Modeling Spatio-temporal Network Computations

1. Modeling Spatio-temporal Network Computations Betsy George, Shashi Shekhar Department of Computer Science and Engineering University of Minnesota Second International Conference on GeoSpatial Semantics, Mexico City, Mexico 29 November 2007

2. Outline • Introduction • Motivation • Problem Statement • Related Work • Contributions • Representation • Conceptual Model • Logical Model • Physical Model • Evaluation of Model using Case studies • Conclusion and Future Work

3. U.P.S. Embraces High-Tech Delivery Methods (July 12, 2007) By Claudia H. Deutsch “The research at U.P.S. is paying off. Last year, it cut 28 million miles from truck routes— in good part by mapping routes that minimize left turns.” 4 PM, November 19, 2007 Sensors on Minneapolis Highway Network periodically report time varying traffic Motivation 1) Transportation network Routing • Varying Congestion Levels and turn restrictions  travel time changes. Some example queries on a time-varying network. 9 PM, November 19, 2007 2) Crime Analysis • Identification of frequent routes (i.e.) Journey to Crime 3) Knowledge discovery from Sensor data. • Spreading Hotspots

4. 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. • Objectives: (i) Expressive power (ii) Minimize storage and computation costs. • Constraints : (i) Changes occur at discrete instants of time, (ii) Logical & Physical independence

5. Key assumptions violated. • Ex., Prefix optimality of shortest paths Challenges • Conflicting Requirements • Expressive Power • Storage Efficiency • New and alternative semantics for common graph operations. • Ex., Shortest Paths are time dependent.

6. Person Related Work In Conceptual Data Model Temporally Enhanced ER Model (TERC)* Degree seekingStudent Non-Degree seekingStudent Alumnus Applicant Donor Added Promoted Graduated Accepted • Models temporal changes in relationships via entity subtypes. * E. Zimayi, C. Parent, S, Spaccapietra, TERC+: A Temporal Conceptual Model, Proceedings of International Symposium on Digital Media Information Base, November 1997

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

8. High Storage Overhead Redundancy of nodes across time-frames Additional edges across time frames. Parking Lot Parking Lot Vehicle Vehicle . . . Limitations of Related Work • Time Expanded Graph • Computationally expensive Algorithms • Increased Network size. • Inadequate support for modeling non-flow parameters and uncertainty on edges. • Lack of physical independence of data • TERC Model t=t1 A B disjoint t=t2 • Model can lead to too many sub-entities t=t3 t=t4 touch • Can affect scalability Relationship between Two Objects at Various Instants Representation using Pictograms • Entities participating in an evolving relationship might not always have subtypes.

9. Our Contributions Time Aggregated Graph (TAG) • Representation • Conceptual Model • Time Aggregation on evolving relationships • Logical Model • Time aggregated Graph • Set of logical operators • Physical Model • Evaluation of Model using Case Studies • Routing Algorithms

10. B t=t1 A t=t2 Touch Disjoint t=t3 t=t4 Inside Covers t=t9 Equals [d,t,o,c,c,v,o,t,d] Overlap Vehicle Parking Lot CoveredBy Contains d – disjoint t – touch o – overlap c – contains v – covers Representation of a Dynamic Relationship Relationship between Two Objects at Various Instants Pictogram with time aggregation Implicit evolving relationships among Spatial Objects • Uses time-aggregation to model evolving relationships.

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

12. [ew1,..,ewT ] | TAG = (N,E,T, [nw1…nwT ], nwi : N RT, ewi : E RT 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 On edge N4-N5 * [2,∞,∞,∞,2] is a time series of attribute; [,1,1,1,1] [1,1,1,1,1] [2,, , ,2] N5 N4 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

13. Graph Elements • Node • Edge • Route • Graph Examples getEdge(N1,N3,1) 2 N2 [,1,1,1,1] [1,1,1,1,1] [2,, , ,2] [1,∞, ∞, ∞, ∞] N4 N5 N1 • Topology_first OperationsExample: getEdge(n1,n2) [2,2,2,2,2] [2,2,2,2,2] N3 • Time_first OperationsExample: getEdge(n1,n2,t) Logical Model • Logical Operators • Accessors • get • Modifiers • insert, update, delete • Predicates • exists insertEdge(N1,N4,1,1)  getEdge(N1,N3) [2,2,2,2,2]

14. Logical Model – A Set of Logical Operators Operators in the Accessor Category Operators in the Modifier Category

15. ∞ N3 N5 N4 N4 N2 V ∞ ∞ ∞ 2 2 N2 [,1,1,1,1] [1,1,1,1,1] [2,, , ,2] N4 N5 N1 V 1 1 1 1 [2,2,2,2,2] [2,2,2,2,2] N3 Attribute time series is stored at the link to each adjacent node. 1 F ‘V’ – Attribute varies over time. ‘F’ – Attribute remains constant 2 2 F F Physical Representation Time-Aggregated Adjacency List N1 N2 N3 N4 N5 • Might use less storage for sparse graphs.

16. N2 [,1,1,1,1] [1,1,1,1,1] [2,, , ,2] N4 N5  N1   [2,2,2,2,2] [2,2,2,2,2] N3 Physical Representation N1 N2 N3 N4 N5 N5 N2 N1 N3 N4 ∞ 1 ∞ ∞ ∞ ∞ 2 ∞ ∞ 2 N1 1 ∞ ∞ ∞ 1 ∞ ∞ ∞ ∞ ∞ N2 N3 ∞ ∞ ∞ 2 ∞ ∞ ∞ ∞ 2 ∞ N4 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ 2 ∞ ∞ ∞ ∞ ∞ ∞ N5 ∞ ∞ ∞ ∞ t=1 t=2 N1 N2 N3 N4 N5 N5 N2 N1 N3 N4 Attributes are stored in a three dimensional matrix. ∞ 1 ∞ ∞ ∞ 1 2 ∞ ∞ 2 N1 1 ∞ ∞ ∞ 1 ∞ ∞ ∞ ∞ ∞ N2 N3 ∞ ∞ ∞ 2 ∞ ∞ ∞ ∞ 2 ∞ One slice on the time dimension  one snapshot N4 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ N5 ∞ ∞ ∞ ∞ t=3 t=4 • More suitable for Time-first (snapshot) operations.

17. N3 N4 N5 N4 N2 N1 N2 N3 N2 N4 [,1,1,1,1] [1,1,1,1,1] ∞ 2 [2,, , ,2] ∞ N4 N5 N1 N5 ∞ 1 [2,2,2,2,2] [2,2,2,2,2] 1 N3 1 1 2 2 2 2 1 2 2 Physical Representation An Implementation of Infinite time series. Time-Aggregated Adjacency List wit sliding windows Time series is stored in a circular queue the link to each adjacent node.

18. (*) 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. 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) • For a Time Expanded Graph, • Storage requirement is O(nT) + O(n+m)T(**) • Experimental Evaluation

19. N2 N2 1 1 1 1 N4 N5 N1 N4 N5 N1 2 2 2 2 N3 N3 N2 t=3 t=2 1 2 N2 N2 N4 N5 N1 1 1 1 1 2 2 2 N4 N4 N5 N5 N1 N1 N3 t=1 2 2 2 2 N3 N3 t=4 t=5 Evaluation using Routing Algorithms 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!!

20. Greedy strategy: • Select the node with the lowest cost to expand. • Traverse every edge at the earliest available time. Shortest Path Algorithm for Given Start Time Algorithm • Every node has a cost ( arrival time at the node). • Complexity of shortest path algorithm is O(m( log T+ log n)) Pseudo code Input: Graph G(N,E), Source s, Destination d, Start time tstart Output: Shortest Path from s to d for start time tstart Insert s in minimum heap Q while Q not empty { u= extract_min (Q); for each node v adjacent to u { t =get_edge_earliest(u,v,cu);  =getEdge (u,v,t); relax(u,v,); insert(Q,v); }}

21. 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). Pseudo code Input: Graph G(N,E), Source s, Destination d, Start time tstart Output: Best start time from s to d for start time tstart Enqueue(Q,s); while Q not empty { u= dequeue(Q); ts =getNode(u); for each node v adjacent to u and for each t{  =getEdge(u,v,t); cu =getNode(u,t); cv=getNode(v,t+ ); if (cu >  + cv), then { update(v,t,  +cv); insert(Q,v); }}

22. 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 • Transportation Network Routing Algorithms • Shortest Path for Fixed Start Time • Shortest Path for Fixed Start Time

23. 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. Future Work • Spatio-temporal Network Databases • Three-Schema Architecture