flexible route planning with contraction hierarchies

8/12/2019 Flexible Route Planning With Contraction Hierarchies

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1 Robert Geisberger, Moritz Kobitzschand Peter Sanders:Flexible Route Planning with Contraction Hierarchies

Faculty for Computer ScienceInstitute for Theoretical Computer Science, Algorithmics II

Institute for Theoretical Computer Science, Algorithmics II

Flexible Route Planning with ContractionHierarchiesRobert Geisberger,Moritz Kobitzschand Peter Sanders {geisberger,kobitzsch,sanders}@kit.edu

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



route planningbecomesubiquitous

route planning servicese. g. Google Maps

wide spread generates widevariety of interests

multiple optimization criterias

possible (e.g. (time|cost) )

a

b c

d e

f(3|6)

(1|2)

(1|1)

(1|2)

(1|2)

(2|1)(3|0)

(3|1)


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Motivation








a

b c

d e

f(3|6)

(1|2)

(1|1)

(1|2)

(1|2)

(2|1)(3|0)

(3|1)

a

f(3|6)


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Motivation








a

b c

d e

f(3|6)

(1|2)

(1|1)

(1|2)

(1|2)

(2|1)(3|0)

(3|1)

a

f(3|6)

a

f(4|4)


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Motivation








a

b c

d e

f(3|6)

(1|2)

(1|1)

(1|2)

(1|2)

(2|1)(3|0)

(3|1)

a

f(3|6)

a

f(4|4)

a

f

(7|2)


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Just imagine...




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Pareto Optimality



find allnon dominated routes

AdominatesB ifA is better orequal toBin every weight term

may result in an exponentialnumber of routes

heuristics necessaryto gain

practical algorithms

a

b c

d e

f

(1|2)

(1|1)

(1|2)

(1|2)

(3|0)

(3|1)

a

f(3|6)

(7|2)

(2|1)

(4|4)


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Pareto Optimality



find allnon dominated routes

AdominatesB ifA is better orequal toBin every weight term

may result in an exponentialnumber of routes

heuristics necessaryto gain

practical algorithms

a

b c

d e

f

(1|2)

(1|1)

(1|2)

(1|2)

(3|0)

(3|1)

a

f(3|6)

(7|2)

(2|3)

(4|6)


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Parameterized Optimality



linear combinationof weightterms

(t|c) t+p c

preprocessingconsidersallpossibleparameter values

queryforfixedparametersingle criteria techniques

parameter range allows to

selectmaximal impactof weightterms

p=0

a

b c

d e

f

(1|2)

(1|1)

(1|2)

(1|2)

(2|1)(3|0)

(3|1)


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(t|c) t+p c





p=0

0

1 1

2 4

3p=0 (3|6)

(1|2) 1

(1|1) 1

(1|2) 1

(1|2) 1

(2|1) 2(3|0) 3

(3|1) 3


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(t|c) t+p c





p=0

0

3 2

5 5

8p=1 (4|4)

(1|2) 3

(1|1) 2

(1|2) 3

(1|2) 3

(2|1) 3(3|0) 3

(3|1) 4


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(t|c) t+p c





p=0

0

5 3

7 6

11p=2 (7|2)

(1|2) 5

(1|1) 3

(1|2) 5

(1|2) 5

(2|1) 4(3|0) 3

(3|1) 5


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Parameterized Weight Functions



thedistance functiond(a, f)for two nodesa, f isconcave

monotone due to positive weight terms

a

b c

d e

f

(1|2)

(1|1)

(1|2)

(1|2)

(2|1)(3|0)

(3|1)

(3|6)

(4|4)

a

f (7|2)

p

d(a, f, p)


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Contraction Hierarchy Routing



order nodes by importance{v1,. . ., vn}

contractin thisorder

preserve original distancesby addingshortcuts

necessity of shortcutsdecided bywitness paths

query only relaxes edges tomore important nodes

a b c d

a

b

c

d

1 2 1

1

2

1

3

node

order


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Contraction Hierarchy Routing



order nodes by importance{v1,. . ., vn}

contractin thisorder

preserve original distancesby addingshortcuts

necessity of shortcutsdecided bywitness paths

query only relaxes edges tomore important nodes

a b c d

a

b

c

d

1 2 1

1

2

1

3

node

order

a d

a

d

b c

c


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Flexible Contraction Hierarchies



shortcuts in flexible scenario ifp :(a, b, c)is the only shortestpath

necessity of shortcuts only oncontinuous intervals storenecessity intervalin edges

necessity interval onaveragedeductable fromtwo single criteria

Dijkstra queriesquery uses only necessary edges

a c

d

(3|1) (3|1)

b

(2|4

) (2|4

)

p

d(a, c, p)

1/3 .


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shortcuts in flexible scenario ifp :(a, b, c)is the only shortestpath

necessity of shortcuts only oncontinuous intervals storenecessity intervalin edges

necessity interval onaveragedeductable fromtwo single criteria

Dijkstra queriesquery uses only necessary edges

a c

d

(3|1) (3|1)

(4|8)b [0,1/3]

p

d(a, c, p)

1/3 .


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single node order not practicalfor wide parameter range

a lot of shortcuts for all

parameter valuessplit parameter intervalwithheuristics

repeat as necessary

bucketsto supportfast

scanningof necessary edges

level


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Core-ALT for flex-CH

10 Robert Geisberger,Moritz Kobitzschand Peter Sanders:Flexible Route Planning with Contraction Hierarchies


core based approach to

speed uppreprocessing(uncontracted core)

speed upquery(contractedcore)

usesALT algorithmon |k|topmostnodes

adaptable to flexible

scenario withlinearinterpolation

p

d(a, f, p)


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Profile Queries



findall possible paths

profile query withmaximal3 #paths 2queries

approximation: recursiononly if improvement largermore than(1 + )possible

p

d(a, f, p)


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Profile Queries






p

d(a, f, p)


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Profile Queries






p

d(a, f, p)


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Profile Queries

11 Robert Geisberger,Moritz Kobitzschand Peter Sanders:Flexible Route Planning with Contraction HierarchiesFaculty for Computer Science





p

d(a, f, p)


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Experimental results

12 Robert Geisberger,Moritz Kobitzschand Peter Sanders:Flexible Route Planning with Contraction HierarchiesFaculty for Computer Science


Table:Preprocessing and query performance for 64 landmarks on the Germanroad network, average over 10 000 queries. (4 692 751 nodes and 10 806 191directed edges)

core preproc space query speedup[hh:mm] [B/node] [ms]

- - 60 2 037.52 10 1 :54 159 2.90 698

uncontracted 5 000 1 :08 167 2.58 789

contracted 10 000 2 :07 183 0.63 3 234

E i l l


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Experimental results

13 Robert Geisberger,Moritz Kobitzschand Peter Sanders:Flexible Route Planning with Contraction Hierarchies Faculty for Computer ScienceInstitute for Theoretical Computer Science, Algorithmics II

0 0.125 0.25 0.5 1 2 4 8 16 32

0

10

20

30

40

50

60

epsilon (%)

# found pathsquery time [ms]

C i t SHARC


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Comparison to SHARC


Table:Approximated profile search on Western Europe

preproc # timealgorithm paths [ms]

flexCH 5 :15 0.00 12.5 21.9flexCH 5 :15 0.01 5.7 6.2

Pareto-SHARC 7 :12 (no guarantee) 5.3 35.4

C l i


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Conclusion


exact flexiblerouting for server scenarios

comparableeven tosingle criteriaspeedupscircumventsproblems of Pareto-optimality

scales wellthrough parameter splitting

Th k Y


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Thank You


Thank you for your attention.

Questions


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Questions


Questions?

Future Work


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Future Work


mobile implementation

applyparallelizationtechniquesmore than two weight terms (?)

flexible route planning with contraction hierarchies

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