1. 1 Time-Aggregated Graphs- Modeling Spatio-temporal Networks August 29, 2008 Department of Computer Science and Engineering University of Minnesota Prof. Shashi Shekhar

2. 2 Selected 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, Proceedings of 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, Volume XI, Special issue of Selected papers from ER 2006, December 2007. B. George, J. Kang, S. Shekhar, STSG: A Data Model for Representation and Knowledge Discovery in Sensor Data, Accepted for publication in Journal of Intelligent Data Analysis. B. George, S. Shekhar, Routing Algorithms in Non-stationary Transportation Network, Proceedings of International Workshop on Computational Transportation Science, Dublin, Ireland, July, 2008. B. George, S. Shekhar, S. Kim, Routing Algorithms in Spatio-temporal Databases, Transactions on Data and Knowledge Engineering (In submission). 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, Proceedings of 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  Routing Algorithms

4. 4 Motivation  Delays at signals, turns, Varying Congestion Levels  travel time changes. 1) Transportation network Routing U.P.S. Embraces High-Tech Delivery Methods (July 12, 2007) By Claudia H. Deutsch “The research at U.P.S. is paying off. ……..— saving roughly three million gallons of fuel in good part by mapping routes that minimize left turns.”  Identification of frequent routes (i.e.) Journey to Crime 2) Crime Analysis 3) Knowledge discovery from Sensor data.  Spreading Hotspots 9 PM, November 19, 2007 4 PM, November 19, 2007 Sensors on Minneapolis Highway Network periodically report time varying traffic 7 PM, November 19, 2007

5. 5 Motivation Signal delays at left turns can cause non-FIFO travel times. Non-FIFO Travel times:  Arrivals at destination are not ordered by the start times.  Can occur due to delays at left turns, multiple lane traffic.. Different congestion levels in different lanes can lead to non-FIFO travel times. Pictures Courtesy: http://safety.transportation.org

6. 6 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) Predictable future (ii) Changes occur at discrete instants of time, (iii) Logical & Physical independence,

7. 7 Problem Definition (contd.)  Predictable Future  Values of edge attributes largely predictable  Operational scenarios – reasonable in the absence of random events (ex., public transportation scheduling)  Assumption not unreasonable in planning scenarios

8. 8 Challenges in Representation  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 (greedy property behind Dijkstra’s algorithm..)

9. 9 Related Work in Representation 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: (2) Time Expanded Graph (TEG) 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 (1) Snapshot Model [Guting04] [Kohler02, Ford65]

10. 10 Limitations of Related Work  High Storage Overhead  Redundancy of nodes across time-frames  Additional edges across time frames in TEG.  Inadequate support for modeling non-flow parameters on edges in TEG.  Lack of physical independence of data in TEG.  Computationally expensive Algorithms  Increased Network size due to redundancy.

11. 11 Outline  Introduction  Motivation  Problem Statement  Related Work  Contributions  Conclusion and Future Work  Representation  Case Studies Routing Algorithms Time Aggregated Graph (TAG)

12. 12 Proposed Approach 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.

13. 13 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

14. 14 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.

15. 15 TAG: Storage Cost Comparison For a TAG of n nodes, m edges and time interval 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 Storage cost of TAG is less than that of TEG if k << m. TAG can benefit from time series compression.

16. 16 Outline  Introduction  Motivation  Problem Statement  Related Work  Contributions  Conclusion and Future Work  Representation  Routing Algorithms Time Aggregated Graph (TAG)

17. 17 Routing Algorithms- Challenges  Violation of optimal prefix property  New and Alternate semantics  Termination of the algorithm: an infinite non-negative cycle over time  Not all optimal paths show optimal prefix property.

18. 18 Routing Algorithms- Challenges t=1 N2 N1 N3 N4 N5 1 1 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=5 N2 N1 N3 N4 N5 1 1 2 2 1 1 2 5 t=4 N2 N1 N3 N4 N5 1 2 2 1 2 N1 1 ∞ 2 1 3 3 3 N2 N5 N3N4 1 1 2 2 ∞ ∞∞ 3 ∞ ∞ ∞ 4 3 1 2 3 ∞ 5 3 1 2 3 8 Solution: Reaches N5 at t=8. Total time = 7 Optimal path: Reach N4 at t=3; Wait for t=4; Reach N5 at t=6 Total time = 5 Find the shortest path travel time from N1 to N5 for start time t = 1.

19. 19 Routing Algorithms – Related Work Limitations: SP-TAG, SP-TAG*,CapeCod Label correcting algorithm over long time periods and large networks is computationally expensive. Predictable Future Unpredictable Future Stationary Non-stationary Dijkstra’s, A*…. General Case Special case (FIFO) LP, Label-correcting Alg. on TEG [Orda91, Kohler02, Pallotino98] [Kanoulas07] LP algorithms are costly.

20. 20 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.

21. 21 Shortest Path Algorithm for Given Start Time Challenges FIFO travel times  Greedy algorithms, A* search (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. Non-FIFO travel times  ATST transformation  Greedy Algorithm

22. 22 Greedy Algorithm (SP-TAG) for FIFO  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) SP-TAG Algorithm for Given Start Time

23. 23 SP-TAG 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.

24. 24 SP-TAG Algorithm for a 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  Computational Complexity [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))

25. 25 Analytical Evaluation * B. C. Dean, Algorithms for Minimum Cost Paths in Time-dependent Networks, Networks 44(1), August 2004.  Computational Complexity  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))

26. 26 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.

27. 27 SP-TAG* (A* based Algorithm) for FIFO Cost function f(n) = g(n) + h(n) g(n) : Actual cost from the source node to node n h(n) : Estimated cost node n to the destination node n Heuristic function h(n) h(n) = Shortest path travel time from ‘n’ to ‘d’ based on least travel time on each edge (Min t=1,2,…,T d ij (t),  ij  E) Lemma 1: Heuristic function h(n) is admissible.  A* search will result in an optimal solution. Lemma 2: Heuristic function h(n) is monotone.  Search is optimal  Closed nodes are not reopened. Actual Cost g(n) : Arrival time at node n

28. 28 SP-TAG* Lemma: Heuristic function h(n) is admissible. Proof: S_TAG : Static network derived from TAG with minimum travel time on each edge. Let P be the shortest path from node i to the destination d. Shortest path travel time SP min =  d pq pq  P min Let P*(t) be the shortest path in TAG that starts at i at time t. P*(t) is a feasible path in S_TAG. SP min =  d pq   d kl and d kl  d kl (t) min pq  P kl  P* min SP min   d kl   d kl(t) = SP(t) min kl  P*

29. 29 SP-TAG* Lemma: Heuristic function h(n) is monotone. Proof: A heuristic is monotone if h(i)  d ij + h(j),  ij  E Since d ij  d ij (t), min Here, SP i-d  d ij (t) + SP j-d must be satisfied. min SP i-d  d ij + SP j-d, else it is a contradiction to the optimality of SP i-d min SP i-d  d ij (t) + SP j-d min

30. 30 Preprocess to find the Initialize A[s] = t start ;  u (  s), A[u] = ∞, f[u] = ∞. f[s] = SP[s]; C =  ; S = {s};. while u  d do u = extract_min(Q); close u (C = C  {u}); S = S – {u}; for each node v adjacent to u do { if f[v] > A[u] + d uv (A[u]) + SP vd (A[u]) A[v] = A[i] + d u,v (A[u]) f[v] = A[u] + d uv (A[u]) + SP vd (A[u]) S = S  {v} if it is not in Q; } SP-TAG * SP i-d for every node i  d min

31. 31 SP-TAG* – Execution Trace To find the shortest path from N1 to N5 for start time t = 1: Heuristic h[ ]= SP min [N1,N2,N3,N4] = [4,3,4,2] f(N1) = g(N1) + h(N1) = 1 + 4 = 5; f[ ] = ∞ N1 is expanded. f(N2) = g(N2) + h(N2) = 3 + 3 = 6 f(N3) = g(N3) + h(N3) = 3 + 4 = 7 N2 is expanded. f(N4) = g(N4) + h(N4) = 4 + 2 = 6 N4 is expanded. f(N5) = g(N5) + h(N5) = 7 + 0 = 7 N1 1 2 1 2 2 N2 N3 N4N5 S_TAG N1 [2,1,1,1,1] [2,2,2,2,2] [1,1,1,1,1] [2,2,2,2,2] [2,3,3,3,2] N2 N3 N4N5 TAG

32. 32 Our Contributions Time Aggregated Graph (TAG)  Shortest Path for a given start time  Analytical & Experimental Evaluation  Representation  Routing Algorithms in general (FIFO & non-FIFO) Networks

33. 33 Related Work – Label Correcting Approach(*) t=1 t=2 t=3t=4 t=5 t=6t=7 N1 N2 N3 N4 N5 t=8 Start time = 1; Start node : N1 Iteration 1: N1_1 selected N1_2 = 2; N2_2 = 2; N3_3 = 3  Selection of node to expand is random. Iteration 2: N2_2 selected N2_3 = 3; N4_3 = 3 Iteration 3: N3_3 selected N3_4 = 4; N4_5 = 5 Iteration.. : N4_3 selected N4_4 = 4; N5_8 = 8... Iteration.. : N4_4 selected N4_5 = 5; N5_6 = 6  Algorithm terminates when no node gets updated. (*) Cherkassky 93,Zhan01, Ziliaskopoulos97  Implementation used the Two-Q version [O(n 2 T 3 (n+m)]

34. 34 Proposed Approach – Key Idea Arrival Time Series Transformation (ATST) the network: N2 N1 N3 N4 N5 [1,1,1,1,1] [2,2,2,2,2] [1,2,5,2,2] N2 N1 N3 N4 N5 [2,3,4,5,6] [3,4,5,6,7] [2,3,4,5,6] [2,4,8,6,7] [3,4,5,6,7] travel times  arrival times at end node  Min. arrival time series Greedy strategy (on cost of node, earliest arrival) works!! N2 N1 N3 N4 N5 [2,3,4,5,6] [3,4,5,6,7] [2,3,4,5,6] [2,4,6,6,7] [3,4,5,6,7] Result is a Stationary TAG. When start time is fixed, earliest arrival  least travel time (Shortest path)

35. 35 SP Algorithm in Non-FIFO Networks (NF-SP-TAG) Greedy strategy on transformed TAG: Cost of a node = Arrival time at the node Expand the node with least cost. Update costs of adjacent nodes. Select Minimum {Cost of edge ij } t ≥ arrival at i N2 N1 N3 N4 N5 [2,3,4,5,6] [3,4,5,6,7] [2,3,4,5,6] [2,4,8,6,6] [3,4,5,6,7] Trace of NF-SP-TAG Algorithm N1 1 ∞ 2 1 3 3 3 N2 N5 N3N4 1 1 2 2 ∞ ∞∞ 3 ∞ ∞ ∞ 4 3 1 2 3 ∞ 5 3 1 2 3 6

36. 36 NF-SP-TAG Algorithm- Pseudocode Pre-process the network. Initialize c[s] = t_start;  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_arrival((u,v), c[u]); 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.

37. 37 NF-SP-TAG Algorithm - Correctness  Earliest arrival for a given start time  Shortest path If it is not, it contradicts “the earliest arrival”.  Algorithm picks the node with the least cost Ensures admissibility.  Algorithm updates the nodes based on the minimum arrival time. NF-SP-TAG Algorithm is correct. Maintains admissibility since Minimum  t  t1 [a ij (t)] Minimum  t  t2 [a ij (t)]≤ for t1 < t2

38. 38 NF-SP-TAG: Analytical Evaluation  Computational Complexity  Complexity of shortest path algorithm is O(m(T+ log n)) [n: Number of nodes, m – Number of edges, T – length of the time series]  For every node extracted,  Earliest arrival lookup – O(T)  Priority queue update – O(log n)  Overall Complexity =  O(degree(v). (T + log n)) = O(m( T+ log n))  Complexity of label correcting algorithm is O(n 2 T 3 (n+m)]

39. 39 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, average node degree. Experiment Platform: CPU: 1.77GHz, RAM: 1GB, OS: UNIX. Experimental Setup

40. 40 Performance Evaluation - Results Experiment 1: Effect of Number of Nodes (Fixed Start Time) Setup: Fixed length of time series = 100 TAG based algorithms are faster than time-expanded graph based algorithms. Experiment 2: Effect of Length of time series. Setup: fixed number of nodes = 786, number of edges = 2106. Experiment 1 Experiment 2

41. 41 Performance Evaluation - Results Experiment 3: Effect of Average Degree of Network. Setup: Length of time series= 240. TAG based algorithms run faster than time-expanded graph based algorithms.

42. 42 Conclusions  Time Aggregated Graph (TAG)  Time series representation of edge/node properties  Non-redundant representation  Often less storage, less computation time  Routing Algorithms  Faster shortest path for fixed start time in general (FIFO & non-FIFO networks.

43. 43 Routing Algorithms – Alternate Semantics 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..

44. 44 Contributions (Broader Picture)  Time Aggregated Graph (TAG)  Routing Algorithms FIFONon-FIFO Fixed Start Time (1) Greedy (SP-TAG) (2) A* search (SP-TAG*) (4) NF-SP-TAG Best Start Time (3) Iterative A* search (TI-SP-TAG*) (5) Label Correcting (BEST) (6) Iterative NF-SP-TAG

45. 45 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. (3) Termination of the algorithm : An infinite non-negative cycle over time Finite time windows are assumed. Costs assumed constant after T. Use Label-correcting approach and node cost series (BEST algorithm) Use performance tuned iterative version of NF-SP-TAG (CP-NF-TAG)

46. 46 CP-NF-BEST (Best Start Time) Key Ideas NF-SP-TAG for each start time Handles non-FIFO travel times Maintains copies of nodes  logical concurrency Terminates when a copy of destination is expanded. Algorithm is correct NF-SP-TAG correctly computes shortest path for every instant. Shortest path is a function of start time and network parameters. Since the algorithm computes the SP for every start time, it finds the least travel time.

47. 47 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 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). C u (t) = min(C u (t),  uv (t) + C v (t+  uv (t) )

48. 48 Best Start Time Shortest Path Algorithm Key Ideas SP-TAG* (A* based) iterated for every start time. Handles FIFO travel times Finds the minimum travel time from all shortest paths Performance optimization : Re-use heuristic costs from previous iterations. Time Iterated SP-TAG* Algorithm for FIFO Networks (TI-SP-TAG*)

49. 49 Future Work  Formulate new algorithms. Incorporate time-dependent turn restrictions in shortest path computation. Develop ‘frequent route discovery’ algorithms based on TAG framework.

50. 50 Thank you.