<s>
A	O
Kinetic	B-Application
Heap	I-Application
is	O
a	O
kinetic	B-Data_Structure
data	I-Data_Structure
structure	I-Data_Structure
,	O
obtained	O
by	O
the	O
kinetization	O
of	O
a	O
heap	B-Application
.	O
</s>
<s>
As	O
a	O
type	O
of	O
kinetic	B-Application
priority	I-Application
queue	I-Application
,	O
it	O
maintains	O
the	O
maximum	O
priority	O
element	O
stored	O
in	O
it	O
.	O
</s>
<s>
The	O
kinetic	B-Application
heap	I-Application
data	O
structure	O
works	O
by	O
storing	O
the	O
elements	O
as	O
a	O
tree	O
that	O
satisfies	O
the	O
following	O
heap	B-Application
property	I-Application
if	O
is	O
a	O
child	B-Data_Structure
node	I-Data_Structure
of	O
,	O
then	O
the	O
priority	O
of	O
the	O
element	O
in	O
must	O
be	O
higher	O
than	O
the	O
priority	O
of	O
the	O
element	O
in	O
.	O
</s>
<s>
This	O
heap	B-Application
property	I-Application
is	O
enforced	O
using	O
certificates	O
along	O
every	O
edge	O
so	O
,	O
like	O
other	O
kinetic	B-Data_Structure
data	I-Data_Structure
structures	I-Data_Structure
,	O
a	O
kinetic	B-Application
heap	I-Application
also	O
contains	O
a	O
priority	O
queue	O
(	O
the	O
event	O
queue	O
)	O
to	O
maintain	O
certificate	O
failure	O
times	O
.	O
</s>
<s>
A	O
regular	O
heap	B-Application
can	O
be	O
kinetized	O
by	O
augmenting	O
with	O
a	O
certificate	O
 [  ] 	O
for	O
every	O
pair	O
of	O
nodes	O
,	O
such	O
that	O
is	O
a	O
child	B-Data_Structure
node	I-Data_Structure
of	O
.	O
</s>
<s>
When	O
a	O
certificate	O
 [  ] 	O
fails	O
,	O
the	O
data	O
structure	O
must	O
swap	O
and	O
in	O
the	O
heap	B-Application
,	O
and	O
update	O
the	O
certificates	O
that	O
each	O
of	O
them	O
was	O
present	O
in	O
.	O
</s>
<s>
For	O
example	O
,	O
if	O
(	O
with	O
child	B-Data_Structure
nodes	I-Data_Structure
and	O
)	O
was	O
a	O
child	B-Data_Structure
node	I-Data_Structure
of	O
(	O
with	O
child	B-Data_Structure
nodes	I-Data_Structure
and	O
and	O
parent	B-Application
node	I-Application
)	O
,	O
and	O
the	O
certificate	O
 [  ] 	O
fails	O
,	O
then	O
the	O
data	O
structure	O
must	O
swap	O
and	O
,	O
then	O
replace	O
the	O
old	O
certificates	O
(	O
and	O
the	O
corresponding	O
scheduled	O
events	O
)	O
[],	O
[],	O
[],	O
[],	O
 [  ] 	O
with	O
new	O
certificates	O
[],	O
[],	O
[],	O
 [  ] 	O
and	O
 [  ] 	O
.	O
</s>
<s>
A	O
kinetic	B-Application
heap	I-Application
supports	O
the	O
following	O
operations	O
:	O
</s>
<s>
(	O
or	O
find-min	O
)	O
:	O
return	O
the	O
(	O
or	O
for	O
a	O
)	O
value	O
stored	O
in	O
the	O
heap	B-Application
at	O
the	O
current	O
virtual	O
time	O
.	O
</s>
<s>
:	O
insert	O
a	O
key	O
into	O
the	O
kinetic	B-Application
heap	I-Application
at	O
the	O
current	O
virtual	O
time	O
,	O
whose	O
value	O
changes	O
as	O
a	O
continuous	O
function	O
of	O
time	O
.	O
</s>
<s>
The	O
deletion	O
is	O
done	O
as	O
in	O
a	O
normal	O
heap	B-Application
in	O
time	O
,	O
but	O
certificates	O
might	O
need	O
to	O
be	O
changed	O
as	O
a	O
result	O
,	O
so	O
the	O
total	O
time	O
for	O
rescheduling	O
certificate	O
failures	O
is	O
.	O
</s>
<s>
Kinetic	B-Application
heaps	I-Application
perform	O
well	O
according	O
to	O
the	O
four	O
metrics	O
(	O
responsiveness	O
,	O
locality	O
,	O
compactness	O
and	O
efficiency	O
)	O
of	O
kinetic	B-Data_Structure
data	I-Data_Structure
structure	I-Data_Structure
quality	O
defined	O
by	O
Basch	O
et	O
al	O
.	O
</s>
<s>
Responsiveness	O
:	O
A	O
kinetic	B-Application
heap	I-Application
is	O
responsive	O
,	O
since	O
each	O
certificate	O
failure	O
causes	O
the	O
concerned	O
keys	O
to	O
be	O
swapped	O
and	O
leads	O
to	O
only	O
few	O
certificates	O
being	O
replaced	O
in	O
the	O
worst	O
case	O
.	O
</s>
<s>
Locality	O
:	O
Each	O
node	O
is	O
present	O
in	O
one	O
certificate	O
each	O
along	O
with	O
its	O
parent	B-Application
node	I-Application
and	O
two	O
child	B-Data_Structure
nodes	I-Data_Structure
(	O
if	O
present	O
)	O
,	O
meaning	O
that	O
each	O
node	O
can	O
be	O
involved	O
in	O
a	O
total	O
of	O
three	O
scheduled	O
events	O
in	O
the	O
worst	O
case	O
,	O
thus	O
kinetic	B-Application
heaps	I-Application
are	O
local	O
.	O
</s>
<s>
Compactness	O
:	O
Each	O
edge	O
in	O
the	O
heap	B-Application
corresponds	O
to	O
exactly	O
one	O
scheduled	O
event	O
,	O
therefore	O
the	O
number	O
of	O
scheduled	O
events	O
is	O
exactly	O
where	O
is	O
the	O
number	O
of	O
nodes	O
in	O
the	O
kinetic	B-Application
heap	I-Application
.	O
</s>
<s>
Thus	O
,	O
kinetic	B-Application
heaps	I-Application
are	O
compact	O
.	O
</s>
<s>
The	O
efficiency	O
of	O
a	O
kinetic	B-Application
heap	I-Application
in	O
the	O
general	O
case	O
is	O
largely	O
unknown	O
.	O
</s>
<s>
However	O
,	O
in	O
the	O
special	O
case	O
of	O
affine	O
motion	O
of	O
the	O
priorities	O
,	O
kinetic	B-Application
heaps	I-Application
are	O
known	O
to	O
be	O
very	O
efficient	O
.	O
</s>
<s>
In	O
this	O
special	O
case	O
,	O
the	O
maximum	O
number	O
of	O
events	O
processed	O
by	O
a	O
kinetic	B-Application
heap	I-Application
can	O
be	O
shown	O
to	O
be	O
exactly	O
the	O
number	O
of	O
edges	O
in	O
the	O
transitive	O
closure	O
of	O
the	O
tree	B-Data_Structure
structure	I-Data_Structure
of	O
the	O
heap	B-Application
,	O
which	O
is	O
for	O
a	O
tree	O
of	O
height	O
.	O
</s>
<s>
If	O
insertions	O
and	O
deletions	O
are	O
made	O
on	O
a	O
kinetic	B-Application
heap	I-Application
that	O
starts	O
empty	O
,	O
the	O
maximum	O
number	O
of	O
events	O
processed	O
is	O
However	O
,	O
this	O
bound	O
is	O
not	O
believed	O
to	O
be	O
tight	O
,	O
and	O
the	O
only	O
known	O
lower	O
bound	O
is	O
.	O
</s>
<s>
This	O
article	O
deals	O
with	O
"	O
simple	O
"	O
kinetic	B-Application
heaps	I-Application
as	O
described	O
above	O
,	O
but	O
other	O
variants	O
have	O
been	O
developed	O
for	O
specialized	O
applications	O
,	O
such	O
as	O
:	O
</s>
<s>
Other	O
heap-like	O
kinetic	B-Application
priority	I-Application
queues	I-Application
are	O
:	O
</s>
