1. 1 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

2. 2 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.

3. 3 Outline  Introduction  Motivation  Problem Statement  Related Work  Contributions  Conclusion and Future Work  Representation  Case Studies Routing Algorithms Sensor Data Representation

4. 4 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

5. 5 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

6. 6 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

7. 7 Related Work Graph-based Models Operations ResearchDatabases Spatial GraphsSpatio-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”.

8. 8 Related Work t=1 N2 N1 N3 N4 N5 1 2 2 2 t=2 N2 N1 N3 N4 N5 1 2 2 1 t=3 N2 N1 N3 N4 N5 1 2 2 1 t=4 N2 N1 N3 N4 N5 1 2 2 1 t=5 N2 N1 N3 N4 N5 1 2 2 2 1 N.. Travel time Node: Edge: Time Expanded Graph t=1 N1 N2 N3 N4 N5 t=2 N1 N2 N3 N4 N5 t=3 N1 N2 N3 N4 N5 t=4 N1 N2 N3 N4 N5 N1 N2 N3 N4 N5 t=5 N1 N2 N3 N4 N5 t=6 N1 N2 N3 N4 N5 t=7 Holdover Edge Transfer Edges Snapshots at t=1,2,3,4,5

9. 9 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

10. 10 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.

11. 11 Limitations of Related Work t=t2 A B t=t3 t=t4 t=t1 Relationship between Two Objects at Various Instants disjoint t=t1 touch B AB A... Representation using Pictograms Time Expanded Graphs cannot model this dynamic relationship since it does not involve a flow.

12. 12 Our Contributions GraphTime Expanded Graph (TEG) & Time Aggregated Graph (TAG) LP Solver (flow networks) 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 PathShortest Path (Fixed Start Time) Shortest Path (Best Start Time) Static Networks Time-variant Networks

13. 13 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

14. 14 Time Aggregated Graph t=1 N2 N1 N3 N4 N5 1 2 2 2 t=2 N2 N1 N3 N4 N5 1 2 2 1 t=3 N2 N1 N3 N4 N5 1 2 2 1 t=4 N2 N1 N3 N4 N5 1 2 2 1 t=5 N2 N1 N3 N4 N5 1 2 2 2 1 N.. 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 N4N5 [m 1,…..,(m T ] m i - travel time at t=i Edge N.. Node  Attributes are aggregated over edges and nodes.

15. 15 Time Aggregated Graph N : Set of nodesE : Set of edgesT : Length of time interval nw i : Time dependent attribute on nodes for time instant i. ew i : 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, [nw 1 …nw T ], [ew 1,..,ew T ] | nw i : N  R T, ew i : E  R T N1 [ ,1,1,1,1] [2,2,2,2,2] [1,1,1,1,1] [2,2,2,2,2] [2, , , ,2] N2 N3 N4N5

16. 16 Case Study -Routing Algorithms t=1 N2 N1 N3 N4 N5 1 2 2 2 t=2 N2 N1 N3 N4 N5 1 2 2 1 t=3 N2 N1 N3 N4 N5 1 2 2 1 t=4 N2 N1 N3 N4 N5 1 2 2 1 t=5 N2 N1 N3 N4 N5 1 2 2 2 1 N.. 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..

17. 17 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 N4N5 N1 For start time t=1 N1- N2- N4- N5 t=1t=2 t=3 wait till t=5 !! t=7 (1) N1-N3-N4 -N5 has non-optimal prefix N1-N3-N4 -N5N1-N2-N4-N5 & are optimal (6 units). N1- N3- N4- N5 t=1 t=3t=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.

18. 18 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.

19. 19 Shortest Path Algorithm for Given Start Time Challenges Assume FIFO travel times. (2) Correctness : Determining when to traverse an edge. N1 1,1,1,1 N2N3 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.

20. 20 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 N4N5 Source: N1; Destination: N5; time: t=1; 1 ∞ ∞ ∞ ∞ 1 3 3 ∞ ∞ 1 3 34 ∞ 1 3 34 ∞ 1 3 34 7 N1 N2N3 N4 N5 (1) (∞) (3) (4) (7)

21. 21 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.

22. 22 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

23. 23 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))

24. 24 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.

25. 25 Best Start Time Shortest Path Algorithm  Finds a start time and a path such that the time spent in the network is minimized. t=1 N2 N1 N3 N4 N5 1 2 2 2 t=2 N2 N1 N3 N4 N5 1 2 2 1 t=3 N2 N1 N3 N4 N5 1 2 2 1 t=4 N2 N1 N3 N4 N5 1 2 2 1 t=5 N2 N1 N3 N4 N5 1 2 2 2 1 N.. Travel time Node: Edge: Start Time: Path : N1 – N2 – N4 – N5 Arrival Time: Time Spent: 1234 5 77 7 89 65 444 A Best Start Time!!

26. 26 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.

27. 27 Best Start Time Shortest Path Label Correcting Vs. Label Setting Algorithms Label Setting Algorithms Label Correcting Algorithms Node expanded Least cost nodeRandom TerminationDestination expandedNo cost updates # ExpansionsOnceMany ComplexityO(n log m)O(n 2 m) * (*) 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.

28. 28 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 N4N5 N5 is selected; Iteration 1: t=1: C N4 (1) > (  N4N5 (1) + C N5 (1+  N4N5 (1))) ∞ > (4 + C N1 (1+4)) C u (t) = min(C u (t),  uv (t) + C v (t+  uv (t) ) (,(, ,, ,, , , )) [4,4,3,3,3]

29. 29 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

30. 30 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 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

31. 31 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 data Mn/DOT basemap for MPLS CBD.

32. 32 Comparison of Storage Cost 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

33. 33 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

34. 34 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.

35. 35 Comparison of Algorithm Complexity For a network of n nodes and m edges and a time interval of length T AlgorithmTime Expanded GraphTime Aggregated Graph Best Start Time Shortest Path O(nT 2 +mT)T) (*) O(n 2 mT) (**) Fixed Start Time Shortest 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.

36. 36 Case Study - Sensor Data Representation Snapshots of a Network at t=1,2,3 Time Aggregated Graph N2 N1 N3 N4 [(1,0.5),(1,0.5), ∞ ] [(1,0.5), ∞,(2,0.5)] [(2,0.6), (2,0.6), (2,0.6)] (Measured Value, Error) Edge Node N.. t = 1 (1,0.5) (2,0.6) (1,0.5) (2,0.6) N1 N2 N3 N4 (1,0.5) (2,0.6) N1 N2 N3 N4 t = 2 (2,0.6) (2,0.5) (2,0.6) N1 N2 N3 N4 t = 3 [(m 1,e 1 )…..,(m T,e T )] m i - measured value at t=i e i - error at t=i Edge N.. Node

37. 37 Case Studies on Sensor TAG (*)  Basic Hotspot Detection  Anomaly Detection  Sensors that display measured values different from expected values are identified.  Sensor nodes that show anomalies (hotspots) are detected through a search of the Sensor TAG  Growing Hotspot Detection  Increase in the number of sensors that report an anomaly is predicted based on the physical attributes modeled on the edges of Sensor TAG. * B. George, J. Kang, S. Shekhar, STSG: A Data Model for Representation of Sand Knowledge Discovery in Sensor Data, Proceedings of ACM SIGKDD Workshop on Knowledge Discovery in Sensor Data, August, 2007.

38. 38 Representation of a Dynamic Relationship A B t=t2 t=t3 t=t4 t=t1 Relationship between Two Objects at Various Instants t=t9 [d,t,o,c,c,v,o,t,d] B A d – disjoint t – touch o – overlap c – contains v – covers Dynamic Relationship expressed in TAG

39. 39 Conclusion GraphTime Expanded Graph (TEG) & Time Aggregated Graph (TAG) LP Solver (flow networks) 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 PathShortest 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)

40. 40 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  Transportation Network Routing Algorithms  Sensor Data Representation

41. 41 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.

42. 42 Future Work  BEST-TAG Algorithm  Performance Tuning  Current Complexity – O(n 2 mT)  Real datasets  Heuristics  Proof of Optimality (all cases)

43. 43 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

44. 44 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

45. 45 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 node Find : Shortest Path from the source to destination Objective: Minimize Computational cost.  Problem Statement For each node v, (degee (u) +1)T costs are maintained. uv Travel time series w1 w2 u v w1 w2 tc(u,v,w1) tc(u,v,w2)

46. 46 Future Work - Algorithms a ib b’ b” c’ Turn restriction – Node Expansion

47. 47 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

48. 48 Future Work Operator SnapshotTime-Aggregate getget(node,time)getNodeSeries(node) getEdgegetEdge(node1,node2,time)getEdgeSeries(node1,node2) get_node_PresencegetNodePresence(node,time)getNodeSeries(node) get_edge_PresencegetEdgePresence(node1,node2,time)getEdgeSeries(node1,node2) get_Graphget_Graph(time)get_Graph() Logical Model A Sample Set of Logical Operators

49. 49 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.

50. 50 Thank you. Questions and Comments ?