<s>
In	O
computer	B-General_Concept
science	I-General_Concept
a	O
retroactive	B-Data_Structure
data	I-Data_Structure
structure	I-Data_Structure
is	O
a	O
data	B-General_Concept
structure	I-General_Concept
which	O
supports	O
efficient	O
modifications	O
to	O
a	O
sequence	O
of	O
operations	O
that	O
have	O
been	O
performed	O
on	O
the	O
structure	O
.	O
</s>
<s>
Manipulation	O
of	O
the	O
past	O
:	O
Changing	O
the	O
past	O
can	O
be	O
helpful	O
in	O
the	O
cases	O
of	O
damage	O
control	O
and	O
retroactive	B-Data_Structure
data	I-Data_Structure
structures	I-Data_Structure
are	O
designed	O
for	O
intentional	O
manipulation	O
of	O
the	O
past	O
.	O
</s>
<s>
The	O
data	B-General_Concept
structure	I-General_Concept
we	O
will	O
use	O
to	O
add	O
the	O
spatial	O
time	O
dimension	O
is	O
a	O
min-heap	O
.	O
</s>
<s>
Notice	O
how	O
the	O
single	O
operation	O
produces	O
a	O
cascading	O
effect	O
which	O
affects	O
the	O
entire	O
data	B-General_Concept
structure	I-General_Concept
.	O
</s>
<s>
At	O
first	O
glance	O
the	O
notion	O
of	O
a	O
retroactive	B-Data_Structure
data	I-Data_Structure
structures	I-Data_Structure
seems	O
very	O
similar	O
to	O
persistent	B-Application
data	I-Application
structures	I-Application
since	O
they	O
both	O
take	O
into	O
account	O
the	O
dimension	O
of	O
time	O
.	O
</s>
<s>
The	O
key	O
difference	O
between	O
persistent	B-Application
data	I-Application
structures	I-Application
and	O
retroactive	B-Data_Structure
data	I-Data_Structure
structures	I-Data_Structure
is	O
how	O
they	O
handle	O
the	O
element	O
of	O
time	O
.	O
</s>
<s>
A	O
persistent	B-Application
data	I-Application
structure	I-Application
maintains	O
several	O
versions	O
of	O
a	O
data	B-General_Concept
structure	I-General_Concept
and	O
operations	O
can	O
be	O
performed	O
on	O
one	O
version	O
to	O
produce	O
another	O
version	O
of	O
the	O
data	B-General_Concept
structure	I-General_Concept
.	O
</s>
<s>
In	O
retroactive	B-Data_Structure
data	I-Data_Structure
structures	I-Data_Structure
we	O
allow	O
changes	O
to	O
be	O
made	O
directly	O
to	O
previous	O
versions	O
.	O
</s>
<s>
Any	O
data	B-General_Concept
structure	I-General_Concept
can	O
be	O
reformulated	O
in	O
a	O
retroactive	O
setting	O
.	O
</s>
<s>
In	O
general	O
the	O
data	B-General_Concept
structure	I-General_Concept
involves	O
a	O
series	O
of	O
updates	O
and	O
queries	O
made	O
over	O
some	O
period	O
of	O
time	O
.	O
</s>
<s>
We	O
define	O
the	O
data	B-General_Concept
structure	I-General_Concept
to	O
be	O
partially	O
retroactive	O
if	O
it	O
can	O
perform	O
update	O
and	O
query	O
operations	O
at	O
the	O
current	O
time	O
and	O
support	O
insertion	O
and	O
deletion	O
operations	O
in	O
the	O
past	O
.	O
</s>
<s>
All	O
retroactive	O
changes	O
on	O
the	O
operational	O
history	O
of	O
the	O
data	B-General_Concept
structure	I-General_Concept
can	O
potentially	O
affect	O
all	O
the	O
operations	O
at	O
the	O
time	O
of	O
the	O
operation	O
to	O
the	O
present	O
.	O
</s>
<s>
The	O
current	O
state	O
of	O
the	O
data	B-General_Concept
structure	I-General_Concept
(	O
i.e.	O
</s>
<s>
We	O
define	O
the	O
data	B-General_Concept
structure	I-General_Concept
to	O
be	O
fully	O
retroactive	O
if	O
in	O
addition	O
to	O
the	O
partially	O
retroactive	O
operations	O
we	O
also	O
allow	O
for	O
one	O
to	O
perform	O
queries	O
about	O
the	O
past	O
.	O
</s>
<s>
The	O
running	O
time	O
of	O
retroactive	B-Data_Structure
data	I-Data_Structure
structures	I-Data_Structure
are	O
based	O
on	O
the	O
number	O
of	O
operations	O
,	O
m	O
,	O
performed	O
on	O
the	O
structure	O
,	O
the	O
number	O
of	O
operations	O
r	O
that	O
were	O
performed	O
before	O
the	O
retroactive	O
operation	O
is	O
performed	O
,	O
and	O
the	O
maximum	O
number	O
of	O
elements	O
n	O
in	O
the	O
structure	O
at	O
any	O
single	O
time	O
.	O
</s>
<s>
The	O
main	O
question	O
regarding	O
automatic	O
retro-activity	O
with	O
respect	O
to	O
data	B-General_Concept
structures	I-General_Concept
is	O
whether	O
or	O
not	O
there	O
is	O
a	O
general	O
technique	O
which	O
can	O
convert	O
any	O
data	B-General_Concept
structure	I-General_Concept
into	O
an	O
efficient	O
retroactive	O
counterpart	O
.	O
</s>
<s>
Once	O
we	O
have	O
rolled	O
back	O
the	O
data	B-General_Concept
structure	I-General_Concept
to	O
the	O
appropriate	O
state	O
we	O
can	O
then	O
apply	O
the	O
retroactive	O
operation	O
to	O
make	O
the	O
change	O
we	O
wanted	O
.	O
</s>
<s>
Once	O
the	O
change	O
is	O
made	O
we	O
must	O
then	O
reapply	O
all	O
the	O
changes	O
we	O
rolled	O
back	O
before	O
to	O
put	O
the	O
data	B-General_Concept
structure	I-General_Concept
into	O
its	O
new	O
state	O
.	O
</s>
<s>
While	O
this	O
can	O
work	O
for	O
any	O
data	B-General_Concept
structure	I-General_Concept
,	O
it	O
is	O
often	O
inefficient	O
and	O
wasteful	O
especially	O
once	O
the	O
number	O
of	O
changes	O
we	O
need	O
to	O
roll-back	O
is	O
large	O
.	O
</s>
<s>
To	O
create	O
an	O
efficient	O
retroactive	B-Data_Structure
data	I-Data_Structure
structure	I-Data_Structure
we	O
must	O
take	O
a	O
look	O
at	O
the	O
properties	O
of	O
the	O
structure	O
itself	O
to	O
determine	O
where	O
speed	O
ups	O
can	O
be	O
realized	O
.	O
</s>
<s>
Thus	O
there	O
is	O
no	O
general	O
way	O
to	O
convert	O
any	O
data	B-General_Concept
structure	I-General_Concept
into	O
an	O
efficient	O
retroactive	O
counterpart	O
.	O
</s>
