<s>
In	O
backtracking	B-Algorithm
algorithms	I-Algorithm
,	O
backjumping	B-Application
is	O
a	O
technique	O
that	O
reduces	O
search	O
space	O
,	O
therefore	O
increasing	O
efficiency	O
.	O
</s>
<s>
While	O
backtracking	B-Algorithm
always	O
goes	O
up	O
one	O
level	O
in	O
the	O
search	B-Data_Structure
tree	I-Data_Structure
when	O
all	O
values	O
for	O
a	O
variable	O
have	O
been	O
tested	O
,	O
backjumping	B-Application
may	O
go	O
up	O
more	O
levels	O
.	O
</s>
<s>
Whenever	O
backtracking	B-Algorithm
has	O
tried	O
all	O
values	O
for	O
a	O
variable	O
without	O
finding	O
any	O
solution	O
,	O
it	O
reconsiders	O
the	O
last	O
of	O
the	O
previously	O
assigned	O
variables	O
,	O
changing	O
its	O
value	O
or	O
further	O
backtracking	B-Algorithm
if	O
no	O
other	O
values	O
are	O
to	O
be	O
tried	O
.	O
</s>
<s>
The	O
algorithm	O
then	O
"	O
goes	O
up	O
"	O
to	O
,	O
changing	O
'	O
s	O
value	O
if	O
possible	O
,	O
backtracking	B-Algorithm
again	O
otherwise	O
.	O
</s>
<s>
Backtracking	B-Algorithm
goes	O
back	O
to	O
,	O
trying	O
to	O
assign	O
it	O
a	O
new	O
value	O
.	O
</s>
<s>
Instead	O
of	O
backtracking	B-Algorithm
,	O
the	O
algorithm	O
makes	O
some	O
further	O
elaboration	O
,	O
proving	O
that	O
the	O
evaluations	O
,	O
,	O
and	O
are	O
not	O
part	O
of	O
any	O
solution	O
.	O
</s>
<s>
The	O
efficiency	O
of	O
a	O
backjumping	B-Application
algorithm	O
depends	O
on	O
how	O
high	O
it	O
is	O
able	O
to	O
backjump	O
.	O
</s>
<s>
In	O
practice	O
,	O
backjumping	B-Application
algorithms	O
use	O
the	O
lowest	O
index	O
they	O
can	O
efficiently	O
prove	O
to	O
be	O
a	O
safe	O
jump	O
.	O
</s>
<s>
These	O
methods	O
have	O
different	O
cost	O
,	O
but	O
a	O
higher	O
cost	O
of	O
finding	O
a	O
higher	O
safe	O
jump	O
may	O
be	O
traded	O
off	O
a	O
reduced	O
amount	O
of	O
search	O
due	O
to	O
skipping	O
parts	O
of	O
the	O
search	B-Data_Structure
tree	I-Data_Structure
.	O
</s>
<s>
The	O
simplest	O
condition	O
in	O
which	O
backjumping	B-Application
is	O
possible	O
is	O
when	O
all	O
values	O
of	O
a	O
variable	O
have	O
been	O
proved	O
inconsistent	O
without	O
further	O
branching	O
.	O
</s>
<s>
In	O
constraint	B-Application
satisfaction	I-Application
,	O
a	O
partial	O
evaluation	O
is	O
consistent	O
if	O
and	O
only	O
if	O
it	O
satisfies	O
all	O
constraints	O
involving	O
the	O
assigned	O
variables	O
,	O
and	O
inconsistent	O
otherwise	O
.	O
</s>
<s>
This	O
happens	O
exactly	O
when	O
the	O
variable	O
is	O
a	O
leaf	O
of	O
the	O
search	B-Data_Structure
tree	I-Data_Structure
(	O
which	O
correspond	O
to	O
nodes	O
having	O
only	O
leaves	O
as	O
children	O
in	O
the	O
figures	O
of	O
this	O
article	O
.	O
)	O
</s>
<s>
The	O
backjumping	B-Application
algorithm	O
by	O
Gaschnig	O
does	O
a	O
backjump	O
only	O
in	O
leaf	O
dead	O
ends	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
it	O
works	O
differently	O
from	O
backtracking	B-Algorithm
only	O
when	O
every	O
possible	O
value	O
of	O
has	O
been	O
tested	O
and	O
resulted	O
inconsistent	O
without	O
the	O
need	O
of	O
branching	O
over	O
another	O
variable	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
it	O
allows	O
for	O
a	O
backjump	O
only	O
at	O
leaf	O
nodes	O
in	O
the	O
search	B-Data_Structure
tree	I-Data_Structure
.	O
</s>
<s>
An	O
internal	O
node	O
of	O
the	O
search	B-Data_Structure
tree	I-Data_Structure
represents	O
an	O
assignment	O
of	O
a	O
variable	O
that	O
is	O
consistent	O
with	O
the	O
previous	O
ones	O
.	O
</s>
<s>
Backjumping	B-Application
at	O
internal	O
nodes	O
cannot	O
be	O
done	O
as	O
for	O
leaf	O
nodes	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
a	O
backjump	O
indicates	O
that	O
the	O
visit	O
of	O
a	O
region	O
of	O
the	O
search	B-Data_Structure
tree	I-Data_Structure
had	O
been	O
a	O
mistake	O
.	O
</s>
<s>
This	O
part	O
of	O
the	O
search	B-Data_Structure
tree	I-Data_Structure
can	O
therefore	O
be	O
ignored	O
when	O
considering	O
a	O
possible	O
backjump	O
from	O
or	O
from	O
one	O
of	O
its	O
ancestors	O
.	O
</s>
<s>
When	O
retracting	O
from	O
a	O
node	O
,	O
this	O
set	O
is	O
removed	O
the	O
variable	O
of	O
the	O
node	O
and	O
collected	O
in	O
the	O
set	O
of	O
the	O
destination	O
of	O
backtracking	B-Algorithm
or	O
backjumping	B-Application
.	O
</s>
<s>
Since	O
nodes	O
that	O
are	O
skipped	O
from	O
backjumping	B-Application
are	O
never	O
retracted	O
from	O
,	O
their	O
sets	O
are	O
automatically	O
ignored	O
.	O
</s>
<s>
The	O
rationale	O
of	O
graph-based	B-Application
backjumping	I-Application
is	O
that	O
a	O
safe	O
jump	O
can	O
be	O
found	O
by	O
checking	O
which	O
of	O
the	O
variables	O
are	O
in	O
a	O
constraint	O
with	O
the	O
variables	O
that	O
are	O
instantiated	O
in	O
leaf	O
nodes	O
.	O
</s>
<s>
The	O
fact	O
that	O
nodes	O
skipped	O
by	O
backjumping	B-Application
can	O
be	O
ignored	O
when	O
considering	O
a	O
further	O
backjump	O
can	O
be	O
exploited	O
by	O
the	O
following	O
algorithm	O
.	O
</s>
<s>
When	O
retracting	O
from	O
a	O
leaf	O
node	O
,	O
the	O
set	O
of	O
variables	O
that	O
are	O
in	O
constraint	O
with	O
it	O
is	O
created	O
and	O
"	O
sent	O
back	O
"	O
to	O
its	O
parent	O
,	O
or	O
ancestor	O
in	O
case	O
of	O
backjumping	B-Application
.	O
</s>
<s>
When	O
further	O
backtracking	B-Algorithm
or	O
backjumping	B-Application
from	O
the	O
node	O
,	O
the	O
variable	O
of	O
the	O
node	O
is	O
removed	O
from	O
this	O
set	O
,	O
and	O
the	O
set	O
is	O
sent	O
to	O
the	O
node	O
that	O
is	O
the	O
destination	O
of	O
backtracking	B-Algorithm
or	O
backjumping	B-Application
.	O
</s>
<s>
Since	O
sets	O
of	O
variables	O
are	O
only	O
sent	O
when	O
retracing	O
from	O
nodes	O
,	O
the	O
sets	O
collected	O
at	O
nodes	O
skipped	O
by	O
backjumping	B-Application
are	O
automatically	O
ignored	O
.	O
</s>
<s>
A	O
still	O
more	O
refined	O
backjumping	B-Application
algorithm	O
,	O
sometimes	O
able	O
to	O
achieve	O
larger	O
backjumps	O
,	O
is	O
based	O
on	O
checking	O
not	O
only	O
the	O
common	O
presence	O
of	O
two	O
variables	O
in	O
the	O
same	O
constraint	O
but	O
also	O
on	O
whether	O
the	O
constraint	O
actually	O
caused	O
inconsistency	O
.	O
</s>
<s>
For	O
this	O
reason	O
,	O
conflict-based	B-Application
backjumping	I-Application
orders	O
constraints	O
in	O
such	O
a	O
way	O
constraints	O
over	O
lower	O
indices	O
variables	O
are	O
preferred	O
over	O
constraints	O
on	O
higher	O
index	O
variables	O
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
algorithm	O
collects	O
,	O
in	O
each	O
node	O
,	O
all	O
sets	O
coming	O
from	O
its	O
descendants	O
that	O
have	O
not	O
been	O
skipped	O
by	O
backjumping	B-Application
.	O
</s>
<s>
When	O
retracting	O
from	O
this	O
node	O
,	O
this	O
set	O
is	O
removed	O
the	O
variable	O
of	O
the	O
node	O
and	O
collected	O
into	O
the	O
destination	O
of	O
backtracking	B-Algorithm
or	O
backjumping	B-Application
.	O
</s>
<s>
Conflict-directed	O
backjumping	B-Application
was	O
proposed	O
for	O
Constraint	B-Application
Satisfaction	I-Application
Problems	I-Application
by	O
Patrick	O
Prosser	O
in	O
his	O
seminal	O
1993	O
paper	O
.	O
</s>
