<s>
In	O
the	O
design	O
of	O
algorithms	O
,	O
partition	B-Data_Structure
refinement	I-Data_Structure
is	O
a	O
technique	O
for	O
representing	O
a	O
partition	O
of	O
a	O
set	O
as	O
a	O
data	B-General_Concept
structure	I-General_Concept
that	O
allows	O
the	O
partition	O
to	O
be	O
refined	O
by	O
splitting	O
its	O
sets	O
into	O
a	O
larger	O
number	O
of	O
smaller	O
sets	O
.	O
</s>
<s>
In	O
that	O
sense	O
it	O
is	O
dual	O
to	O
the	O
union-find	B-Algorithm
data	I-Algorithm
structure	I-Algorithm
,	O
which	O
also	O
maintains	O
a	O
partition	O
into	O
disjoint	B-Algorithm
sets	I-Algorithm
but	O
in	O
which	O
the	O
operations	O
merge	O
pairs	O
of	O
sets	O
.	O
</s>
<s>
In	O
some	O
applications	O
of	O
partition	B-Data_Structure
refinement	I-Data_Structure
,	O
such	O
as	O
lexicographic	B-Algorithm
breadth-first	I-Algorithm
search	I-Algorithm
,	O
the	O
data	B-General_Concept
structure	I-General_Concept
maintains	O
as	O
well	O
an	O
ordering	O
on	O
the	O
sets	O
in	O
the	O
partition	O
.	O
</s>
<s>
Partition	B-Data_Structure
refinement	I-Data_Structure
forms	O
a	O
key	O
component	O
of	O
several	O
efficient	O
algorithms	O
on	O
graphs	O
and	O
finite	B-Architecture
automata	I-Architecture
,	O
including	O
DFA	B-General_Concept
minimization	I-General_Concept
,	O
the	O
Coffman	B-Operating_System
–	I-Operating_System
Graham	I-Operating_System
algorithm	I-Operating_System
for	O
parallel	O
scheduling	O
,	O
and	O
lexicographic	B-Algorithm
breadth-first	I-Algorithm
search	I-Algorithm
of	O
graphs	O
.	O
</s>
<s>
A	O
partition	B-Data_Structure
refinement	I-Data_Structure
algorithm	O
maintains	O
a	O
family	O
of	O
disjoint	B-Algorithm
sets	I-Algorithm
.	O
</s>
<s>
At	O
the	O
start	O
of	O
the	O
algorithm	O
,	O
this	O
family	O
contains	O
a	O
single	O
set	O
of	O
all	O
the	O
elements	O
in	O
the	O
data	B-General_Concept
structure	I-General_Concept
.	O
</s>
<s>
Such	O
an	O
algorithm	O
may	O
be	O
implemented	O
efficiently	O
by	O
maintaining	O
data	B-General_Concept
structures	I-General_Concept
representing	O
the	O
following	O
information	O
:	O
</s>
<s>
Associated	O
with	O
each	O
set	O
,	O
a	O
collection	B-Application
of	O
its	O
elements	O
of	O
,	O
in	O
a	O
form	O
such	O
as	O
a	O
doubly	B-Data_Structure
linked	I-Data_Structure
list	I-Data_Structure
or	O
array	B-Data_Structure
data	I-Data_Structure
structure	I-Data_Structure
that	O
allows	O
for	O
rapid	O
deletion	O
of	O
individual	O
elements	O
from	O
the	O
collection	B-Application
.	O
</s>
<s>
Alternatively	O
,	O
this	O
component	O
of	O
the	O
data	B-General_Concept
structure	I-General_Concept
may	O
be	O
represented	O
by	O
storing	O
all	O
of	O
the	O
elements	O
of	O
all	O
of	O
the	O
sets	O
in	O
a	O
single	O
array	O
,	O
sorted	O
by	O
the	O
identity	O
of	O
the	O
set	O
they	O
belong	O
to	O
,	O
and	O
by	O
representing	O
the	O
collection	B-Application
of	O
elements	O
in	O
any	O
set	O
by	O
its	O
starting	O
and	O
ending	O
positions	O
in	O
this	O
array	O
.	O
</s>
<s>
The	O
time	O
to	O
perform	O
a	O
single	O
refinement	O
operations	O
in	O
this	O
way	O
is	O
,	O
independent	O
of	O
the	O
number	O
of	O
elements	O
in	O
the	O
family	O
of	O
sets	O
and	O
also	O
independent	O
of	O
the	O
total	O
number	O
of	O
sets	O
in	O
the	O
data	B-General_Concept
structure	I-General_Concept
.	O
</s>
<s>
An	O
early	O
application	O
of	O
partition	B-Data_Structure
refinement	I-Data_Structure
was	O
in	O
an	O
algorithm	O
by	O
for	O
DFA	B-General_Concept
minimization	I-General_Concept
.	O
</s>
<s>
In	O
this	O
problem	O
,	O
one	O
is	O
given	O
as	O
input	O
a	O
deterministic	B-General_Concept
finite	I-General_Concept
automaton	I-General_Concept
,	O
and	O
must	O
find	O
an	O
equivalent	O
automaton	O
with	O
as	O
few	O
states	O
as	O
possible	O
.	O
</s>
<s>
Initially	O
,	O
there	O
are	O
two	O
subsets	O
,	O
one	O
containing	O
all	O
the	O
accepting	B-Architecture
states	I-Architecture
of	O
the	O
automaton	O
and	O
one	O
containing	O
the	O
remaining	O
states	O
.	O
</s>
<s>
Partition	B-Data_Structure
refinement	I-Data_Structure
was	O
applied	O
by	O
in	O
an	O
efficient	O
implementation	O
of	O
the	O
Coffman	B-Operating_System
–	I-Operating_System
Graham	I-Operating_System
algorithm	I-Operating_System
for	O
parallel	O
scheduling	O
.	O
</s>
<s>
Sethi	O
showed	O
that	O
it	O
could	O
be	O
used	O
to	O
construct	O
a	O
lexicographically	O
ordered	O
topological	B-Algorithm
sort	I-Algorithm
of	O
a	O
given	O
directed	O
acyclic	O
graph	O
in	O
linear	O
time	O
;	O
this	O
lexicographic	O
topological	B-Algorithm
ordering	I-Algorithm
is	O
one	O
of	O
the	O
key	O
steps	O
of	O
the	O
Coffman	B-Operating_System
–	I-Operating_System
Graham	I-Operating_System
algorithm	I-Operating_System
.	O
</s>
<s>
In	O
this	O
application	O
,	O
the	O
elements	O
of	O
the	O
disjoint	B-Algorithm
sets	I-Algorithm
are	O
vertices	O
of	O
the	O
input	O
graph	O
and	O
the	O
sets	O
used	O
to	O
refine	O
the	O
partition	O
are	O
sets	O
of	O
neighbors	O
of	O
vertices	O
.	O
</s>
<s>
Partition	B-Data_Structure
refinement	I-Data_Structure
also	O
forms	O
a	O
key	O
step	O
in	O
lexicographic	B-Algorithm
breadth-first	I-Algorithm
search	I-Algorithm
,	O
a	O
graph	O
search	O
algorithm	O
with	O
applications	O
in	O
the	O
recognition	O
of	O
chordal	O
graphs	O
and	O
several	O
other	O
important	O
classes	O
of	O
graphs	O
.	O
</s>
<s>
Again	O
,	O
the	O
disjoint	B-Algorithm
set	I-Algorithm
elements	O
are	O
vertices	O
and	O
the	O
set	O
represent	O
sets	O
of	O
neighbors	O
,	O
so	O
the	O
algorithm	O
takes	O
linear	O
time	O
.	O
</s>
