<s>
In	O
computer	B-General_Concept
science	I-General_Concept
,	O
best	B-General_Concept
,	I-General_Concept
worst	I-General_Concept
,	I-General_Concept
and	I-General_Concept
average	I-General_Concept
cases	I-General_Concept
of	O
a	O
given	O
algorithm	O
express	O
what	O
the	O
resource	B-General_Concept
usage	O
is	O
at	O
least	O
,	O
at	O
most	O
and	O
on	O
average	O
,	O
respectively	O
.	O
</s>
<s>
Usually	O
the	O
resource	B-General_Concept
being	O
considered	O
is	O
running	O
time	O
,	O
i.e.	O
</s>
<s>
time	O
complexity	O
,	O
but	O
could	O
also	O
be	O
memory	O
or	O
some	O
other	O
resource	B-General_Concept
.	O
</s>
<s>
Best	B-General_Concept
case	I-General_Concept
is	O
the	O
function	O
which	O
performs	O
the	O
minimum	O
number	O
of	O
steps	O
on	O
input	O
data	O
of	O
n	O
elements	O
.	O
</s>
<s>
Worst	B-General_Concept
case	I-General_Concept
is	O
the	O
function	O
which	O
performs	O
the	O
maximum	O
number	O
of	O
steps	O
on	O
input	O
data	O
of	O
size	O
n	O
.	O
Average	B-General_Concept
case	I-General_Concept
is	O
the	O
function	O
which	O
performs	O
an	O
average	O
number	O
of	O
steps	O
on	O
input	O
data	O
of	O
n	O
elements	O
.	O
</s>
<s>
In	O
real-time	B-General_Concept
computing	I-General_Concept
,	O
the	O
worst-case	B-General_Concept
execution	I-General_Concept
time	I-General_Concept
is	O
often	O
of	O
particular	O
concern	O
since	O
it	O
is	O
important	O
to	O
know	O
how	O
much	O
time	O
might	O
be	O
needed	O
in	O
the	O
worst	B-General_Concept
case	I-General_Concept
to	O
guarantee	O
that	O
the	O
algorithm	O
will	O
always	O
finish	O
on	O
time	O
.	O
</s>
<s>
Average	B-General_Concept
performance	I-General_Concept
and	O
worst-case	B-General_Concept
performance	I-General_Concept
are	O
the	O
most	O
used	O
in	O
algorithm	O
analysis	O
.	O
</s>
<s>
Less	O
widely	O
found	O
is	O
best-case	B-General_Concept
performance	I-General_Concept
,	O
but	O
it	O
does	O
have	O
uses	O
:	O
for	O
example	O
,	O
where	O
the	O
best	B-General_Concept
cases	I-General_Concept
of	O
individual	O
tasks	O
are	O
known	O
,	O
they	O
can	O
be	O
used	O
to	O
improve	O
the	O
accuracy	O
of	O
an	O
overall	O
worst-case	B-General_Concept
analysis	O
.	O
</s>
<s>
Computer	B-General_Concept
scientists	I-General_Concept
use	O
probabilistic	B-General_Concept
analysis	I-General_Concept
techniques	O
,	O
especially	O
expected	O
value	O
,	O
to	O
determine	O
expected	O
running	O
times	O
.	O
</s>
<s>
The	O
terms	O
are	O
used	O
in	O
other	O
contexts	O
;	O
for	O
example	O
the	O
worst	O
-	O
and	O
best-case	O
outcome	O
of	O
an	O
epidemic	O
,	O
worst-case	B-General_Concept
temperature	O
to	O
which	O
an	O
electronic	O
circuit	O
element	O
is	O
exposed	O
,	O
etc	O
.	O
</s>
<s>
Where	O
components	O
of	O
specified	O
tolerance	O
are	O
used	O
,	O
devices	O
must	O
be	O
designed	O
to	O
work	O
properly	O
with	O
the	O
worst-case	B-General_Concept
combination	O
of	O
tolerances	O
and	O
external	O
conditions	O
.	O
</s>
<s>
The	O
term	O
best-case	B-General_Concept
performance	I-General_Concept
is	O
used	O
in	O
computer	B-General_Concept
science	I-General_Concept
to	O
describe	O
an	O
algorithm	O
's	O
behavior	O
under	O
optimal	O
conditions	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
best	B-General_Concept
case	I-General_Concept
for	O
a	O
simple	O
linear	B-Algorithm
search	I-Algorithm
on	O
a	O
list	O
occurs	O
when	O
the	O
desired	O
element	O
is	O
the	O
first	O
element	O
of	O
the	O
list	O
.	O
</s>
<s>
Development	O
and	O
choice	O
of	O
algorithms	O
is	O
rarely	O
based	O
on	O
best-case	B-General_Concept
performance	I-General_Concept
:	O
most	O
academic	O
and	O
commercial	O
enterprises	O
are	O
more	O
interested	O
in	O
improving	O
Average-case	B-General_Concept
complexity	O
and	O
worst-case	B-General_Concept
performance	I-General_Concept
.	O
</s>
<s>
Worst-case	B-General_Concept
performance	I-General_Concept
analysis	O
and	O
average-case	B-General_Concept
performance	O
analysis	O
have	O
some	O
similarities	O
,	O
but	O
in	O
practice	O
usually	O
require	O
different	O
tools	O
and	O
approaches	O
.	O
</s>
<s>
Similarly	O
,	O
even	O
when	O
a	O
sensible	O
description	O
of	O
a	O
particular	O
"	O
average	B-General_Concept
case	I-General_Concept
"	O
(	O
which	O
will	O
probably	O
only	O
be	O
applicable	O
for	O
some	O
uses	O
of	O
the	O
algorithm	O
)	O
is	O
possible	O
,	O
they	O
tend	O
to	O
result	O
in	O
more	O
difficult	O
analysis	O
of	O
equations	O
.	O
</s>
<s>
Worst-case	B-General_Concept
analysis	O
gives	O
a	O
safe	O
analysis	O
(	O
the	O
worst	B-General_Concept
case	I-General_Concept
is	O
never	O
underestimated	O
)	O
,	O
but	O
one	O
which	O
can	O
be	O
overly	O
pessimistic	O
,	O
since	O
there	O
may	O
be	O
no	O
(	O
realistic	O
)	O
input	O
that	O
would	O
take	O
this	O
many	O
steps	O
.	O
</s>
<s>
One	O
modern	O
approach	O
in	O
academic	O
theory	O
to	O
bridge	O
the	O
gap	O
between	O
worst-case	B-General_Concept
and	O
average-case	B-General_Concept
analysis	I-General_Concept
is	O
called	O
smoothed	O
analysis	O
.	O
</s>
<s>
When	O
analyzing	O
algorithms	O
which	O
often	O
take	O
a	O
small	O
time	O
to	O
complete	O
,	O
but	O
periodically	O
require	O
a	O
much	O
larger	O
time	O
,	O
amortized	B-General_Concept
analysis	I-General_Concept
can	O
be	O
used	O
to	O
determine	O
the	O
worst-case	B-General_Concept
running	O
time	O
over	O
a	O
(	O
possibly	O
infinite	O
)	O
series	O
of	O
operations	O
.	O
</s>
<s>
This	O
amortized	B-General_Concept
cost	O
can	O
be	O
much	O
closer	O
to	O
the	O
average	O
cost	O
,	O
while	O
still	O
providing	O
a	O
guaranteed	O
upper	O
limit	O
on	O
the	O
running	O
time	O
.	O
</s>
<s>
online	B-Algorithm
algorithms	I-Algorithm
are	O
frequently	O
based	O
on	O
amortized	B-General_Concept
analysis	I-General_Concept
.	O
</s>
<s>
The	O
worst-case	B-General_Concept
analysis	O
is	O
related	O
to	O
the	O
worst-case	B-General_Concept
complexity	I-General_Concept
.	O
</s>
<s>
Many	O
algorithms	O
with	O
bad	O
worst-case	B-General_Concept
performance	I-General_Concept
have	O
good	O
average-case	B-General_Concept
performance	O
.	O
</s>
<s>
Here	O
methods	O
like	O
random	O
self-reducibility	O
can	O
be	O
used	O
for	O
some	O
specific	O
problems	O
to	O
show	O
that	O
the	O
worst	B-General_Concept
case	I-General_Concept
is	O
no	O
harder	O
than	O
the	O
average	B-General_Concept
case	I-General_Concept
,	O
or	O
,	O
equivalently	O
,	O
that	O
the	O
average	B-General_Concept
case	I-General_Concept
is	O
no	O
easier	O
than	O
the	O
worst	B-General_Concept
case	I-General_Concept
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
some	O
data	O
structures	O
like	O
hash	B-Algorithm
tables	I-Algorithm
have	O
very	O
poor	O
worst-case	B-General_Concept
behaviors	O
,	O
but	O
a	O
well	O
written	O
hash	B-Algorithm
table	I-Algorithm
of	O
sufficient	O
size	O
will	O
statistically	O
never	O
give	O
the	O
worst	B-General_Concept
case	I-General_Concept
;	O
the	O
average	O
number	O
of	O
operations	O
performed	O
follows	O
an	O
exponential	O
decay	O
curve	O
,	O
and	O
so	O
the	O
run	O
time	O
of	O
an	O
operation	O
is	O
statistically	O
bounded	O
.	O
</s>
<s>
Algorithm	O
Data	O
structure	O
Time	O
complexity:Best	O
Time	O
complexity:Average	O
Time	O
complexity:Worst	O
Space	O
complexity:Worst	O
Quick	B-Algorithm
sort	I-Algorithm
Array	O
O( n	O
log(n )	O
)	O
O( n	O
log(n )	O
)	O
O(n2 )	O
O(n )	O
Merge	O
sort	O
Array	O
O( n	O
log(n )	O
)	O
O( n	O
log(n )	O
)	O
O( n	O
log(n )	O
)	O
O(n )	O
Heap	O
sort	O
Array	O
O( n	O
log(n )	O
)	O
O( n	O
log(n )	O
)	O
O( n	O
log(n )	O
)	O
O(1 )	O
Smooth	O
sort	O
Array	O
O(n )	O
O( n	O
log(n )	O
)	O
O( n	O
log(n )	O
)	O
O(1 )	O
Bubble	O
sort	O
Array	O
O(n )	O
O(n2 )	O
O(n2 )	O
O(1 )	O
Insertion	B-Algorithm
sort	I-Algorithm
Array	O
O(n )	O
O(n2 )	O
O(n2 )	O
O(1 )	O
Selection	O
sort	O
Array	O
O(n2 )	O
O(n2 )	O
O(n2 )	O
O(1 )	O
Bogo	O
sort	O
Array	O
O(n )	O
O( n	O
n	O
!	O
)	O
</s>
<s>
Insertion	B-Algorithm
sort	I-Algorithm
applied	O
to	O
a	O
list	O
of	O
n	O
elements	O
,	O
assumed	O
to	O
be	O
all	O
different	O
and	O
initially	O
in	O
random	O
order	O
.	O
</s>
<s>
Working	O
out	O
the	O
resulting	O
average-case	B-General_Concept
running	O
time	O
yields	O
a	O
quadratic	O
function	O
of	O
the	O
input	O
size	O
,	O
just	O
like	O
the	O
worst-case	B-General_Concept
running	O
time	O
.	O
</s>
<s>
Quicksort	B-Algorithm
applied	O
to	O
a	O
list	O
of	O
n	O
elements	O
,	O
again	O
assumed	O
to	O
be	O
all	O
different	O
and	O
initially	O
in	O
random	O
order	O
.	O
</s>
<s>
This	O
popular	O
sorting	B-Algorithm
algorithm	I-Algorithm
has	O
an	O
average-case	B-General_Concept
performance	O
of	O
O(nlog(n )	O
)	O
,	O
which	O
contributes	O
to	O
making	O
it	O
a	O
very	O
fast	O
algorithm	O
in	O
practice	O
.	O
</s>
<s>
But	O
given	O
a	O
worst-case	B-General_Concept
input	O
,	O
its	O
performance	O
degrades	O
to	O
O(n2 )	O
.	O
</s>
<s>
Also	O
,	O
when	O
implemented	O
with	O
the	O
"	O
shortest	O
first	O
"	O
policy	O
,	O
the	O
worst-case	B-General_Concept
space	O
complexity	O
is	O
instead	O
bounded	O
byO(log(n )	O
)	O
.	O
</s>
<s>
In	O
the	O
worst	B-General_Concept
case	I-General_Concept
this	O
leads	O
to	O
O( ∞	O
)	O
time	O
,	O
an	O
infinite	O
loop	O
.	O
</s>
<s>
Linear	B-Algorithm
search	I-Algorithm
on	O
a	O
list	O
of	O
n	O
elements	O
.	O
</s>
<s>
In	O
the	O
absolute	O
worst	B-General_Concept
case	I-General_Concept
,	O
the	O
search	O
must	O
visit	O
every	O
element	O
once	O
.	O
</s>
