<s>
In	O
computer	B-General_Concept
science	I-General_Concept
,	O
the	O
method	O
of	O
contraction	B-Protocol
hierarchies	I-Protocol
is	O
a	O
speed-up	B-Operating_System
technique	I-Operating_System
for	O
finding	O
the	O
shortest-path	O
in	O
a	O
graph	O
.	O
</s>
<s>
The	O
most	O
intuitive	O
applications	O
are	O
car-navigation	B-Application
systems	I-Application
:	O
a	O
user	O
wants	O
to	O
drive	O
from	O
to	O
using	O
the	O
quickest	O
possible	O
route	O
.	O
</s>
<s>
The	O
shortest	O
path	O
in	O
a	O
graph	O
can	O
be	O
computed	O
using	O
Dijkstra	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
but	O
,	O
given	O
that	O
road	O
networks	O
consist	O
of	O
tens	O
of	O
millions	O
of	O
vertices	O
,	O
this	O
is	O
impractical	O
.	O
</s>
<s>
Contraction	B-Protocol
hierarchies	I-Protocol
is	O
a	O
speed-up	O
method	O
optimized	O
to	O
exploit	O
properties	O
of	O
graphs	O
representing	O
road	O
networks	O
.	O
</s>
<s>
Contraction	B-Protocol
hierarchies	I-Protocol
do	O
not	O
know	O
about	O
which	O
roads	O
humans	O
consider	O
"	O
important	O
"	O
(	O
e.g.	O
</s>
<s>
highways	O
)	O
,	O
but	O
they	O
are	O
provided	O
with	O
the	O
graph	O
as	O
input	O
and	O
are	O
able	O
to	O
assign	O
importance	O
to	O
vertices	O
using	O
heuristics	B-Algorithm
.	O
</s>
<s>
Contraction	B-Protocol
hierarchies	I-Protocol
are	O
not	O
only	O
applied	O
to	O
speed-up	O
algorithms	O
in	O
car-navigation	B-Application
systems	I-Application
but	O
also	O
in	O
web-based	O
route	B-Application
planners	I-Application
,	O
traffic	O
simulation	O
,	O
and	O
logistics	O
optimization	O
.	O
</s>
<s>
Implementations	O
of	O
the	O
algorithm	O
are	O
publicly	O
available	O
as	O
open	B-Application
source	I-Application
software	I-Application
.	O
</s>
<s>
The	O
contraction	B-Protocol
hierarchies	I-Protocol
(	O
CH	O
)	O
algorithm	O
is	O
a	O
two-phase	O
approach	O
to	O
the	O
shortest	O
path	O
problem	O
consisting	O
of	O
a	O
preprocessing	O
phase	O
and	O
a	O
query	O
phase	O
.	O
</s>
<s>
CHs	O
rely	O
on	O
shortcuts	O
to	O
achieve	O
this	O
speedup	B-Operating_System
.	O
</s>
<s>
The	O
contraction	B-Protocol
hierarchies	I-Protocol
algorithm	O
has	O
no	O
knowledge	O
about	O
road	O
types	O
but	O
is	O
able	O
to	O
determine	O
which	O
shortcuts	O
have	O
to	O
be	O
created	O
using	O
the	O
graph	O
alone	O
as	O
input	O
.	O
</s>
<s>
As	O
optimal	O
node	O
ordering	O
is	O
NP-complete	O
,	O
heuristics	B-Algorithm
are	O
used	O
.	O
</s>
<s>
Bottom-up	O
and	O
top-down	O
heuristics	B-Algorithm
exist	O
.	O
</s>
<s>
On	O
one	O
hand	O
,	O
the	O
computationally	O
cheaper	O
bottom-up	O
heuristics	B-Algorithm
decide	O
the	O
order	O
in	O
which	O
to	O
contract	O
the	O
vertices	O
in	O
a	O
greedy	B-Algorithm
fashion	O
;	O
this	O
means	O
the	O
order	O
is	O
not	O
known	O
in	O
advance	O
but	O
rather	O
the	O
next	O
node	O
is	O
selected	O
for	O
contraction	O
after	O
the	O
previous	O
contraction	O
has	O
been	O
completed	O
.	O
</s>
<s>
Top-down	O
heuristics	B-Algorithm
on	O
the	O
other	O
hand	O
precompute	O
the	O
whole	O
node	O
ordering	O
before	O
the	O
first	O
node	O
is	O
contracted	O
.	O
</s>
<s>
In	O
bottom-up	O
heuristics	B-Algorithm
,	O
a	O
combination	O
of	O
factors	O
is	O
used	O
to	O
select	O
the	O
next	O
vertex	O
for	O
contraction	O
.	O
</s>
<s>
Top-down	O
heuristics	B-Algorithm
,	O
on	O
the	O
other	O
hand	O
,	O
yield	O
better	O
results	O
but	O
need	O
more	O
preprocessing	O
time	O
.	O
</s>
<s>
This	O
can	O
be	O
approximated	B-Algorithm
using	O
nested	B-Algorithm
dissections	I-Algorithm
.	O
</s>
<s>
To	O
compute	O
a	O
nested	B-Algorithm
dissection	I-Algorithm
,	O
one	O
recursively	O
separates	O
a	O
graph	O
into	O
two	O
parts	O
,	O
which	O
are	O
themselves	O
then	O
separated	O
into	O
two	O
parts	O
and	O
so	O
on	O
.	O
</s>
<s>
Place	O
all	O
nodes	O
last	O
in	O
the	O
node	O
ordering	O
and	O
then	O
recursively	O
compute	O
the	O
nested	B-Algorithm
dissection	I-Algorithm
for	O
and	O
,	O
the	O
intuition	O
being	O
that	O
all	O
queries	O
from	O
one	O
half	O
of	O
the	O
graph	O
to	O
the	O
other	O
half	O
of	O
the	O
graph	O
need	O
to	O
pass	O
through	O
the	O
small	O
separator	O
and	O
therefore	O
nodes	O
in	O
this	O
separator	O
are	O
of	O
high	O
importance	O
.	O
</s>
<s>
Nested	B-Algorithm
dissections	I-Algorithm
can	O
be	O
efficiently	O
calculated	O
on	O
road	O
networks	O
because	O
of	O
their	O
small	O
separators	O
.	O
</s>
<s>
For	O
this	O
to	O
work	O
,	O
the	O
contraction	O
order	O
has	O
to	O
be	O
computed	O
using	O
metric-independent	O
nested	B-Algorithm
dissections	I-Algorithm
.	O
</s>
<s>
This	O
is	O
called	O
one-to-one	O
shortest	O
path	O
and	O
is	O
used	O
for	O
example	O
in	O
car-navigation	B-Application
systems	I-Application
.	O
</s>
<s>
As	O
Dijkstra	B-Algorithm
’s	I-Algorithm
algorithm	I-Algorithm
visits	O
each	O
edge	O
exactly	O
once	O
and	O
therefore	O
runs	O
in	O
linear	O
time	O
it	O
is	O
theoretically	O
optimal	O
.	O
</s>
<s>
Dijkstra	B-Algorithm
’s	I-Algorithm
algorithm	I-Algorithm
,	O
however	O
,	O
is	O
hard	O
to	O
parallelize	B-Operating_System
and	O
is	O
not	O
cache-optimal	B-General_Concept
because	O
of	O
its	O
bad	O
locality	O
.	O
</s>
<s>
CHs	O
can	O
be	O
used	O
for	O
a	O
more	O
cache-optimal	B-General_Concept
implementation	O
.	O
</s>
<s>
In	O
production	O
,	O
car-navigation	B-Application
systems	I-Application
should	O
be	O
able	O
to	O
compute	O
fastest	O
travel	O
routes	O
using	O
predicted	O
traffic	O
information	O
and	O
display	O
alternative	O
routes	O
.	O
</s>
<s>
CHs	O
can	O
be	O
extended	O
to	O
optimize	O
multiple	O
metrics	O
at	O
the	O
same	O
time	O
;	O
this	O
is	O
called	O
multi-criteria	O
route	B-Application
planning	I-Application
.	O
</s>
<s>
A	O
number	O
of	O
bounds	O
have	O
been	O
established	O
on	O
the	O
preprocessing	O
and	O
query	O
performance	O
of	O
contraction	B-Protocol
hierarchies	I-Protocol
.	O
</s>
<s>
The	O
first	O
analysis	O
of	O
contraction	B-Protocol
hierarchy	I-Protocol
performance	O
relies	O
in	O
part	O
on	O
a	O
quantity	O
known	O
as	O
the	O
highway	O
dimension	O
.	O
</s>
<s>
Calculating	O
the	O
exact	O
value	O
of	O
the	O
highway	O
dimension	O
is	O
NP-hard	O
and	O
W[1]-hard,	O
but	O
for	O
grids	O
it	O
is	O
known	O
that	O
the	O
highway	O
dimension	O
.	O
</s>
<s>
An	O
alternative	O
analysis	O
was	O
presented	O
in	O
the	O
Customizable	O
Contraction	B-Protocol
Hierarchy	I-Protocol
line	O
of	O
work	O
.	O
</s>
<s>
+	O
Query	O
Time	O
Complexity	O
of	O
Contraction	B-Protocol
Hierarchies	I-Protocol
Algorithm/Analysis	O
Technique	O
Year	O
Time	O
Complexity	O
Notes	O
Bounded	O
Growth	O
Graphs	O
2018	O
Customizable	O
Contraction	B-Protocol
Hierarchies	I-Protocol
2013-2018	O
or	O
.	O
</s>
