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8/12/2019 Time Table Ch Slides

1/23

1 Robert Geisberger:Contraction of Timetable Networks with Realistic Transfers

Faculty for Computer ScienceInstitute for Theoretical Computer Science, Algorithmics II

Institute for Theoretical Computer Science, Algorithmics II

Contraction of Timetable Networks withRealistic TransfersRobert Geisberger [email protected]

KIT University of the State of Baden-Wrttemberg and

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


2/23

Motivation



Route planning intime-dependent road networksis fast.Speed-uptechniques gainthree orders of magnitudeovertime-dependent Dijkstra (earliest arrival queries).

Contraction Hierarchies [ALENEX09, SEA10]SHARC [ESA08, SEA10]

Road network is modelled as graph withtime-dependent edgeweightsthat maparrival time travel time.

v

u

1

2

s


3/23

Motivation



Route planning inpublic transportation networksis slow.

Speed-uptechniques gainone order of magnitude.

SHARC [ESA08, ATMOS09]

Network is still modelled as graph withtime-dependent edgeweightsthat maparrival time travel time.

Parallel edges necessary to modelrealistic transfers(minimumtransfer buffer for each station).

1 2

A B

T Ttransfer 5 min.


4/23

Motivation



Node contraction, a very successful technique for route network

performs worse when parallel edges are involved:

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

Contractnodes in this order, nodev is contracted by

foreachpair(u, v)and(v, w)of edgesdo

ifu, v, wis a unique shortest paththenaddshortcut(u, w)with weightw(u, v, w)

Queryrelaxes only edgesto more important nodes

valid due to shortcuts.

1

1

1

1

1

v

2 1 2

2

6

1

4

3

5

3

35

8

s

tnode

order

2


5/23

Motivation



Node contraction, a very successful technique for route network

performs worse when parallel edges are involved:

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

Contractnodes in this order, nodev is contracted by

foreachpair(u, v)and(v, w)of edgesdo

ifu, v, wis a unique shortest paththenaddshortcut(u, w)with weightw(u, v, w)

Queryrelaxes only edgesto more important nodes

valid due to shortcuts.

2

2

1

1

1

1

1

v

2 1 2

2

6

1

4

3

5

3

35

8

s

tnode

order

2


6/23

Motivation



Node contraction, a very successful technique for route networkperforms worse when parallel edges are involved:

Shortcuts can multiply: aincoming parallel edges and boutgoingparallel edges may result ina bparallel shortcuts.

T T T

A B C


7/23

Motivation



Node contraction, a very successful technique for route networkperforms worse when parallel edges are involved:

Shortcuts can multiply: aincoming parallel edges and boutgoingparallel edges may result ina bparallel shortcuts.

T T

A C


8/23

Station graph modelFirst contribution



1:1mapping betweennodes and stations.No parallel edges.

Each edge stores a set of connections, no FIFO-property required.

Store additionaltrain information to respect transfer buffers.

departure train departure time arrival time arrival train1 09:30 10:15 12 09:45 10:15 22 10:45 11:10 3

A B

Another station model was independently developed by Berger et al.[ATMOS09], but requires parallel edges and the FIFO-property.


9/23

Station graph modelDominant connections



We say that a connectionPdominatesa connectionQif we canreplaceQbyP, i.e.

Pdoesnot depart before Qand doesnot arrive after Q.

When theirdeparture trainsdiffer, there has to be enough time totransfer from the train of Qto the train of P.

The same has to hold forarrival trains.

10:30 10:32

10:35

train 1 train 1

Ptrain 2

Q

replace

transfer


10/23

Station graph modelOperations



Store onlydominantset of connections with each edge.Asearchcomputes dominant sets of connections.

Required operations:

Linkthe connections of two incident edges.

Build theminimumof two sets of connections between the samestation pair.

ffg

f * g g

Both operations are almost linear in the number of connections.


11/23

Station graph modelProfile query



Compute a dominant set ofall connectionsbetween a pair ofstations(A, B).

Dijkstra-likelabel correctingalgorithm based on new link andminimum operation. Priority queue key: minimum duration.

A C 1. link

2. minimum

B

Combine link and minimum operation to avoid some copying.


12/23

Station graph modelTime query



Compute thefastest connectionbetween a pair of stations (A,

B)not departing earlier than time .

Problem:Subpath-optimality not given when we only look at time.

We need to compute for each train the earliest arrival time, and onlydrop connections that arrive transfer buffer or more minutes later.

8:04

8:02

8:42

8:448:

458:

40

train 1

train 2

train 3

train 2

train 4

9:23

9:25

A B

8:00

Ctransfer 5 min.


13/23

Timetable Contraction HierarchiesSecond contribution



Most difficult part ispreprocessing. Highlevel:1 Assign each node apriorityon how attractive it is to contract it.2 Contractthe most attractive node.3 Updatethe priorities of the neighbors of the contracted node.4

Repeatfrom Step 2 until all nodes are contracted.Step 3 is the most time-consuming, as it performs a simulatedcontraction for each neighbor to compute the number of necessaryshortcuts (used for node priority).

Problem:For road networks,min-max-searchhelps to speedup

contraction. But maximum on timetable networks is mostlytoo high,e.g. when there is no service during the night.


14/23

Timetable Contraction HierarchiesPreprocessing



Store for each remaining node theset of necessary shortcuts.

Feasible, as there are much less nodes as in road networks.After contractionof a node, we need toupdatetheses sets.

Only the endpoints of added shortcuts are affected (neighbors).At most one forward profile search and one backward profile search fromeach neighbor are necessary.

v

u w

y x

v

u wnew shortcutat contraction ofv

potentially neces-sary shortcut forcontraction of u

potentially necessaryshortcut for contrac-tion of w


15/23

Timetable Contraction HierarchiesQuery



Profile queryis performed bidirectional from source and target.

Time querydoes not know the arrival time. Two-phase approach:1 BackwardBFS from the target usingdownward edges.2 Forwardquery usingupward edges, and then useddownward edges.

t

time = ?time =

s

u

Additional optimizations important for road networks(min-max-search, stall-on-demand) bring no advantage.


16/23

ExperimentsSetup



Environment: Intel Xeon X5550 at 2.67 GHz

Networks:

longdistance connections of Europe

localtraffic in Berlin/Brandenburg in Germany

time-dependent station based factornetwork nodes edges nodes edges nodes edges

long 550 975 1 488 978 30 517 88 091 18.1 16.9local 228 874 599 406 12 069 33 473 16.9 17.9


17/23

ExperimentsStation graph model



query #delete speed time speedtype model mins up [ms] up

lo

ng

time time-dep. 259 506 - 54.3 -

station 14 504 17.9 9.4 5.8

profile time-dep. 1 949 940 - 1994 -station 48 216 40.4 242 8.2

local time

time-dep. 112 683 - 20.9 -station 5 969 18.9 4.0 5.2

profile

time-dep. 1 167 630 - 1263 -

station 33 592 34.8 215 5.9


18/23

ExperimentsTimetable Contraction Hierarchies



PREPROC. QUERYtime edge type #del. speed time speed

[s] inc. mins up [ms] up

long

619 86% time 183 79 0.2 43.5

profile 251 192 3.4 71.4

local

685 128% time 186 32 0.4 9.2

profile 426 79 24.2 8.9


19/23




Comparison with time-dependent SHARC [ESA08] on networklong.



ecoSHARC

2268 74% time 32 575 8 7.4 7.2profile 181 782 11 415.0 5.4

genSHARC

18 522 74% time 8 771 30 2.0 26.6

profile 55 306 35 114.7 19.5

CH 619 86%

time 183 79 0.2 43.5

profile 251 192 3.4 71.4

We scaled timings of SHARC based on plain Dijkstra timings.


20/23




Comparison with time-dependent SHARC [ESA08] on networklong.



ecoSHARC

2 268 74% time 32 575 8 7.4 7.2profile 181 782 11 415.0 5.4

genSHARC

18 522 74% time 8 771 30 2.0 26.6

profile 55 306 35 114.7 19.5

CH 619 86% time 183 1418 0.2 251.4

profile 251 7769 3.4 586.5

We scaled timings of SHARC based on plain Dijkstra timings.


21/23

Conclusion



Station graph model is superiorto time-dependent model for thegiven scenario.

Node contraction works, as hierarchy is better visible due to 1:1mapping between nodes and stations.

Timetable Contraction Hierarchies have preprocessing time of afewminuteswith query times ofhalf a millisecond.

Open work:

Support formulti-criteriascenarios, that e.g. respect number of

transfers.Combination withgoal-directedtechniques.


22/23

Thank You



Thank you for your attention.


23/23

Questions

21 Robert Geisberger:Contraction of Timetable Networks with Realistic TransfersFaculty for Computer Science

Institute for Theoretical Computer Science, Algorithmics II

Questions?

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