<s>
A	O
skew	B-Application
heap	I-Application
(	O
or	O
self-adjusting	B-Application
heap	I-Application
)	O
is	O
a	O
heap	B-Application
data	I-Application
structure	I-Application
implemented	O
as	O
a	O
binary	O
tree	O
.	O
</s>
<s>
Skew	B-Application
heaps	I-Application
are	O
advantageous	O
because	O
of	O
their	O
ability	O
to	O
merge	O
more	O
quickly	O
than	O
binary	B-Application
heaps	I-Application
.	O
</s>
<s>
In	O
contrast	O
with	O
binary	B-Application
heaps	I-Application
,	O
there	O
are	O
no	O
structural	O
constraints	O
,	O
so	O
there	O
is	O
no	O
guarantee	O
that	O
the	O
height	O
of	O
the	O
tree	O
is	O
logarithmic	O
.	O
</s>
<s>
Every	O
operation	O
(	O
add	O
,	O
remove_min	O
,	O
merge	O
)	O
on	O
two	O
skew	B-Application
heaps	I-Application
must	O
be	O
done	O
using	O
a	O
special	O
skew	B-Application
heap	I-Application
merge	O
.	O
</s>
<s>
A	O
skew	B-Application
heap	I-Application
is	O
a	O
self-adjusting	O
form	O
of	O
a	O
leftist	B-Application
heap	I-Application
which	O
attempts	O
to	O
maintain	O
balance	O
by	O
unconditionally	O
swapping	O
all	O
nodes	O
in	O
the	O
merge	O
path	O
when	O
merging	O
two	O
heaps	B-Application
.	O
</s>
<s>
With	O
no	O
structural	O
constraints	O
,	O
it	O
may	O
seem	O
that	O
a	O
skew	B-Application
heap	I-Application
would	O
be	O
horribly	O
inefficient	O
.	O
</s>
<s>
However	O
,	O
amortized	B-General_Concept
complexity	I-General_Concept
analysis	I-General_Concept
can	O
be	O
used	O
to	O
demonstrate	O
that	O
all	O
operations	O
on	O
a	O
skew	B-Application
heap	I-Application
can	O
be	O
done	O
in	O
O(log n )	O
.	O
</s>
<s>
In	O
fact	O
,	O
with	O
denoting	O
the	O
golden	O
ratio	O
,	O
the	O
exact	O
amortized	B-General_Concept
complexity	I-General_Concept
is	O
known	O
to	O
be	O
logφ	O
n	O
(	O
approximately	O
1.44	O
log2	O
n	O
)	O
.	O
</s>
<s>
Skew	B-Application
heaps	I-Application
may	O
be	O
described	O
with	O
the	O
following	O
recursive	O
definition	O
:	O
</s>
<s>
A	O
heap	B-Application
with	O
only	O
one	O
element	O
is	O
a	O
skew	B-Application
heap	I-Application
.	O
</s>
<s>
The	O
result	O
of	O
skew	O
merging	O
two	O
skew	B-Application
heaps	I-Application
and	O
is	O
also	O
a	O
skew	B-Application
heap	I-Application
.	O
</s>
<s>
When	O
two	O
skew	B-Application
heaps	I-Application
are	O
to	O
be	O
merged	O
,	O
we	O
can	O
use	O
a	O
similar	O
process	O
as	O
the	O
merge	O
of	O
two	O
leftist	B-Application
heaps	I-Application
:	O
</s>
<s>
Compare	O
roots	O
of	O
two	O
heaps	B-Application
;	O
let	O
be	O
the	O
heap	B-Application
with	O
the	O
smaller	O
root	O
,	O
and	O
q	O
be	O
the	O
other	O
heap	B-Application
.	O
</s>
<s>
Let	O
be	O
the	O
name	O
of	O
the	O
resulting	O
new	O
heap	B-Application
.	O
</s>
<s>
Split	O
each	O
heap	B-Application
into	O
subtrees	O
by	O
cutting	O
every	O
path	O
.	O
</s>
<s>
Adding	O
a	O
value	O
to	O
a	O
skew	B-Application
heap	I-Application
is	O
like	O
merging	O
a	O
tree	O
with	O
one	O
node	O
together	O
with	O
the	O
original	O
tree	O
.	O
</s>
<s>
Removing	O
the	O
first	O
value	O
in	O
a	O
heap	B-Application
can	O
be	O
accomplished	O
by	O
removing	O
the	O
root	O
and	O
merging	O
its	O
child	O
subtrees	O
.	O
</s>
<s>
In	O
many	O
functional	O
languages	O
,	O
skew	B-Application
heaps	I-Application
become	O
extremely	O
simple	O
to	O
implement	O
.	O
</s>
