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1 Robert Geisberger:Heuristic Contraction Hierarchies with Approximation Guarantee

Institute for Theoretical InformaticsAlgorithms II

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Institute for Theoretical Informatics, Algorithms II

Heuristic Contraction Hierarchies withApproximation GuaranteeRobert Geisberger [email protected]

Dennis Schieferdecker [email protected]

KIT University of the State of Baden-Wrttemberg and

National Large-scale Research Center of the Helmholtz Association www.kit.edu


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Motivation



My domain: Route planning inroad networks.

Input:GraphG= (V, E)with edge weight function c :E R+.

Task: Compute theshortest path distance d(s, t).

Previous work:

Perform apreprocessingstep.

Several algorithms with different tradeoffs ofpreprocessing time,preprocessing space, andquery speedup.

algorithm space preproc. speedup

[B/n] [min]Dijkstra 0 0 1contraction hierarchies -4 32 41 041transit nodes + edge flags 320 229 3 327 372


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Motivation



Arising Question:Can we transfer these results toother graph classes?

Problems:

We justobserve good performance.

Wemiss hard theoryexplaining the good performance.Some of our algorithms have in the worst case aruntime worse thanDijkstraon arbitrary graphs.

Our contribution:

We applied our algorithms on other promising graph classes,observed their performance, and looked for potential optimizations.

The result is an approximate version of contraction hierarchies thatimproves the performance on certain graph classes.


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Contraction Hierarchies (CH)

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Related Work



Contraction Hierarchies (CH) [WEA 2008]CH is based on the concept ofnode contraction: removing a nodeand adding shortcuts topreserveshortest paths distances.

in more detail:

Ordernodes by importance,V ={1, 2,. . . , n}.

Contractnodes in this order, nodev is contracted by

foreachpair(u, v)and(v, w)of edgesdoifu, v, wis a unique shortest paththen

addshortcut(u,

w)with weightc

(u,

v,

w)

Queryrelaxes only edges to more important nodes.valid due to shortcuts


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Example: Construction



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Construction



To identify necessary shortcuts,

performlocal witness searchfrom nodeuwith incoming edge(u, v),

ignorenodevat search,

andadd shortcut(u, w)iff no witness is found orc(witness) > c(u, v) +c(v, w).

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Construction



To identify necessary shortcuts,

performlocal witness searchfrom nodeuwith incoming edge(u, v),

ignorenodevat search,

andadd shortcut(u, w)iff no witness is found orc(witness) > c(u, v) +c(v, w).

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Query



The query is a modifiedbidirectionalDijkstra algorithm.

upward graph G := (V, E)withE :={(u, v) E |u< v}downward graph G := (V, E)withE :={(u, v) E |u> v}

We perform a forward search in G and a backward search inG.

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Query






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Query






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S b k K l h


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Saarbrcken to Karlsruhe299 edges compressed to 13 shortcuts.

S b k K l h


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Saarbrcken to Karlsruhe

316 settled nodes and 951 relaxed edges


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Our Contribution



Approximate Contraction Hierachies (apxCH)Path found by query algorithm has a maximumstretch of(1 + ).

Improves performanceon certain instances.

(a) input (b) CH (c) apxCH-10%


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Approximate Contraction HierarchiesBasic idea



Avoid a shortcut, even when a witness is (1 + )times longer.

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no shortcut


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12 Robert Geisberger:Heuristic Contraction Hierarchies with Approximation GuaranteeInstitute for Theoretical Informatics

Algorithms II


But:straightforward implementation can causeerrors to stack.

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12 Robert Geisberger:Heuristic Contraction Hierarchies with Approximation GuaranteeInstitute for Theoretical Informatics

Algorithms II



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no shortcut

A i C i Hi hi


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12 Robert Geisberger:Heuristic Contraction Hierarchies with Approximation Guarantee Institute for Theoretical InformaticsAlgorithms II



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no shortcut

A i t C t ti Hi hi


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missing shortcut

A i t C t ti Hi hi


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Approximate Contraction HierarchiesDetails


Store asecond edge weight cwith each edge (initially c := c).

Add shortcut iff c(witness) > (1 + )c(shortcut).

Update c(x, y) :=min

c(x, y),c(shortcut) c(x,y)c(witness)

for all edges

(x, y)on the witness path that prevented the shortcut. c(witness) c(shortcut)

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(5)

(5)

(5) (6)

(8)(4)

A i t C t ti Hi hi


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for all edges


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no shortcut

(5)

(5)

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Approximate Contraction Hierarchies


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for all edges


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no shortcut

(5)

(5)

(4 611 ) (5 511 )

(8)(4)



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for all edges


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eorder

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(5)

(5)

(4 611 ) (5 511 )

(8)(4)



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for all edges


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shortcut 11

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(5)

(5)

(4 611 ) (5 511 )

(8)(4)

Combination with Goal directed


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Combination with Goal-directedTechniques


Techniques: A*, ALT, Arc flags.

Goal-direction only on thecore of 5% highest ordered nodes.

Previous algorithms[Bauer et al. 2008] work almost directlywithapproximate node contraction.

A* and ALT can beweighted with(1 + )[Pohl 1970].

s t

Experimental Results


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sensor network, 1 000 000 nodes, average degree 10

preproc. query[s] [B/n] #settled [ms] error

bidir. Dijkstra 0 0 326 597 127.1 -bidir. ALT-a64 194 512 3 173 2.8 -

unidir. A* 0 16 57 385 36.6 -unidir. WA*-10% 0 16 1 234 1.0 1.25%unidir. WA*-21% 0 16 724 0.7 2.87%

CH 1 887 0 2 969 4.0 -apxCH-1% 993 -4 2 742 2.7 0.16%

apxCH-10% 474 -18 2 584 1.9 2.17%apxCHALT-10% 489 7 215 0.3 2.16%apxCHALT-10% W-10% 489 7 102 0.2 3.56%



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CH 1 887 0 2 969 4.0 -apxCH-1% 993 -4 2 742 2.7 0.16%




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CH 1 887 0 2 969 4.0 -apxCH-1% 993 -4 2 742 2.7 0.16%




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road network of Western Europe, 18 million nodes, 42.2 million edges


TRAVE

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TIME

CH 1 050 -1 430 0.206 -

apxCH-10% 1 099 -2 430 0.199 0.40%CHASE 13 421 7 42 0.028 -apxCHASE-10% 11 977 5 42 0.026 0.40%

DISTANCE CH 1 258 0 1 333 1.198 -

apxCH-10% 950 0 1 248 0.873 1.32%

CHASE 84 759 15 59 0.058 -apxCHASE-10% 33 147 10 62 0.048 1.32%



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road network of Western Europe, 18 million nodes, 42.2 million edges


TRAVE

L

TIME

CH 1 050 -1 430 0.206 -

apxCH-10% 1 099 -2 430 0.199 0.40%CHASE 13 421 7 42 0.028 -apxCHASE-10% 11 977 5 42 0.026 0.40%

DISTANCE CH 1 258 0 1 333 1.198 -

apxCH-10% 950 0 1 248 0.873 1.32%

CHASE 84 759 15 59 0.058 -apxCHASE-10% 33 147 10 62 0.048 1.32%

Conclusion


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Conclusion

17 Robert Geisberger:

Heuristic Contraction Hierarchies with Approximation GuaranteeInstitute for Theoretical Informatics

Algorithms II

Approximationextends the application rangeof contraction

hierarchies beyond road networks.Further speed-up achieved withgoal-direction.


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18 Robert Geisberger:

Heuristic Contraction Hierarchies with Approximation GuaranteeInstitute for Theoretical Informatics

Algorithms II

Thanksfor your attention.

Anyquestion?

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