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Time-Aggregated Graphs- Modeling Spatio-temporal Networks

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

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Time-Aggregated Graphs- Modeling Spatio-temporal Networks

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

  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. Outline • Introduction • Motivation • Problem Statement • Related Work • Contributions • Representation • Routing Algorithms • Conclusion and Future Work

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

  5. Motivation 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. Signal delays at left turns cancause non-FIFO travel times. Pictures Courtesy: http://safety.transportation.org

  6. Problem Definition • Input : a) A Spatial Network b) Temporal changes of the network topology and parameters. • Output : A model that supports efficient correct algorithms for computing the query results. • Objective : Minimize storage and computation costs. • Constraints : (i) Predictable future (ii) Changes occur at discrete instants of time, (iii) Logical & Physical independence,

  7. Key assumptions violated. • Ex., Prefix optimality of shortest paths • (greedy property behind Dijkstra’s algorithm..) Challenges in Representation • Conflicting Requirements • Expressive Power • Storage Efficiency • New and alternative semantics for common graph operations. • Ex.,Shortest Paths are time dependent.

  8. N2 N2 N2 1 1 1 1 1 2 N4 N4 N5 N1 N5 N1 N4 N5 N1 2 2 2 2 2 2 N3 N3 N3 t=2 t=3 t=1 N2 N2 1 1 1 1 2 Node: N4 N5 N4 N1 N5 N1 2 2 Edge: 2 N3 N3 2 N.. t=4 t=5 Travel time Holdover Edge N1 N1 N1 N1 N1 N1 N1 N2 N2 N2 N2 N2 N2 N2 Transfer Edges N3 N3 N3 N3 N3 N3 N3 N4 N4 N4 N4 N4 N4 N4 N5 N5 N5 N5 N5 N5 N5 t=3 t=4 t=6 t=7 t=1 t=5 t=2 Related Work in Representation (1) Snapshot Model [Guting04] (2) Time Expanded Graph (TEG) [Kohler02, Ford65]

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

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

  11. N2 N2 N2 1 1 1 1 1 2 N4 N5 N4 N1 N5 N4 N1 N5 N1 2 2 2 2 2 2 N3 N3 N3 t=3 t=2 t=1 N.. N2 N2 1 Node: 1 1 1 N.. 2 N4 N5 N4 N1 N5 N1 Edge: Travel time 2 2 2 2 N3 N3 t=4 t=5 Proposed Approach Snapshots of a Network at t=1,2,3,4,5 Time Aggregated Graph • Attributes are aggregated over edges and nodes. N2 Node [,1,1,1,1] [1,1,1,1,1] [2,, , ,2] N4 N5 N1 Edge [m1,…..,(mT] [2,2,2,2,2] [2,2,2,2,2] N3 mi- travel time at t=i

  12. 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. N2 [,1,1,1,1] On edge N4-N5 * [2,∞,∞,∞,2] is a time series of attribute; [1,1,1,1,1] [2,, , ,2] N4 N5 N1 * At t=1, the edge has an attribute value of 2. [2,2,2,2,2] [2,2,2,2,2] * At t=2, the ‘∞’ can indicate the absence of connectivity between the nodes at t=2. N3 Time Aggregated Graph [ew1,..,ewT ] | TAG = (N,E,T, [nw1…nwT ], nwi : N RT, ewi : E RT

  13. Performance Evaluation: Dataset Minneapolis CBD [1/2, 1, 2, 3 miles radii] • Road data • Mn/DOT basemap for MPLS CBD.

  14. (*) All edge and node parameters might not display time-dependence. (**) D. Sawitski, Implicit Maximization of Flows over Time, Technical Report (R:01276),University of Dortmund, 2004. 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) • 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.

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

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

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

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

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

  20. Start time = 1; Start node : N1 Iteration 1: N1_1 selected N1_2 = 2; N2_2 = 2; N3_3 = 3 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 Related Work – Label Correcting Approach(*) • Selection of node to expand is random. • Algorithm terminates when no node gets updated. N1 N2 N3 N4 N5 t=8 t=3 t=4 t=6 t=7 t=2 t=1 t=5 • Implementation used the Two-Q version [O(n2T 3(n+m)] (*) Cherkassky 93,Zhan01, Ziliaskopoulos97

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

  22. N2 Select Minimum {Cost of edge ij } [2,3,4,5,6] [2,3,4,5,6] t ≥ arrival at i [2,4,8,6,6] N4 N5 N1 [3,4,5,6,7] [3,4,5,6,7] N3 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. Trace of NF-SP-TAG Algorithm N1 N2 N3 N4 N5 1 ∞ ∞ 1 ∞ ∞ 2 1 ∞ 2 ∞ 3 3 3 2 3 1 ∞ 4 3 2 3 1 ∞ 5 3 2 3 1 6

  23. 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.

  24. for t1 < t2 [aij(t)] ≤ [aij(t)] Minimum Minimum t  t1 t  t2 NF-SP-TAG Algorithm - Correctness NF-SP-TAG Algorithm is correct. • 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. Maintains admissibility since

  25. NF-SP-TAG: Analytical Evaluation • Computational Complexity [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 shortest path algorithm is O(m(T+ log n)) • Complexity of label correcting algorithm is O(n2T3(n+m)]

  26. Length of Time Series Real Dataset (without time series) Time Series Generation Road network with travel time series Network Expansion TAG Based Algorithms Shortest Path Algorithms on Time Expanded Graph Run-time Run-time Data Analysis Performance Evaluation: Experiment Design 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 Time expanded network

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

  28. 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.

  29. 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.

  30. N2 N2 N2 1 1 1 1 1 2 N4 N5 N4 N1 N5 N4 N1 N5 N1 2 2 2 2 2 2 N3 N3 N3 t=3 t=2 t=1 N2 N2 1 1 1 1 2 N4 N5 N4 N1 N5 N1 2 2 2 2 N3 N3 Node: t=4 N.. t=5 Edge: Travel time Routing Algorithms – Alternate Semantics Finding the shortest path from N1 to N5.. Start at t=3: Start at t=1: Shortest Path is N1-N2-N4-N5; Travel time is 4 units. Shortest Path is N1-N3-N4-N5; Travel time is 6 units. Fixed Start Time Shortest Path Least Travel Time (Best Start Time) Shortest Path is dependent on start time!!

  31. Contributions (Broader Picture) • Time Aggregated Graph (TAG) • Routing Algorithms

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

  33. Thank you.

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