-Algorithm
table	I-Algorithm
)	O
,	O
by	O
summing	O
over	O
all	O
the	O
blocks	O
that	O
could	O
exist	O
(	O
rather	O
than	O
the	O
ones	O
that	O
actually	O
exist	O
in	O
a	O
given	O
state	O
of	O
the	O
table	O
)	O
,	O
and	O
multiplying	O
the	O
term	O
for	O
each	O
potential	O
block	O
by	O
the	O
probability	O
that	O
the	O
block	O
is	O
actually	O
occupied	O
.	O
</s>
<s>
at	O
least	O
elements	O
have	O
hash	B-Error_Name
values	I-Error_Name
that	O
lie	O
within	O
a	O
block	O
of	O
cells	O
of	O
length	O
.	O
</s>
<s>
Because	O
linear	B-Algorithm
probing	I-Algorithm
is	O
especially	O
sensitive	O
to	O
unevenly	O
distributed	O
hash	B-Error_Name
values	I-Error_Name
,	O
it	O
is	O
important	O
to	O
combine	O
it	O
with	O
a	O
high-quality	O
hash	B-Algorithm
function	I-Algorithm
that	O
does	O
not	O
produce	O
such	O
irregularities	O
.	O
</s>
<s>
The	O
analysis	O
above	O
assumes	O
that	O
each	O
key	O
's	O
hash	B-Algorithm
is	O
a	O
random	O
number	O
independent	O
of	O
the	O
hashes	B-Algorithm
of	O
all	O
the	O
other	O
keys	O
.	O
</s>
<s>
However	O
,	O
random	O
or	O
pseudorandom	B-Error_Name
hash	B-Error_Name
values	I-Error_Name
may	O
be	O
used	O
when	O
hashing	O
objects	O
by	O
their	O
identity	O
rather	O
than	O
by	O
their	O
value	O
.	O
</s>
<s>
For	O
instance	O
,	O
this	O
is	O
done	O
using	O
linear	B-Algorithm
probing	I-Algorithm
by	O
the	O
IdentityHashMap	O
class	O
of	O
the	O
Java	B-Language
collections	I-Language
framework	I-Language
.	O
</s>
<s>
The	O
hash	B-Error_Name
value	I-Error_Name
that	O
this	O
class	O
associates	O
with	O
each	O
object	O
,	O
its	O
identityHashCode	O
,	O
is	O
guaranteed	O
to	O
remain	O
fixed	O
for	O
the	O
lifetime	O
of	O
an	O
object	O
but	O
is	O
otherwise	O
arbitrary	O
.	O
</s>
<s>
Because	O
the	O
identityHashCode	O
is	O
constructed	O
only	O
once	O
per	O
object	O
,	O
and	O
is	O
not	O
required	O
to	O
be	O
related	O
to	O
the	O
object	O
's	O
address	O
or	O
value	O
,	O
its	O
construction	O
may	O
involve	O
slower	O
computations	O
such	O
as	O
the	O
call	O
to	O
a	O
random	O
or	O
pseudorandom	B-Error_Name
number	I-Error_Name
generator	O
.	O
</s>
<s>
For	O
instance	O
,	O
Java	O
8	O
uses	O
an	O
Xorshift	B-Algorithm
pseudorandom	B-Error_Name
number	I-Error_Name
generator	O
to	O
construct	O
these	O
values	O
.	O
</s>
<s>
For	O
most	O
applications	O
of	O
hashing	O
,	O
it	O
is	O
necessary	O
to	O
compute	O
the	O
hash	B-Algorithm
function	I-Algorithm
for	O
each	O
value	O
every	O
time	O
that	O
it	O
is	O
hashed	O
,	O
rather	O
than	O
once	O
when	O
its	O
object	O
is	O
created	O
.	O
</s>
<s>
In	O
such	O
applications	O
,	O
random	O
or	O
pseudorandom	B-Error_Name
numbers	I-Error_Name
cannot	O
be	O
used	O
as	O
hash	B-Error_Name
values	I-Error_Name
,	O
because	O
then	O
different	O
objects	O
with	O
the	O
same	O
value	O
would	O
have	O
different	O
hashes	B-Algorithm
.	O
</s>
<s>
And	O
cryptographic	B-Algorithm
hash	I-Algorithm
functions	I-Algorithm
(	O
which	O
are	O
designed	O
to	O
be	O
computationally	O
indistinguishable	O
from	O
truly	O
random	O
functions	O
)	O
are	O
usually	O
too	O
slow	O
to	O
be	O
used	O
in	O
hash	B-Algorithm
tables	I-Algorithm
.	O
</s>
<s>
Instead	O
,	O
other	O
methods	O
for	O
constructing	O
hash	B-Algorithm
functions	I-Algorithm
have	O
been	O
devised	O
.	O
</s>
<s>
These	O
methods	O
compute	O
the	O
hash	B-Algorithm
function	I-Algorithm
quickly	O
,	O
and	O
can	O
be	O
proven	O
to	O
work	O
well	O
with	O
linear	B-Algorithm
probing	I-Algorithm
.	O
</s>
<s>
In	O
particular	O
,	O
linear	B-Algorithm
probing	I-Algorithm
has	O
been	O
analyzed	O
from	O
the	O
framework	O
of	O
-independent	O
hashing	O
,	O
a	O
class	O
of	O
hash	B-Algorithm
functions	I-Algorithm
that	O
are	O
initialized	O
from	O
a	O
small	O
random	O
seed	O
and	O
that	O
are	O
equally	O
likely	O
to	O
map	O
any	O
-tuple	O
of	O
distinct	O
keys	O
to	O
any	O
-tuple	O
of	O
indexes	O
.	O
</s>
<s>
The	O
parameter	O
can	O
be	O
thought	O
of	O
as	O
a	O
measure	O
of	O
hash	B-Algorithm
function	I-Algorithm
quality	O
:	O
the	O
larger	O
is	O
,	O
the	O
more	O
time	O
it	O
will	O
take	O
to	O
compute	O
the	O
hash	B-Algorithm
function	I-Algorithm
but	O
it	O
will	O
behave	O
more	O
similarly	O
to	O
completely	O
random	O
functions	O
.	O
</s>
<s>
For	O
linear	B-Algorithm
probing	I-Algorithm
,	O
5-independence	O
is	O
enough	O
to	O
guarantee	O
constant	O
expected	O
time	O
per	O
operation	O
,	O
</s>
<s>
while	O
some	O
4-independent	O
hash	B-Algorithm
functions	I-Algorithm
perform	O
badly	O
,	O
taking	O
up	O
to	O
logarithmic	O
time	O
per	O
operation	O
.	O
</s>
<s>
Another	O
method	O
of	O
constructing	O
hash	B-Algorithm
functions	I-Algorithm
with	O
both	O
high	O
quality	O
and	O
practical	O
speed	O
is	O
tabulation	B-Algorithm
hashing	I-Algorithm
.	O
</s>
<s>
In	O
this	O
method	O
,	O
the	O
hash	B-Error_Name
value	I-Error_Name
for	O
a	O
key	O
is	O
computed	O
by	O
using	O
each	O
byte	O
of	O
the	O
key	O
as	O
an	O
index	O
into	O
a	O
table	O
of	O
random	O
numbers	O
(	O
with	O
a	O
different	O
table	O
for	O
each	O
byte	O
position	O
)	O
.	O
</s>
<s>
Hash	B-Algorithm
functions	I-Algorithm
constructed	O
this	O
way	O
are	O
only	O
3-independent	O
.	O
</s>
<s>
Nevertheless	O
,	O
linear	B-Algorithm
probing	I-Algorithm
using	O
these	O
hash	B-Algorithm
functions	I-Algorithm
takes	O
constant	O
expected	O
time	O
per	O
operation	O
.	O
</s>
<s>
Both	O
tabulation	B-Algorithm
hashing	I-Algorithm
and	O
standard	O
methods	O
for	O
generating	O
5-independent	O
hash	B-Algorithm
functions	I-Algorithm
are	O
limited	O
to	O
keys	O
that	O
have	O
a	O
fixed	O
number	O
of	O
bits	O
.	O
</s>
<s>
To	O
handle	O
strings	O
or	O
other	O
types	O
of	O
variable-length	O
keys	O
,	O
it	O
is	O
possible	O
to	O
compose	B-Application
a	O
simpler	O
universal	B-Algorithm
hashing	I-Algorithm
technique	O
that	O
maps	O
the	O
keys	O
to	O
intermediate	O
values	O
and	O
a	O
higher	O
quality	O
(	O
5-independent	O
or	O
tabulation	O
)	O
hash	B-Algorithm
function	I-Algorithm
that	O
maps	O
the	O
intermediate	O
values	O
to	O
hash	B-Algorithm
table	I-Algorithm
indices	O
.	O
</s>
<s>
found	O
that	O
the	O
Multiply-Shift	O
family	O
of	O
hash	B-Algorithm
functions	I-Algorithm
(	O
defined	O
as	O
)	O
was	O
"	O
the	O
fastest	O
hash	B-Algorithm
function	I-Algorithm
when	O
integrated	O
with	O
all	O
hashing	O
schemes	O
,	O
i.e.	O
,	O
producing	O
the	O
highest	O
throughputs	O
and	O
also	O
of	O
good	O
quality	O
"	O
whereas	O
tabulation	B-Algorithm
hashing	I-Algorithm
produced	O
"	O
the	O
lowest	O
throughput	O
"	O
.	O
</s>
<s>
They	O
also	O
found	O
MurmurHash	B-Error_Name
to	O
be	O
superior	O
than	O
tabulation	B-Algorithm
hashing	I-Algorithm
:	O
"	O
By	O
studying	O
the	O
results	O
provided	O
by	O
Mult	O
and	O
Murmur	O
,	O
we	O
think	O
that	O
the	O
trade-off	O
for	O
by	O
tabulation	O
(	O
...	O
)	O
is	O
less	O
attractive	O
in	O
practice	O
"	O
.	O
</s>
<s>
The	O
idea	O
of	O
an	O
associative	B-Application
array	I-Application
that	O
allows	O
data	O
to	O
be	O
accessed	O
by	O
its	O
value	O
rather	O
than	O
by	O
its	O
address	O
dates	O
back	O
to	O
the	O
mid-1940s	O
in	O
the	O
work	O
of	O
Konrad	O
Zuse	O
and	O
Vannevar	O
Bush	O
,	O
but	O
hash	B-Algorithm
tables	I-Algorithm
were	O
not	O
described	O
until	O
1953	O
,	O
in	O
an	O
IBM	O
memorandum	O
by	O
Hans	O
Peter	O
Luhn	O
.	O
</s>
<s>
Luhn	O
used	O
a	O
different	O
collision	O
resolution	O
method	O
,	O
chaining	O
,	O
rather	O
than	O
linear	B-Algorithm
probing	I-Algorithm
.	O
</s>
<s>
summarizes	O
the	O
early	O
history	O
of	O
linear	B-Algorithm
probing	I-Algorithm
.	O
</s>
<s>
It	O
was	O
the	O
first	O
open	B-Algorithm
addressing	I-Algorithm
method	O
,	O
and	O
was	O
originally	O
synonymous	O
with	O
open	B-Algorithm
addressing	I-Algorithm
.	O
</s>
<s>
According	O
to	O
Knuth	O
,	O
it	O
was	O
first	O
used	O
by	O
Gene	O
Amdahl	O
,	O
Elaine	O
M	O
.	O
McGraw	O
(	O
née	O
Boehme	O
)	O
,	O
and	O
Arthur	O
Samuel	O
in	O
1954	O
,	O
in	O
an	O
assembler	B-Language
program	I-Language
for	O
the	O
IBM	B-Device
701	I-Device
computer	O
.	O
</s>
<s>
The	O
first	O
published	O
description	O
of	O
linear	B-Algorithm
probing	I-Algorithm
is	O
by	O
,	O
who	O
also	O
credits	O
Samuel	O
,	O
Amdahl	O
,	O
and	O
Boehme	O
but	O
adds	O
that	O
"	O
the	O
system	O
is	O
so	O
natural	O
,	O
that	O
it	O
very	O
likely	O
may	O
have	O
been	O
conceived	O
independently	O
by	O
others	O
either	O
before	O
or	O
since	O
that	O
time	O
"	O
.	O
</s>
<s>
The	O
first	O
theoretical	O
analysis	O
of	O
linear	B-Algorithm
probing	I-Algorithm
,	O
showing	O
that	O
it	O
takes	O
constant	O
expected	O
time	O
per	O
operation	O
with	O
random	O
hash	B-Algorithm
functions	I-Algorithm
,	O
was	O
given	O
by	O
Knuth	O
.	O
</s>
<s>
Significant	O
later	O
developments	O
include	O
a	O
more	O
detailed	O
analysis	O
of	O
the	O
probability	O
distribution	O
of	O
the	O
running	O
time	O
,	O
and	O
the	O
proof	O
that	O
linear	B-Algorithm
probing	I-Algorithm
runs	O
in	O
constant	O
time	O
per	O
operation	O
with	O
practically	O
usable	O
hash	B-Algorithm
functions	I-Algorithm
rather	O
than	O
with	O
the	O
idealized	O
random	O
functions	O
assumed	O
by	O
earlier	O
analysis	O
.	O
</s>
