Project presentation
1
Agenda
 Motivation
 Problem Statement
 Related Work
 Proposed Solution
 Hierarchical routing theory
2
Motivation
Many Applications of network routing
Examples: Online Map service,
phone service, transportation
navigation service

Identification of frequent routes


Crime Analysis
Identification of congested routes

Network Planning
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Motivation
Existing work on transportation network routing
 Based on constant edge value.
In real world
 Travel time of road segment changes over time.
People are interested in various routing queries
Question: Can we build a model
which can support various spatiotemporal network routing queries?
I94 @ Hamline Ave at 8AM & 10AM
*U. Demiryurek, F. B. Kashani, and C. Shahabi. Towards k-nearest neighbor search in time
dependent spatial network databases. In Proceedings of DNIS, 2010
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Problem Statement
Input:
 A spatial network G=(N,E).
 Temporal changes of the network topology and parameters.
Output:
 A model to process routing queries in spatio-temporal network.
Objective:
 Minimize storage and computation cost.
Constraints:
 Spatio-temporal network and pre-computed information are stored in
secondary memory.
 Changes occur at discrete instants of time.
 Allow wait at intermediate nodes of a path.
 Routing is based on Lagrange path.
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Key Concept
 Graph G= (N, E): a directed flat graph consisting of a node set N, and
an edge set E.
 Fragment: a sub-graph of G, which consists a subset of nodes and
edges of G.
 Boundary node: a node that has neighbors in more than one
fragment.
 Hierarchical graph: a two-level representation of the original graph.
The base-level is composed of a set of disjoint fragments
The higher-level called boundary graph, is comprised of the boundary
nodes
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Challenges
New semantics for spatial networks
 Optimal paths are time dependent
Key assumptions violated
 Prefix optimality of shortest paths (Non-FIFO travel time)
Conflicting Requirements
 Minimum Storage Cost
 Computational Efficiency
[1,1,1,1]
B
[1,1,1,1]
[1,1,3,1]
A
C
[2,2,2,2]
D
E
[1,1,1,1]
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Related Work


Static Model [HEPV’98, HiTi’02, Highway’ 07]

Does not model temporal variations in the network parameters

Supports queries such as shortest path in static networks

Pre-compute and store information
Spatio-temporal Model for specific query [Voronoi diagram’ 10]

Designed for specific query such as K nearest neighbors

Not scalable to other spatio-temporal network routing queries
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Contributions
Hierarchical model
 Support different routing queries in spatio-temporal network.
 Less storage cost, less computation time.
Hierarchical routing theory in spatio-temporal network
Evaluate model by different spatio-temporal routing queries.
 Shortest path query.
 Best start time query.
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Proposed Solution
Input: spatio-temporal network
snapshots at t=1,2,3,4,5
Node:
A
Edge:
travel time
t=1
t=2
t=3
t=4
t=5
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Proposed Solution
Output: hierarchical graph & pre-computed information
(a) Hierarchical graph overview
(b) two fragments created at base level graph
(c) boundary nodes identified and
pushed to higher level
(d) boundary graph contains only boundary
nodes
Shortest path cost:
[m1,…..,mT]
mi- travel time at t=i
Partitioned
sub network:
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Proposed Solution
Base level graph & pre-computed information
Fragment 1
Shortest path cost:
[m1,…..,mT]
mi- travel time at t=i
Partitioned
sub network:
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Proposed Solution
Base level graph & pre-computed information
Fragment 2
Shortest path cost:
[m1,…..,mT]
mi- travel time at t=i
Partitioned
sub network:
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Proposed Solution
Higher level graph & pre-computed information
Boundary graph
Shortest path cost:
[m1,…..,mT]
mi- travel time at t=i
Partitioned
sub network:
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Agenda
 Motivation
 Problem Statement
 Related Work
 Proposed Solution
 Hierarchical routing theory
15
Intuition of hierarchical model
Hierarchical routing theory:
SP(A,G)= Fragment(i).SP(A,C)+BG.SP(C,E)+Fragment(j).SP(E,G) where
∀ni,nj niϵBN(i)∧nj∈BN(j)∧(SPC(A,C)+SPC(C,E)+SPC(E,G))≤(SPC(A,ni)+SPC(ni, nj)+SPC(nj,G))
----------------------------------------------------------------------------------------------A ϵ Fragment i, G ϵ Fragment j, i ≠ j
C ϵ BN(Fragment i), E ϵ BN(Fragment j)
BN(Fragment i): boundary nodes set of Fragment i
SP(A,G): shortest path from A to G
SPC(A,G): shortest path cost from A to G
BG: boundary graph
-----------------------------------------------------------------------------------------------
Find Shortest path from A to G
SP(A,G)=SP(A,C)+SP(C,E)+SP(E,G)
*Materialization Trade-Offs in Hierarchical Shortest Path Algorithms, S. Shekhar, A. Fetterer, and B. Goyal,
Proc. Intl. Symp. on Large Spatial Databases, Springer Verlag (Lecture Notes in Computer Science), (1997).16
Hierarchical routing theory in Spatio-temporal network
 Shortest path: given a start time, start and end node, travel along the









shortest path has the earliest arrival time
Path cost: given a path with start time, path cost is the arrival time
minus start time
P(p,q,t0): a path from p to q start at time t0
SP(p,q,t0): shortest path from node p to node q start at time t0
PC(p,q,t0): path cost from node p to node q start at time t0
SPC(p,q,t0): shortest path cost from p to q start at time t0
∆t: wait time
G: original graph
BG: boundary graph
BN(Gi): boundary nodes of fragment Gi
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Hierarchical routing theory in Spatio-temporal network
Theorem
1 G.PC(p,q,t0)=G.SPC(p,q,t0)+ ∆t, ∆t is wait time at node q
2 G.SPC(p,q,t0)=BG.SPC(p,q,t0), p, q ∈BG
G.PC(A,C,T2)=4
BG.SPC(C,E,T1)=2
G.SPC(C,E,T1)=2
G.SPC(A,C,T2)=2
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Hierarchical routing theory in Spatio-temporal network
Theorem:
Let P = {G1,G2,…,Gp} be a partition of original graph G, BG be the boundary graph. For
node s ∈ Node set of Gu, node d ∈ Node set of Gv, where 1≤u,v≤p, and u≠v. Start time
is fixed at t1. Then
SP(s,d,t1)=Fragment(Gu).SP(s,ni,t1)+BG.SP(ni,nj,t2)+Fragment(Gv).SP(nj,d,t3)
ni ∈BN(Gu), nj ∈BN(Gv)
t2=t1+SPC(s,ni,t1)+ ∆t1, t3=t2+SPC(ni,nj,t2)+ ∆t2
∆t1 is wait time at ni, ∆t2 is wait time at nj.
Find SP(A,G,t1)
t1
t2
t3
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Shortest path algorithm in spatio-temporal network
Shortest path algorithm in spatio-temporal network
Time expended graph
T1
T2
T3
T4
T5
T6
T7
T8
A
C
E
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Find SP(A,E,T2) :SP(A,E,T2)=SP(A,C,T2)+SP(C,E,T5)
T1
T2
T3
T4
T5
T6
T7
T8
A
C
E
initial
T2
T4
T5
T7
Node ID
Arrival time
Parent node
Parent start time Status
A
A
C
C
T2
T2
T4
Infinite
T4
--A
-A
--T2
-T2
Explored
closed
closed
Toexplore
Explored
closed
E
E
T7
Infinite
T7
C
-C
T5
-T5
explored
Toexplore
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Summary
What we have done
 Hierarchical model
 Hierarchical routing theory in spatio-temporal network
Future work
 Study data structure support the hierarchical model
 Optimize algorithm and storage cost
 Study impact of network update
 Experiments
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References
1. Materialization Trade-Offs in Hierarchical Shortest Path Algorithms, S. Shekhar, A. Fetterer,
and B. Goyal, Proc. Intl. Symp. on Large Spatial Databases, Springer Verlag. 1997
2. Betsy George, Shashi Shekhar, Time Aggregated Graphs for Modeling Spatio-temporal
Network, Journal on Semantics of Data (Editors: J.F. Roddick, S. Spaccapietra), Vol XI,
December, 2007
3. Fast object search on road networks, C.K. Lee, A. Wang-Chien Lee, and Beihua Zheng ,
Proceedings of the 12th International Conference on Extending Database Technology: Advances
in Database Technology. Vol 360. 2009
4. Ugur Demiryurek, Farnoush Banaei-Kashani, and Cyrus Shahabi. Efficient K-Nearest
Neighbor Search in Time-Dependent Spatial Networks. 2010
5. Hierarchical Encoded Path Views for Path Query Processing: An Optimal Model and Its
Performance Evaluation, Ning Jing, Yun-Wu Huang, Elke A. Rundensteiner, IEEE Transactions on
Knowledge and Data Eng., May/June 1998 (Vol. 10, No. 3)
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