Outline - SSTD 2011
Transcript of Outline - SSTD 2011
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Online Computation of Fastest Path in Online Computation of Fastest Path in TimeTime--Dependent Spatial NetworksDependent Spatial Networksp pp p
Ugur DemiryurekUgur Demiryurek11, , FarnoushFarnoush BanaeiBanaei--KashaniKashani11, , Cyrus ShahabiCyrus Shahabi11, and , and AnandAnand RanganathanRanganathan22
University of Southern CaliforniaUniversity of Southern California11
and IBM T.J. Watson Research Centerand IBM T.J. Watson Research Center22
Outline
MotivationMotivationProblem DefinitionProblem DefinitionRelated WorkRelated WorkTimeTime--dependent Fastest Path Computationdependent Fastest Path ComputationPerformance EvaluationPerformance Evaluation
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Conclusion and Future WorkConclusion and Future Work
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Motivation
What is the fastest path to
?
• Growing Popularity of Online Map Services
100 Million hits per month (http://googlemobile.blogspot.com/)
Which
?
What is the fastest path to
? is nearby??
Road networks can be very large, e.g., 45M segments for North America
Motivation•• Existing FP and FP Existing FP and FP PrecomputationPrecomputation Techniques Techniques
–– Based on the Based on the constant constant edge weights for each edgeedge weights for each edge
The path recommendation from online map applications remains the same throughout the day regardless of the departure-time from the source (i.e., query time)
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Motivation•• Existing FP and FP Existing FP and FP PrecomputationPrecomputation Techniques Techniques
–– Based on the Based on the constant constant edge weights for each edgeedge weights for each edge
•• In RealIn Real worldworld•• In RealIn Real--worldworld–– The weight of an edge is a function of time, i.e.,The weight of an edge is a function of time, i.e., timetime--dependent.dependent.–– ArrivalArrival--time to an edge determines the traveltime to an edge determines the travel--time on that edgetime on that edge..
5:00 PM8 :30 AM
Monday travel‐time on a segment of I‐10 in LA (generated based on two years of historical traffic sensor data)
Pictures courtesy : http://www.wfrc.org/cms
Motivation•• TimeTime--dependent Fastest Pathdependent Fastest Path
–– Recommends different paths for different departureRecommends different paths for different departure--timestimes
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Outline
MotivationMotivationProblem DefinitionProblem DefinitionRelated WorkRelated WorkTimeTime--dependent Fastest Path dependent Fastest Path ComputationComputation
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Performance EvaluationPerformance EvaluationConclusion and Future WorkConclusion and Future Work
Problem Definition
s
•• Given a timeGiven a time--dependent spatial network dependent spatial network where where edge weights are function of timeedge weights are function of time
wij(t)
Source s and Destination d
Time-dependent Fastest Path (TDFP)
sij( )
t
w3
w1
w2
d
TDFP (s, d, t_s) with respect to s, d and query time t_s finds minimum travel time path among all paths between s and d
Travel-time from s to any destination is time-dependent, i.e., changes based on the departure time.
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t6 n4
Problem Definition
t2
t5
4n3
n4
n4
Path={n1,n3,n4}, Cost=6Path={n1,n2,n4}, Cost=5
SP={n1,n2,n4}
t0
2 1
22
n4
n3
n2
n1
3n2
Path={n1,n3,n4}, Cost=3
Challenges•• Input Size for Precomputation: Input Size for Precomputation:
–– SuperSuper--polynomial number of shortest polynomial number of shortest pp p yp ypaths between any pair of nodespaths between any pair of nodes
n4
t
w12(t), w34(t)
( )
5
w24(t)
10
15
15
t
( )
n2
n1
n2
n1 n4
n
source destination
–– The shortest path The shortest path is not unique in TDis not unique in TD--SN SN and changes with the departure timeand changes with the departure time
–– Recall:SPRecall:SP is unique in static road networks.is unique in static road networks.–– The The lowerlower--envelopeenvelope can have can have super-
polynomial pieces [Dean’04,Foschini’11]pieces [Dean’04,Foschini’11]28
35
fp3fp1
fp2fpi (cost)
204
w23(t)
15
25
825
w13(t)
10
tt
n3
1n3
fp1=f24(f12(t)) fp2=f34(f23(f12(t)))fp3=f34(f13(t))
p y p [ , ]p [ , ]
fpi : total traveltotal travel--time to destinationtime to destination3 10 20
17
t (departure time)
15
7
t : depature time
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Challenges•• Availability of the timeAvailability of the time--dependent dependent
edge weights (i.e., traveledge weights (i.e., travel--time) datatime) data–– NavteqNavteq and and TeleatlasTeleatlas
•• Recently released the timeRecently released the time--dependent traveldependent travel--times for road times for road networks in North Americanetworks in North America
–– Government Agencies Government Agencies •• LA Metro and USC (LA Metro and USC (MetransMetrans andand•• LA Metro and USC (LA Metro and USC (MetransMetrans and and
IMSC) IMSC)
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Outline
MotivationMotivationProblem DefinitionProblem DefinitionRelated WorkRelated WorkTimeTime--dependent Fastest Path dependent Fastest Path ComputationComputation
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Performance EvaluationPerformance EvaluationConclusion and Future WorkConclusion and Future Work
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Related Work
TD R d N kS i R d N k
Shortest Path
TD-Road NetworkStatic Road Network
• Dijkstra
• A* Precomputation:•Geometric speed-up techniques for finding SP, [Wagner et al.,ESA'03]
•A* Search Meets Graph Theory [Goldberg,SODA’05]
•Engineering fast route planning algorithms, [Sanders et al., WEA’07]
• Hierarchical routing in road networks [Geisberger et al WEA’08 Sanders ESA’06]Hierarchical routing in road networks, [Geisberger et al., WEA 08, Sanders ESA 06]
•Scalable network distance browsing [Samet et al., SIGMOD’08]
Related Work
TD R d N k
Shortest Path
S i R d N k TD-Road Network
• Dreyfus [OR’69] (Dijkstra Variant)
• Orda and Rom, [JACM’90] (Bellman F.)
• George and Shekhar [SSTD’07] (TAG)
• Time-dependent SHARC [ Delling et al., ESA’08]
• Time-dependent CH [Batz et al., ALENEX’08]
Precomputation:
Static Road Network
• Time-dependent ALT [Nannicini et al., WEA’09]
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Outline
MotivationMotivationProblem DefinitionProblem DefinitionRelated WorkRelated WorkTimeTime--dependent Fastest Path dependent Fastest Path ComputationComputation
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Performance EvaluationPerformance EvaluationConclusion and Future WorkConclusion and Future Work
TD Fastest Path
•• Generalize A* algorithm proposed for static spatial Generalize A* algorithm proposed for static spatial networks to timenetworks to time--dependent road networksdependent road networksnetworks to timenetworks to time dependent road networksdependent road networks
•• Dijkstra vs. A* Dijkstra vs. A*
d
ssA* SearchDijkstraDijkstra
d
16
increasing cost s
d
h(v) =(vi,d) vivj
Optimality Condition:h(v) should not overestimate the actual distance between v and d
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TD Fastest Path•• Generalize A* algorithm proposed for static spatial networks to timeGeneralize A* algorithm proposed for static spatial networks to time--
dependent road networksdependent road networks
sd Bidirectional
A* Algorithm
Time-dependent bidirectional A* is not straight forward:Challenge 1: The distance between any node v and d is time-dependent, hence need a good h(v) Challenge 2: Start the backward search from the arrival-time at the destination td , but td cannot be determined at the query time
TD Fastest Path
•• Proposed solution: Proposed solution: PrecomputationPrecomputation Phase:Phase:–– PrecomputationPrecomputation Phase: Phase:
•• Partition the road network into nonPartition the road network into non--overlapping overlapping partitions partitions
•• Precompute Precompute lowerlower--bound intra and inter bound intra and inter distance labels distance labels within and across the partitionswithin and across the partitions
–– Online Phase: Online Phase: •• Use the Use the precomputedprecomputed distance labels as a heuristic distance labels as a heuristic
function in the bidirectional timefunction in the bidirectional time--dependent A* dependent A* searchsearch
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TD Fastest Path: Precomputationwij(t) Upper-bound
travel time s
• TD Spatial Network Property
•• TimeTime--independentindependent GraphsGraphs
t
t a e t ew3
Lower-bound travel time
w1
s
dTDFP (s,d,t)
pp pp–– LowerLower--bound Graph bound Graph –– G G , LTT, LTT
•• where edge weights are minimum possible weights where edge weights are minimum possible weights
–– UpperUpper--bound Graphbound Graph-- G , UTTG , UTT•• where edge weights are maximum possible weights where edge weights are maximum possible weights
DLTT(q,p)< TDSP(q,p,t)<DUTT(q,p)
TD Fastest Path: Precomputation
•• Partition the road network to nonPartition the road network to non--overlapping partitions overlapping partitions [Gonzalez, VLDB’07][Gonzalez, VLDB’07]
S1 S2S3
S4S5
S6S7
S10
[Gonzalez, VLDB 07][Gonzalez, VLDB 07]
Border Nodes
S8S9 S11
Our algorithm yields correct results with all non-overlapping partitioning algorithms
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TD Fastest Path: Precomputation
•• Compute Compute intra and inter distance labels intra and inter distance labels within and within and across the partitions based onacross the partitions based on LowerLower--bound Graphbound Graph GGacross the partitions based on across the partitions based on LowerLower bound Graphbound Graph GG
S d
b1
b2
b3
b4
S1 S2
• Only store the minimum of nodeOnly store the minimum of node--toto--border, borderborder, border--toto--border, border, and borderand border--toto--node travel timesnode travel times
h(s) = LTT (s, b1) + LTT (b1, b3) + LTT (b4, d) <= TDSP(s,d,ts) Challenge 1
TD Fastest Path: Precomputation
Node Partition Node-to-Border
Border-to-Node
Border Border Distance Distance
• Distance Labels
Border Noden1 S1 b1,5 b1,7
n2 S1 b2,6 b3,4
…. …. …. ….
n41 S9 b17,3 b15,6
nn Sk bu,,x bv,y
b1 b3 14 12
b1 b4 18 15
b1 b15 12 9
…. …. ….
bn bk x y
Node-to-Border (Intra) Border-to-Border (Inter)
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TD Fastest Path
•• Bidirectional TimeBidirectional Time--dependent A* dependent A* –– Forward Search: TimeForward Search: Time--dependent A* using h(v) found based ondependent A* using h(v) found based on–– Forward Search: TimeForward Search: Time--dependent A using h(v) found based on dependent A using h(v) found based on
distance labels distance labels –– Backward Search : TimeBackward Search : Time--independent A* based on the reverse independent A* based on the reverse
lowerlower--bound graph bound graph GG . Note: h(v) is still valid . Note: h(v) is still valid
s dTDSP(s,u,ts) LTT(u,d)
u
Challenge 2
Cannot stop! TDSP(Cannot stop! TDSP(s,u,ts,u,tss)+ LTT()+ LTT(uu dd, , ttuu) < TDSP() < TDSP(s,d,ts,d,tss))
TD Fastest Path
•• Bidirectional TimeBidirectional Time--dependent A* dependent A* –– Forward Search: TimeForward Search: Time--dependent A* using h(v) found based ondependent A* using h(v) found based on–– Forward Search: TimeForward Search: Time--dependent A using h(v) found based on dependent A using h(v) found based on
distance labels distance labels –– Backward Search : TimeBackward Search : Time--independent A* based on the reverse independent A* based on the reverse
lowerlower--bound graph bound graph GG . Note: h(v) is still valid. . Note: h(v) is still valid.
s dTDSP(s,u,ts) LTT(u,d)
u
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TD Fastest Path
•• Bidirectional TimeBidirectional Time--dependent A* dependent A* –– Forward Search: TimeForward Search: Time--dependent A* using h(v) found based ondependent A* using h(v) found based on–– Forward Search: TimeForward Search: Time--dependent A using h(v) found based on dependent A using h(v) found based on
distance labels distance labels –– Backward Search : TimeBackward Search : Time--independent A* based on the reverse independent A* based on the reverse
lowerlower--bound graph bound graph GG . Note: h(v) is still valid. . Note: h(v) is still valid.
s d
TDSP( d t )TDSP(s,d,ts)
Continue the search only within the nodes found by backward search Continue the search only within the nodes found by backward search (see Section:5.2)(see Section:5.2)
Outline
MotivationMotivationProblem DefinitionProblem DefinitionRelated WorkRelated WorkTimeTime--dependent Fastest Path dependent Fastest Path ComputationComputation
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Performance EvaluationPerformance EvaluationConclusion and Future WorkConclusion and Future Work
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Experimental Evaluation• DataSetsDataSets: : (obtained from (obtained from NavteqNavteq))
•• Los Angeles (LA) road network, Los Angeles (LA) road network, 304304,,162 162 nodesnodesnodes nodes •• California (CA) road network, California (CA) road network, 11,,965965,,300 300 nodes nodes
• Experimental Setup:•• A server with A server with 22..7 7 GHz Pent. Duo Core GHz Pent. Duo Core
Proc. and Proc. and 1212GB RAMGB RAM•• Source s destination d and departureSource s destination d and departure
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•• Source s, destination d and departure Source s, destination d and departure time time ttss are determined uniformly at randomare determined uniformly at random
•• Average results computed from Average results computed from 1000 1000 random srandom s--d queries d queries
Experimental Evaluation
•• TimeTime--dependent Data dependent Data GenerationGenerationGenerationGeneration–– 6500 Sensors on freeways and 6500 Sensors on freeways and
arterials in LAarterials in LA•• 1 sensor/reading per minute 1 sensor/reading per minute •• Collecting and archiving past 2 Collecting and archiving past 2
years years
–– Spatially and temporally aggregateSpatially and temporally aggregateSpatially and temporally aggregate Spatially and temporally aggregate the sensor data by assigning the sensor data by assigning interpolation points (for each 5 interpolation points (for each 5 minutes)minutes)
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Experimental Evaluation
•• Comparison with TDComparison with TD--ALTALT–– TDTD--ALT: Determine 64ALT: Determine 64TDTD ALT: Determine 64 ALT: Determine 64
landmarks based on landmarks based on maxCovermaxCover(best known landmark selection (best known landmark selection algorithm) algorithm)
–– TDFP: Divide CA network to 60 TDFP: Divide CA network to 60 partitions partitions
Storage:Storage:
•
Response Time:TD-ALT very loose bounds based on therandomly selected s and d, and hencethe large search space.
Storage: Storage: • TD-ALT attaches each node an array of 64 elements. Total Storage = 63 MB for CA
• TDFP consumes, for each node, an array of 2 elements + border-to-border distance labels. Total Storage=8.5 MB for CA
Experimental Evaluation
•• SpeedSpeed--up up vsvs Distance Distance •• LowerLower--bound Qualitybound QualityUnidirectional vs Bidirectional wrt
lowerlower--bound quality = bound quality = δ(δ(u,vu,v)/ d()/ d(u,vu,v))
N ïN ï dd (( )/)/ dd
Unidirectional vs Bidirectional wrtdistance between s and d?
Naïve: Naïve: ddeuceuc((u,vu,v)/ )/ max_speedmax_speedALT:ALT: Landmark based Landmark based DL: DL: Distance label based Distance label based
Speed-up is significantly more especially for long distance queries
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Experimental Evaluation•• More Comparison More Comparison
Outline
MotivationMotivationProblem DefinitionProblem DefinitionRelated WorkRelated WorkTimeTime--dependent Fastest Path dependent Fastest Path ComputationComputation
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Performance EvaluationPerformance EvaluationConclusion and Future WorkConclusion and Future Work
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Conclusion•• From static road networks to more realistic timeFrom static road networks to more realistic time--dependent road dependent road
networks where edge weights are timenetworks where edge weights are time--varyingvarying•• PrePre--computation is challenging in timecomputation is challenging in time--dependent road networks dependent road networks
(Super(Super--polynomial input size)polynomial input size)•• Proposed TimeProposed Time--dependent bidirectional A* based on inter and intra dependent bidirectional A* based on inter and intra
distance labelsdistance labels•• Plan to work on Plan to work on
–– Incremental algorithms to support rapid network edge weight Incremental algorithms to support rapid network edge weight changes (e.g., due to accidents)changes (e.g., due to accidents)changes (e.g., due to accidents)changes (e.g., due to accidents)
–– Better bounds to expedite the searchBetter bounds to expedite the search–– Different network Different network partitioning techniques partitioning techniques
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Thank You!
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