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1 Shashi Shekhar McKnight Distinguished Uninversity Professor University of Minnesota www.cs.umn.edu/~shekhar, www.spatial.cs.umn.edu www.cs.umn.edu/~shekhar Spatio-Temporal Networks: A GIS Perspective A Provocation at Visualizing Network Dynamics Workshop (11/4-6/2008) Supporting NATO Research Task Group IST-059/RTG-025 Outline Brief overview of my research group Recent NGA NURI Grant Network Dynamics Representation Provocation: Time Aggregated Graphs

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2 Spatial Databases: Example Projects only in old plan Only in new plan In both plans Evacutation Route Planning Parallelize Range Queries Storing graphs in disk blocksShortest Paths

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3 Spatial Data Mining: Example Projects Nest locationsDistance to open water Vegetation durability Water depth Location prediction: nesting sitesSpatial outliers: sensor (#9) on I-35Co-location PatternsTele connections

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4 1. Books Spatial Databases: A Tour, Prentice Hall, 2003 Encyclopedia of GIS, Springer, 2008 Service Activities 2. Journals GeoInformatica: An Intl. Journal on Advances in Computer Sc. for GIS

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5 Outline Brief overview of my research Recent NGA NURI Grant Network Dynamics – Representations Provocation: Time Aggregated Graphs

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6 Dynamic Purpose aware Graph Data Models for Representing and Reasoning about Composite Networks Investigators: Shashi Shekhar,(U Minnesota) Start Date: August 2008 Motivation: Complex and Fluid Spatio-temporal Structures Challenge 1: Composite Networks Challenge 2: Time-variant Problem Definition Inputs: (i) Complicated Feature datasets (ii) A set of intelligence analysis tasks Output: Data Model for representation and reasoning Objective Function: Semantic expressiveness Constraints: Computational resources

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7 Composite Networks Example: Money Laundering – ATM, Transportation (Road, Subway) State of the Art: Graph Theory Time Geography: event-process Network Engines Critical Barriers: Composite Multi-purpose networks Time-variance Approach: 1.Decompose composite networks into single purpose networks 2.Role ( network entities, e.g. bridge ) is a bridge an obstacle or a link ? 3.Time aggregated graphs Manhattan Money Laundering Incident

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8 Adding Roles, Purposes to Network Data Model Proposed Extension Existing Graph model (Oracle) Primitive Analysis Questions: What is overall purpose of each component network? What are network-element role-types (e.g. nodes, edges, obstacles, etc.) ? What are instances of each element role-types? What are the operations on element-types, roles, purposes and network? Approach: Purpose Aware Graphs (PAG) Tasks: T1: Conceptual Model for PAGT2: Data types, Operators T3: Query Processing algorithmsT4:Purpose and Role Taxonomy T5: Validation

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9 Challenge 2: Time-variant, Fluid Networks Syria's Suspected Nuclear Facility Source: New York Times and Digital Globe Basic Modelling Questions: What is the variation of the role of a node or an edge over time? Where is a purpose changed or where does re- purposing occur? What are the nodes and edges that causes the re-purposing of a network? What are the nodes and edges that are part of a series of re-purposing? Proposed Approach: Dynamic-Purpose Aware Graphs (DPAG) Tasks G1: Event and Process Model for DPAG G2: Data type, query operators on DPAG G3: Algorithms for DPAG G4: Storage and Access Methods for DPAG G5: Validation

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10 Outline Brief overview of my research Recent NGA NURI Grant Network Dynamics – Representations Provocation: Time Aggregated Graphs

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11 Motivation Delays at signals, turns, Varying Congestion Levels travel time changes. 1) Transportation network Routing 2) Crime Analysis Identification of frequent routes (i.e.) Journey to Crime 3) Dynamic Social Network Analysis Emerging leaders or dense sub-networks, Cells with increased chatter, 4) 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

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12 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,

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13 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

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14 Challenges in Representation Conflicting Requirements Expressive Power Storage Efficiency New and alternative semantics for common graph operations. What is the best start time ? Shortest Paths are time dependent. Emerging, Dissipating, periodic, spreading, … Key assumptions violated. Ex., Prefix optimality of shortest paths (greedy property behind Dijkstra’s algorithm..)

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15 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]

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16 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.

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17 Outline Brief overview of my research Recent NGA NURI Grant Network Dynamics – Representations Provocation Representation: Time Aggregated Graphs Example Analysis: Shortest Path

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18 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.

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19 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

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20 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.

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21 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.

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22 Outline Brief overview of my research Recent NGA NURI Grant Network Dynamics – Representations Provocation Representation: Time Aggregated Graphs Example Analysis: Shortest Path

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23 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.

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24 Challenges: Lack of Dynamic Programming Principle 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 Naïve 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.

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25 Challenge of Non-FIFO Travel Times 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

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26 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.

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27 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)]

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28 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)

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29 Routing – New Semantics (Best Start Time) 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..

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30 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

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31 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.

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