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Time-Aggregated Graphs-Modeling Spatio-temporal Networks
September 7, 2007
Betsy George
Department of Computer Science and Engineering University of Minnesota
Advisor : Prof. Shashi Shekhar
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Publications
Time Aggregated Graphs B. George, S. Shekhar, Time Aggregated Graphs for Modeling Spatio-temporal
Networks-An Extended Abstract, Proceedings of Workshops (CoMoGIS) at International Conference on Conceptual Modeling, (ER2006) 2006. (Best Paper Award)
B. George, S. Kim, S. Shekhar, Spatio-temporal Network Databases and Routing Algorithms: A Summary of Results, Proceedings of International Symposium on Spatial and Temporal Databases (SSTD07), July, 2007.
B. George, J. Kang, S. Shekhar, STSG: A Data Model for Representation and Knowledge Discovery in Sensor Data, Proceedings of Workshop on Knowledge Discovery from Sensor data at the International Conference on Knowledge Discovery and Data Mining (KDD) Conference, August 2007. (Best Paper Award).
B. George, S. Shekhar, Modeling Spatio-temporal Network Computations: A Summary of Results, Accepted for presentation at the Second International Conference on GeoSpatial Semantics (GeoS2007), 2007.
B. George, S. Shekhar, Time Aggregated Graphs for Modeling Spatio-temporal Networks, Journal on Semantics of Data (In second review) , Special issue of Selected papers from ER 2006.
Evacuation Planning Q Lu, B. George, S. Shekhar, Capacity Constrained Routing Algorithms for
Evacuation Planning: A Summary of Results, Proceedings of International Symposium on Spatial and Temporal Databases (SSTD05), August, 2005.
S. Kim, B. George, S. Shekhar, Evacuation Route Planning: Scalable Algorithms, Accepted for presentation at ACM International Symposium on Advances in Geographic Information Systems (ACMGIS07), November, 2007.
Q Lu, B. George, S. Shekhar, Capacity Constrained Routing Algorithms for Evacuation Planning, International Journal of Semantic Computing, Volume 1, No. 2, June 2007.
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Outline
Introduction Motivation Problem Statement Related Work
Contributions
Conclusion and Future Work
Representation Case Studies
Routing Algorithms
Sensor Data Representation
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Motivation
accurate computation of frequent routing queries.
Varying Congestion Levels and turn restrictions travel time changes.
Examples: Transportation network Routing, Crime pattern analysis, knowledge discovery from Sensor data.
Many Applications…
I94 @ Hamline Ave at 8AM & 10AM
Traffic sensors on Twin-Cities, MN Road Network monitor traffic levels/travel time on the road network. (Courtesy: MN-DoT (www.dot.state.mn.us) )
Identification of frequent routes
Crime Analysis
Identification of congested routes
Network Planning
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Problem Definition
Input : a) A Spatial Network b) Temporal changes of the network topology and parameters.
Objective : Minimize storage and computation costs.
Output : A model that supports efficient correct algorithms for computing the query results.
Constraints : (i) Changes occur at discrete instants of time, (ii) Logical & Physical independence
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Challenges
Conflicting Requirements
Expressive Power
Storage Efficiency
New and alternative semantics for common graph operations.
Ex., Shortest Paths are time dependent.
Key assumptions violated.
Ex., Prefix optimality of shortest paths
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Related Work
Graph-based Models
Operations Research
Databases
Spatial Graphs Spatio-temporal Graphs(Time Aggregated
Graphs)
Flow networks ( Time Expanded Graphs)
Spatial Graphs [Erwig’94, Guting’96, Mouratidis’06, Shekhar’97] Does not model temporal variations in the network topology, parameters Supports operations such as shortest path computation on static graphs Maintains connectivity of link-node networks
Flow Networks (Time expanded Graphs)[Ford’58, Kaufman’93, Kohler’02,Dean’04]
Models time-dependent flow networks Maintains a copy of the graph for each time instant. Cannot model scenarios where edge parameter does not represent a
“flow”.
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Related Work
t=1
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1N..
Travel time
Node:
Edge:
Time Expanded Graph
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Holdover Edge
Transfer Edges
Snapshots at t=1,2,3,4,5
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Related Work
Shortest Paths in Time Expanded Graphs
LP solvers (NETFLO, RELAX IV) provide support for Shortest Path Computation.
Models the time-expanded graph as an Uncapacitated flow network.
E : set of edges in the TEG
C(e) : Edge Cost
x(e) =1 if edge e is taken
= 0, otherwise
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Limitations of Related Work
High Storage Overhead Redundancy of nodes across time-frames Additional edges across time frames.
Inadequate support for modeling non-flow parameters and uncertainty on edges.
Time Expanded Graph
Lack of physical independence of data.
Computationally expensive Algorithms Increased Network size.
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Our Contributions
Graph Time Expanded Graph (TEG) & Time Aggregated Graph (TAG)
LP Solver (flow networks)
Flow algorithms based on LP
Flow algorithms based on LP
Label Correcting Algorithms
Two-Q Algorithm,..
BEST-TAG Algorithm
Label Setting Algorithms
Dijkstra’s Algorithm,..
SP-TAG Algorithm
Lack of optimal prefix
Shortest Path Shortest Path (Fixed Start
Time)
Shortest Path (Best Start Time)
Static Networks
Time-variant Networks
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Our Contributions
Time Aggregated Graph (TAG)
Shortest Path for the ‘best’ start time
Shortest Path for a given start time
Analytical & Experimental Evaluation
Representation Case Studies
Routing Algorithms
Sensor Data Representation
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Time Aggregated Graph
t=1
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t=5
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1N..
Travel time
Node:
Edge:
Snapshots of a Network at t=1,2,3,4,5
Time Aggregated Graph
N1
[,1,1,1,1]
[2,2,2,2,2]
[1,1,1,1,1]
[2,2,2,2,2]
[2,, , ,2]
N2
N3
N4 N5
[m1,…..,(mT]
mi- travel time at t=i
Edge
N..
Node
Attributes are aggregated over edges and nodes.
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Time Aggregated Graph
N : Set of nodes E : Set of edges T : Length of time interval
nwi: Time dependent attribute on nodes for time instant i.
ewi: Time dependent attribute on edges for time instant i.
On edge N4-N5
* [2,∞,∞,∞,2] is a time series of attribute;
* At t=2, the ‘∞’ can indicate the absence of connectivity between the nodes at t=2.
* At t=1, the edge has an attribute value of 2.
TAG = (N,E,T, [nw1…nwT ],
[ew1,..,ewT ] |nwi : N RT, ewi : E RT
N1
[,1,1,1,1]
[2,2,2,2,2]
[1,1,1,1,1]
[2,2,2,2,2]
[2,, , ,2]
N2
N3
N4 N5
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Case Study -Routing Algorithms
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Travel time
Node:
Edge:
Start at t=1:Shortest Path is N1-N3-N4-N5;
Travel time is 6 units.
Start at t=3:Shortest Path is N1-N2-N4-N5;
Travel time is 4 units.
Shortest Path is dependent on start time!!
Fixed Start Time Shortest Path Least Travel Time (Best Start Time)
Finding the shortest path from N1 to N5..
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Shortest Path Algorithm for Given Start Time
Challenges(1) Not all shortest paths
show optimal substructure.
[1,1,1,1,1]
[2,2,2,2,2]
[1,1,1,1,1]
[2,2,2,2,2]
[2,, , ,2]
N2
N3
N4 N5N1For start time t=1N1- N2- N4- N5t=1
t=2
t=3
wait till t=5 !!
t=7
(1)
N1-N3-N4 -N5 has non-optimal prefix
N1-N3-N4 -N5 N1-N2-N4 -N5 & are optimal (6 units).
N1- N3- N4- N5t=1
t=3
t=5
t=7
(2)
Lemma: At least one optimal path satisfies the optimal substructure property. N1-N2-N4-N5 in the example has optimal prefixes.
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Shortest Path Algorithm for Given Start Time
Challenge-1
Lemma: At least one optimal path satisfies the optimal substructure property.
Proof:
• For a given start time, the non-optimal substructure is due to waits at intermediate nodes.
• For the path from ‘s’ to ‘d’, let ‘u’ be an intermediate, wait node. • Append the optimal path from ‘s’ to ‘u’ to the path from ‘u’ to ‘d’ allowing wait at ‘u’.
• This path is optimal. (by Contradiction)
(1) Not all shortest paths show optimal substructure.
Greedy algorithm can be used to find the shortest path.
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Shortest Path Algorithm for Given Start Time
Challenges
Assume FIFO travel times.
(2) Correctness : Determining when to traverse an edge.
N11,1,1,1 N2 N3
1,3,1,2
When to traverse the edge N2-N3 for start time t=1 at N1?Traversing N2-N3 as soon as N2 is reached, would give sub optimal solution.
(3) Termination of the algorithm : An infinite non-negative cycle over time
Finite time windows are assumed.
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Shortest Path Algorithm for Given Start Time
Algorithm
Every node has a cost ( arrival time at the node). Greedy strategy:
Select the node with the lowest cost to expand.
Traverse every edge at the earliest available time.
N1
[,1,1,1,1]
[2,2,2,2,2]
[1,1,1,1,1]
[2,2,2,2,2]
[2,, , ,2]
N2
N3
N4 N5
Source: N1; Destination: N5; time: t=1;
1 ∞ ∞ ∞ ∞
1 3 3 ∞ ∞
1 3 3 4 ∞
1 3 3 4 ∞
1 3 3 4 7
N1 N2 N3 N4 N5
(1)
(∞)
(∞)
(∞) (∞)
(3)
(3)
(4) (7)
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Shortest Path Algorithm for Given Start Time
Initialize c[s] = 0; v ( s), c[v] = ∞. Insert s in the priority queue Q. while Q is not empty do u = extract_min(Q); close u (C = C {u}) for each node v adjacent to u do { t = min_t((u,v), c[u]); // min_t finds the earliest departure time for (u,v) If t + u,v(t) < c[v] c[v] = t + u,v(t) parent[v] = u insert v in Q if it is not in Q; } Update Q.
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Shortest Path Algorithm for Given Start Time
Correctness of the Algorithm (Optimality of the result)
The SP-TAG is correct under the assumption of FIFO travel times and finite time windows.
Lack of optimal substructure of some shortest paths is due to a potential wait at an intermediate node. Algorithm picks the path that shows optimal substructure and allows waits.
Lemma: When a node is closed, the cost associated with the node is the shortest path cost.
Based on proof for Dijkstra’s algorithm. Difference - Earliest availability of edge
- Admissible guarantees optimality
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Analytical Evaluation
* B. C. Dean, Algorithms for Minimum Cost Paths in Time-dependent Networks, Networks 44(1), August 2004.
Computational Complexity
Dijkstra’s Cost Model extended to include the dynamic nature of edge presence.
Each edge traversal Binary search to find the earliest departure O(log T )
Complexity of shortest path algorithm is O(m( log T+ log n))
[n: Number of nodes, m – Number of edges, T – length of the time series]
For every node extracted, Earliest edge lookup – O(log T) Priority queue update – O(log n) Overall Complexity = O(degree(v). (log T + log n)) = O(m( log T+ log n))
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Analytical Evaluation
Complexity of Shortest Path algorithm based on TAG is O(m( log T+ log n))
Complexity of Shortest Path Algorithm based on Time Expanded Graph is O(nT log T+mT) (*)
Lemma : Time-aggregated graph performs asymptotically better than time expanded graphs when log (n) < T log (T).
* B. C. Dean, Algorithms for Minimum Cost Paths in Time-dependent Networks, Networks 44(1), August 2004.
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Best Start Time Shortest Path Algorithm
Finds a start time and a path such that the time spent in the network is minimized.
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1N..
Travel time
Node:
Edge:
Start Time:
Path : N1 – N2 – N4 – N5
Arrival Time:
Time Spent:
1 2 3 4 5
7 7 7 8 9
6 5 4 4 4
A Best Start Time!!
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Best Start Time Shortest Path Algorithm
Challenges(1) Best Start Time shortest paths
need not have optimal prefixes.
N1[1,2,2,2,2,2] [2,∞, ∞,
∞,2,2]N2 N3
Optimal solution for the shortest path from N1 to N3 is suboptimal for N1 to N2 due to the wait at N2.
(2) Correctness: Lack of FIFO property.
Use Label-correcting approach instead Greedy methods.
Use node cost series instead of a scalar node cost.
(3) Termination of the algorithm : An infinite non-negative cycle over time
Finite time windows are assumed.
Costs assumed constant after T.
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Best Start Time Shortest Path
Label Correcting Vs. Label Setting Algorithms
Label Setting Algorithms
Label Correcting Algorithms
Node expanded
Least cost node Random
Termination Destination expanded No cost updates
# Expansions Once Many
Complexity O(n log m) O(n2m) *
(*) Two-Q Algorithm
Data Structure used – Pair of queues Q1, Q2 Q1 – Set of nodes scanned (expanded) before (repeated expansion)
Q2 – Set of nodes not scanned before (first expansion) Nodes from Q1 are given preference
* S. Pallottino, Shortest Path Methods: Complexity, Interrelations and New Propositions, Networks, 14:257-267, 1984.
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Best Start Time Shortest Path Algorithm
Algorithm: Each node has a cost series.
Node to be expanded is selected at random. Every entry in the cost series of ‘adjacent’ nodes are updated (if there is an improvement in the existing cost).
N1
[0,0,0,0,0]
N2
N3
N4 N5
N5 is selected;
Iteration 1: t=1:CN4(1) > (N4N5(1) + CN5(1+ N4N5(1)))
∞ > (4 + CN1(1+4))
Cu(t) = min(Cu(t), uv(t) + Cv(t+ uv(t) ) (,, , ,)
[4,4,3,3,3]
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Best Start Time Shortest Path Algorithm
Key Ideas
Label correcting Algorithm for every time instant
Handles non-FIFO travel times
Finds the minimum travel time from all shortest paths
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Performance Evaluation: Experiment Design
Network Expansion
TAG Based Algorithms Shortest Path Algorithms on Time
Expanded Graph
Data Analysis
Length of Time Series
Real Dataset (without time
series) Road network with travel time series
Run-time Run-time
Time Series Generation
Time expanded network
Goals
1. Compare TAG based algorithms with algorithms based on time expanded graphs (e.g. NETFLO):
- Performance: Run-time
2. Test effect of independent parameters on performance: - Number of nodes, Length of time series
Experiment Platform: CPU: 1.77GHz, RAM: 1GB, OS: UNIX.
Experimental Setup
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Performance Evaluation: Dataset
Minneapolis CBD [1/2, 1, 2, 3 miles radii]
Dataset # Nodes # Edges
1.(MPLS -1/2)
111 287
2. (MPLS -1 mi)
277 674
3.(MPLS - 2
mi)
562 1443
4.(MPLS - 3
mi)
786 2106
Road dataMn/DOT basemap for MPLS CBD.
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Comparison of Storage Cost
Memory(Length of time series=150)
100
1100
2100
3100
4100
5100
111 277 562 786
No: of nodes
Sto
rag
e u
nit
s (K
B)
TAG
TEXP
For a TAG of n nodes, m edges and time interval of length T, If there are k edge time series in the TAG , storage required for time
series is O(kT). (*) Storage requirement for TAG is O(n+m+kT)
(**) D. Sawitski, Implicit Maximization of Flows over Time, Technical Report (R:01276),University of Dortmund, 2004.
(*) All edge and node parameters might not display time-dependence.
For a Time Expanded Graph,
Storage requirement is O(nT) + O(n+m)T (**)
Experimental Evaluation
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Performance Evaluation :Experiment Results 1
Experiment 1: Effect of Number of Nodes
Setup: Fixed length of time series = 100
• TAG based algorithms are faster than time-expanded graph based algorithms.
Shortest Path – Given Start Time Shortest Path – Best Start Time
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Performance Evaluation : Experiment Results 2
Experiment 2: Effect of Length of time series.
Setup: fixed number of nodes = 786, number of edges = 2106.
Shortest Path – Given Start Time Shortest Path – Best Start Time
• TAG based algorithms run faster than time-expanded graph based algorithms.
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Comparison of Algorithm Complexity
For a network of n nodes and m edges and a time interval of length T
Algorithm Time Expanded Graph
Time Aggregated Graph
Best Start TimeShortest Path
O(nT2+mT)T) (*) O(n2mT)(**)
Fixed Start TimeShortest Path
O(nT log T+mT) (*) O(m log T+log n) (**)
(*) B.C. Dean, Algorithms for Minimum Cost Paths in Time-Dependent Networks, Networks, 44(1) pages (41-46), 2004.
(**) B. George, S. Kim, S. Shekhar, Spatio-temporal Network Databases and Routing Algorithms: A Summary of Results, Proceedings of International Symposium on Spatial and Temporal Databases (SSTD’07), July 2007.
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Conclusion
Graph Time Expanded Graph (TEG) & Time Aggregated Graph (TAG)
LP Solver (flow networks)
Flow algorithms based on LP
Flow algorithms based on LP
Label Correcting Algorithms
Two-Q Algorithm,..
BEST-TAG Algorithm
Label Setting Algorithms
Dijkstra’s Algorithm,..
SP-TAG Algorithm
Lack of optimal prefix
Shortest Path Shortest Path (Fixed Start
Time)
Shortest Path (Best Start Time)
Static Networks
Time-variant Networks
Key Insights Fixed Start time shortest paths – Greedy strategy gives optimal solutions. Flexible Start time – Greedy strategy need not give optimal solution.
(Label correcting method)
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Conclusions
Time Aggregated Graph (TAG) Time series representation of edge/node properties Non-redundant representation Often less storage, less computation time
Evaluation of the Model using Case Studies
Shortest Path for Fixed Start Time
Shortest Path for Fixed Start Time
Transportation Network Routing Algorithms
Sensor Data Representation
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Future Work
Algorithms Performance Tuning of Best Start Time Algorithm Incorporate capacities on nodes/edges and
develop optimal algorithms for Evacuation Planning.
Incorporate time-dependent turn restrictions in shortest path computation.
Develop ‘frequent route discovery’ algorithms based on TAG framework.
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Future Work
BEST-TAG Algorithm
Performance Tuning
Current Complexity – O(n2mT) Real datasets Heuristics
Proof of Optimality (all cases)
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Future Work - Algorithms
Evacuation Planning
Given: (i) A transportation network, a directed graph G = ( N, E ) (ii) Capacity constraints for each edge and node, (iii) Time-dependent travel time for each edge, (iv) Number of evacuees and source nodes (v) Evacuation destinations.
Find : Evacuation plan consisting of a set of origin-destination routes & scheduling of evacuees on each route.
Objective: Minimize evacuation egress time, Computational cost. Optimize evacuation time subject to time-dependent travel times & Capacity constraints.
Problem Statement
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Future Work - Algorithms
Frequent route discovery Algorithm Motivation: Crime Analysis Effective patrolling
Routes are time-dependent
Time-dependent schedule of Public transportation
Route discovery on Spatio-temporal networks (Journey-to-crime)*
Explore TAG as a model for Spatio-temporal network data Spatio-temporal data mining.
Crime data is Spatio-temporal
* CrimeStat 3.0, Ned Levine & Associates
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Future Work - Algorithms
Shortest Path with time-dependent turn restrictions
Given: (i) A transportation network, a directed graph G = ( N, E ) (ii) Time-dependent travel time for each edge, (iii) Time-dependent turn costs (iv) Source node, Destination nodeFind : Shortest Path from the source to destination
Objective: Minimize Computational cost.
Problem Statement
For each node v, (degee (u) +1)T costs are maintained.
u v
Travel time series
w1w2
u v
w1
w2
tc(u,v,w1)
tc(u,v,w2)
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Future Work
Spatio-temporal Network Databases
Conceptual level Extend Pictogram-enhanced ER model.
Logical level Formulate a complete set of logical operators
Physical level Add spatial properties to nodes, edges. Design indexing methods for time-aggregated graph. Explore the possibility of infinite time windows. Formulate new algorithms.
Persistence Shared Interrelated
Three-Schema Architecture
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References
ESRI, ArcGIS Network Analyst, 2006. Oracle, Oracle Spatial 10g, August 2005. M. Erwig, R.H. Guting, Explicit Graphs in a Functional Model for Spatial
Databases, IEEE Transactions on Knowledge and Data Engineering, 6(5), 1994.
S. Shekhar, D. Liu, Connectivity Clustered Access Method for Networks and Network Analysis, IEEE Transactions on Knowledge and Data Engineering, January, 1997.
L.R. Ford, D.R. Fulkerson, Constructing maximal dynamic flows from static flows, Operations Research, 6:419-433, 1958.
E. Kohler, K. Langtau and M. Skutella, Time expanded graphs for time-dependent travel times, Proc. 10th Annual European Symposium on Algorithms, 2002.
D.E. Kaufman, R.L. Smith, Fastest Path in Time-dependent Networks for Intelligent Vehicle Highway Systems Applications, IVHS Journal, 1(1), 1993.
K. Mouratidis, M. Yiu, D. Papadias, N. Mamoulis. Continuous Nearest Neighbor Monitoring in Road Networks. Proceedings of the Very Large Data Bases Conference (VLDB), pp. 43-54, Seoul, Korea, Sept. 12 - Sept. 15, 2006.
B.C. Dean, Algorithms for Minimum Cost Paths in Time-Dependent Networks, Networks, 44(1) pages (41-46), 2004.
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Thank you.
Questions and Comments ?