<s>
(	O
also	O
known	O
in	O
applied	O
mathematics	O
as	O
the	O
cardinality	B-Algorithm
estimation	I-Algorithm
problem	O
)	O
is	O
the	O
problem	O
of	O
finding	O
the	O
number	O
of	O
distinct	O
elements	O
in	O
a	O
data	O
stream	O
with	O
repeated	O
elements	O
.	O
</s>
<s>
The	O
elements	O
might	O
represent	O
IP	B-Protocol
addresses	I-Protocol
of	O
packets	O
passing	O
through	O
a	O
router	B-Protocol
,	O
unique	O
visitors	O
to	O
a	O
web	O
site	O
,	O
elements	O
in	O
a	O
large	O
database	O
,	O
motifs	O
in	O
a	O
DNA	O
sequence	O
,	O
or	O
elements	O
of	O
RFID/sensor	O
networks	O
.	O
</s>
<s>
An	O
example	O
of	O
an	O
instance	O
for	O
the	O
cardinality	B-Algorithm
estimation	I-Algorithm
problem	O
is	O
the	O
stream	O
:	O
.	O
</s>
<s>
Initialize	O
an	O
efficient	O
dictionary	O
data	O
structure	O
,	O
,	O
such	O
as	O
hash	B-Error_Name
table	O
or	O
search	O
tree	O
in	O
which	O
insertion	O
and	O
membership	O
can	O
be	O
performed	O
quickly	O
.	O
</s>
<s>
State-of-the-art	O
estimators	O
hash	B-Error_Name
every	O
element	O
into	O
a	O
low-dimensional	O
data	O
sketch	O
using	O
a	O
hash	B-Error_Name
function	I-Error_Name
,	O
.	O
</s>
<s>
The	O
estimator	O
of	O
choice	O
in	O
practice	O
is	O
the	O
HyperLogLog	B-Algorithm
algorithm	O
.	O
</s>
<s>
The	O
hash	B-Error_Name
function	I-Error_Name
guarantees	O
that	O
is	O
identical	O
for	O
all	O
the	O
appearances	O
of	O
.	O
</s>
<s>
As	O
an	O
application	O
example	O
,	O
could	O
be	O
IP	B-Protocol
packets	O
received	O
by	O
a	O
server	O
.	O
</s>
<s>
Each	O
packet	O
belongs	O
to	O
one	O
of	O
IP	B-Protocol
flows	O
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
HyperLogLog	B-Algorithm
algorithm	O
can	O
be	O
extended	O
to	O
solve	O
the	O
weighted	O
problem	O
.	O
</s>
<s>
The	O
extended	O
HyperLogLog	B-Algorithm
algorithm	O
offers	O
the	O
best	O
performance	O
,	O
in	O
terms	O
of	O
statistical	O
accuracy	O
and	O
memory	O
usage	O
,	O
among	O
all	O
the	O
other	O
known	O
algorithms	O
for	O
the	O
weighted	O
problem	O
.	O
</s>
