Transport Optimization Finland 2009

8/12/2019 Transport Optimization Finland 2009

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Algorithm Engineering for Large Graphs

Fast Route Planning

Veit Batz,Robert Geisberger, Dennis Luxen,

Peter Sanders, Christian Vetter

Universitt Karlsruhe (TH)


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Sanders et al.: Route Planning 2

Route Planning

?

Goals: exactshortest paths inlarge (time-dependent) road networks

fast queries(point-to-point, many-to-many)

fast preprocessing

low spaceconsumption

fast updateoperations

Applications:

route planning systems in the internet, car navigation systems,

ride sharing, traffic simulation, logistics optimisation


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Contraction Hierarchies

[WEA 08]

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 edges

to more important nodes

valid due to shortcuts

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tnode

order

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[WEA 08]

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 edges

to more important nodes

valid due to shortcuts

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tnode

order

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Node Order

usepriority queueof nodes, nodev is weighted with a linear

combination of:

edge difference:#shortcuts#edgesincident tov

uniformity:e.g. #deleted neighbors

. . .

integrated construction and ordering:

1. popnodev on top of the priority queue

2. contractnodev

3. update weightsof remaining nodes


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Saarbrcken to Karlsruhe299 edges compressed to 13 shortcuts.


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Saarbrcken to Karlsruhe

316 settled nodes and 951 relaxed edges


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foundationfor our other methods

conceptuallyvery simple

handlesdynamic scenarios

Static scenario:

7.5 minpreprocessing

0.21 msto determine the path length

0.56 msto determine a complete path description

little space consumption (23 bytes/node)


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Transit-Node Routing

[DIMACS Challenge 06, ALENEX 07, Science 07]

joint work with H. Bast, S. Funke, D. Matijevic

very fast queries

(down to1.7s, 3 000 000 times faster than D IJKSTRA)

winnerof the 9th DIMACS Implementation Challenge

more preprocessing time (2:37h) and space (263 bytes/node) needed

s t

SciAm50 Award


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Many-to-Many Shortest Paths

joint work with S. Knopp, F. Schulz, D. Wagner

[ALENEX 07]

efficientmany-to-many variantof

hierarchical bidirectional algorithms

10 00010 000 table in10s

S


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input:sourcesS={s1, . . . ,sn}andtargetsT={t1, . . . ,tm}

naive algorithm a: performmin(n,m)Dijkstra one-to-many

searches

n=m=10000:10000 5s13.9h

naive algorithm b: performn mTNR-queries

n=m=10000:10000 10000 1.7s=170s

better algorithm: exploithierarchicalnature of CH

S

T


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performn forward-upwardsearches from eachsi

storethe distanced= (si,v)of each reached nodev in buckets

then performm backward-upwardsearches from eachtj

scanbuckets at each reached node

correctnessof CH ensures that

d(si,tj) = minvreached

((si,v) +(v,tj))

si tjsi, dd

. . .

v


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Ride Sharing

Current approaches:

match only ride offers withidenticalstart/destination (perfect fit)

sometimes radial search around start/destination

Our approach:

driver picks passenger up and gives him a ride to his destination

find the driver with theminimal detour(reasonable fit)

Efficient algorithm:

adaption of the many-to-many algorithm

matches a request to 100 000 offers in25 ms


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Turn Penalties

convert node-based graph toedge-based graph

applyspeedup technique, e.g. CH

Germany:1.812 minpreprocessing,200422squery

e1 e3

e4e2

e5 e6

e7

v1 v2 v3

v4

v1

e1

v3

e7

e2

e3

v2

e4

e5 e6

v4

node-based graph edge-based graph


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Dynamic Scenarios

change entirecost function

(e.g., use different speed profile)

change afew edge weights

(e.g., due to a traffic jam)


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Dynamic Scenarios

change afew edge weights

server scenario:if something changes,

updatethe preprocessed data structures

answermanysubsequent queries veryfast

mobile scenario:if something changes,

itdoes not payto update the data structures performsingleprudent query that

takes changed situation into account


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Mobile Contraction Hierarchies[ESA 08]

preprocess data on a personal computer

highly compressedblocked graph representation 8 bytes/node

compactroute reconstruction data structure + 8 bytes/node

experiments on a Nokia N800 at 400 MHz

cold querywith empty block cache 56ms

compute complete path 73ms

recomputation, e.g. if driver took the wrong exit 14ms

query after 1 000edge-weight changes, e.g. traffic jams 699 ms


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Time-Dependent Route Planning

edge weights aretravel time functions:

- {time of daytravel time}

- piecewise linear

- FIFO-propertywaiting does not help

query(s,t,0)start, target,departure time

looking for:

a fastest route froms totdependingon

0

Earliest ArrivalProblem


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Travel Time Functions

we need three operations

evaluation: f() O(1) time

merging:min(f,g) O(|f|+ |g|)time

chaining: f g(f afterg) O(|f|+ |g|)time

note:min(f,g)and f ghave O(|f|+ |g|)points each.

increase of complexity

g f

f * g


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Time-Dependent Dijkstra

Only onedifference to standard Dijkstra:

Cost of relaxed edge(u,v)depends...

...onshortest path tou.

s

v

u

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Profile Search

Modified Dijkstra:

Node labels are travel time functions

Edge relaxation: fnew:=min(fold,fu,v fu)

PQ key ismin fu

A label correctingalgorithm

fu,v

fu

fold

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u


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Min-Max-Label Search

Approximate version of profile search:

Computesupperand lower bounds

Node labels are pairsmmu:= (minfu,maxfu)

Edge relaxation:

mmnew:=min(mmold,mmu+ (minfu,v,maxfu,v))

PQ key is the lower bound

A label correctingalgorithm

) =min( ,

fu,v

s

v

u

mm

mm

u

old


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Time-Dependent Contraction Hierarchies

two major challenges:

1. contractionduring precomputation

witnesses can be found byprofile search

...which isstraightforward

...butincredibly slow!

do somethingmore intelligent!

2. bidirectional search

problem: arrival timenot known

...but can besolved


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Restricted Profile Search

phase 1:restricts the search space

min-max-labelsearch

might already find awitness

if not:marka corridor of nodes:

initiallymarknodew

for each nodev markonly thosetwo predecessors

corresponding to the upper / lower bound

phase 2:profile searchonly usingmarkednodes

u

v

w

MinMaxLabel Search

Profile Search


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Bidirectional Time-Dependent Search

phase 1:two alternating searches:

forward:time-dependent Dijkstra

backward:min-max-label search

meeting points are candidates

phase 2:from allcandidates...

...do time-dependentmany-to-one forward Dijkstra

...only usingvisited edges

...using min/max distances toprunesearch

time = 0

s t

time = ?

u


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Experimental Comparision

PREPROCESSING QUERIES

time space time speed

input algorithm [h:m] [B/n] [ms] up

Germany TCH timed ord 1:48 + 0:14 743 1.19 1 242

midweek TCH min ord 0:05 + 0:20 1 029 1.22 1 212

Germany TCH timed ord. 0:38 + 0:07 177 1.07 1 321

Sunday TCH min ord. 0:05 + 0:06 248 0.71 1 980


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Parallel Precomputation

contraction:

contract maximum independent sets of nodes, i.e. nodes that are

least importantin their1 hop neighborhood, in parallel

add shortcuts even incase of equality

node order:

use the current priority terms in the priority queue

use2-3 hop neighborhoodfor good results

use priority terms thatrarely decreaseon update

6.5x speedup on 8 cores


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Summary

static routingin road networks is easy

applications that require massive amount or routing

instantaneous mobile routing

techniques for advanced models

time-dependentrouting is fast

bidirectional time-dependent search

fast queries

fast (parallel) precomputation


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Current / Future Work

Multiple objectivefunctions and restrictions (bridge height,. . . )

Multicriteriaoptimization (cost, time,. . . )

Integrate individual and public transportation

Other objectivesfor time-dependent travel

Routing driventraffic simulation

Real-timetraffic processing for optimal global routing


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Ultimate Routing in Road Networks?

Massive floating car data accurate current situation

Past data+traffic model+real time simulation

Nash euqilibrium predicting near future

time dependent routing in Nashequilibrium

realistic traffic-adaptive routing

Yet another step further

traffic steeringtowards a social optimum


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Macroscopic Traffic SimulationGoals:

fastsimulation of traffic in large road networks

basedon shortest paths

exploitspeedup techniques

Status of implementation:

timeindependentversion as student project

timedependentversion under development

Basis for equilibria computation


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Nash Equilibria in Road Networks

Computation:Iterativesimulation with adapted edge weights

Basic approach (simplified):

Permute set ofs t-pairs

For eachs t-pair (until equilibrium is reached)

-compute pathandupdate weightson its edges

Goals and applications:

Develop model fornear future predictionsof road traffic

Providerealistic traffic-adaptive routing

Traffic steering towardssocial optimum


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Multi-Criteria Routing

multipleoptimization criterias

e.g. distance, time, costs

flexibilityat route calculation time

e.g. individual vehicle speeds

diversityof results

e.g. calculate Pareto-optimal results

roundtripswith scenic value

e.g. for tourists


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Current State

adopt contraction hierarchies to multi-criteria:

modifiythe contraction so the query stays simple

add allnecessary shortcutsduring contraction

do this by modifying thelocal search

linear combination oftwo:x + aywitha[l,u]

label is now a function of x (see timedependent CH)

linear combination ofmore:a1x1+ + anxn withai[li,ui]

Pareto-optimal(may add too many shortcuts)

too many shortcutsneeded when done naive


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Challenges

current speedup-techniques largely rely onhierarchy

every optimization criterion has a specific influence on the

hierarchyof a road network

e.g. finding thefastestroute contains more hierarchy than finding

theshortestroute

however multiple criteriainterferewith hierarchy, but the algorithm

should workfastonlargegraphs

e.g. motorways drop in the hierarchy because of road tolls

new algorithmic ideas necessary

Documents

Transport Optimization Finland 2009