1. Project presentation 1

2. Agenda Motivation Problem Statement Related Work Proposed Solution Hierarchical routing theory 2

3. 3 Motivation Many Applications of network routing Examples: Online Map service, phone service, transportation navigation service  Identification of frequent routes  Crime Analysis  Identification of congested routes  Network Planning

4. Motivation Existing work on transportation network routing Based on constant edge value. In real world Travel time of road segment changes over time. People are interested in various routing queries 4 *U. Demiryurek, F. B. Kashani, and C. Shahabi. Towards k-nearest neighbor search in time dependent spatial network databases. In Proceedings of DNIS, 2010 I94 @ Hamline Ave at 8AM & 10AM Question: Can we build a model which can support various spatio- temporal network routing queries?

5. Problem Statement Input:  A spatial network G=(N,E).  Temporal changes of the network topology and parameters. Output:  A model to process routing queries in spatio-temporal network. Objective:  Minimize storage and computation cost. Constraints:  Spatio-temporal network and pre-computed information are stored in secondary memory.  Changes occur at discrete instants of time.  Allow wait at intermediate nodes of a path.  Routing is based on Lagrange path. 5

6. Key Concept Graph G= (N, E): a directed flat graph consisting of a node set N, and an edge set E. Fragment: a sub-graph of G, which consists a subset of nodes and edges of G. Boundary node: a node that has neighbors in more than one fragment. Hierarchical graph: a two-level representation of the original graph. The base-level is composed of a set of disjoint fragments The higher-level called boundary graph, is comprised of the boundary nodes 6

7. Challenges New semantics for spatial networks Optimal paths are time dependent Key assumptions violated Prefix optimality of shortest paths (Non-FIFO travel time) Conflicting Requirements Minimum Storage Cost Computational Efficiency 7 A A B B C C [1,1,1,1] D D [2,2,2,2][1,1,1,1] E E [1,1,3,1]

8. Related Work 8  Static Model [HEPV’98, HiTi’02, Highway’ 07] Does not model temporal variations in the network parameters Supports queries such as shortest path in static networks Pre-compute and store information  Spatio-temporal Model for specific query [Voronoi diagram’ 10] Designed for specific query such as K nearest neighbors Not scalable to other spatio-temporal network routing queries

9. Contributions Hierarchical model Support different routing queries in spatio-temporal network. Less storage cost, less computation time. Hierarchical routing theory in spatio-temporal network Evaluate model by different spatio-temporal routing queries. Shortest path query. Best start time query. 9

10. Proposed Solution 10 Input: spatio-temporal network snapshots at t=1,2,3,4,5 A A Node: travel time Edge: t=1 t=2 t=3t=4 t=5

11. Proposed Solution 11 Output: hierarchical graph & pre-computed information [m 1,…..,m T ] m i - travel time at t=i Shortest path cost: (a) Hierarchical graph overview(b) two fragments created at base level graph (c) boundary nodes identified and pushed to higher level (d) boundary graph contains only boundary nodes Partitioned sub network:

12. Proposed Solution 12 Base level graph & pre-computed information [m 1,…..,m T ] m i - travel time at t=i Partitioned sub network: Shortest path cost: Fragment 1

13. Proposed Solution 13 Base level graph & pre-computed information [m 1,…..,m T ] m i - travel time at t=i Partitioned sub network: Shortest path cost: Fragment 2

14. Proposed Solution 14 Higher level graph & pre-computed information [m 1,…..,m T ] m i - travel time at t=i Partitioned sub network: Shortest path cost: Boundary graph

15. Agenda Motivation Problem Statement Related Work Proposed Solution Hierarchical routing theory 15

16. 16 Intuition of hierarchical model Hierarchical routing theory: SP(A,G)= Fragment(i).SP(A,C)+BG.SP(C,E)+Fragment(j).SP(E,G) where ∀ ni,nj niBN(i) ∧ nj ∈ BN(j) ∧ (SPC(A,C)+SPC(C,E)+SPC(E,G))≤(SPC(A,ni)+SPC(ni, nj)+SPC(nj,G)) ----------------------------------------------------------------------------------------------- A Fragment i, G Fragment j, i ≠ j C BN(Fragment i), E BN(Fragment j) BN(Fragment i): boundary nodes set of Fragment i SP(A,G): shortest path from A to G SPC(A,G): shortest path cost from A to G BG: boundary graph ----------------------------------------------------------------------------------------------- SP(A,G)=SP(A,C)+SP(C,E)+SP(E,G) Find Shortest path from A to G *Materialization Trade-Offs in Hierarchical Shortest Path Algorithms, S. Shekhar, A. Fetterer, and B. Goyal, Proc. Intl. Symp. on Large Spatial Databases, Springer Verlag (Lecture Notes in Computer Science), (1997).

17. 17 Hierarchical routing theory in Spatio-temporal network Shortest path: given a start time, start and end node, travel along the shortest path has the earliest arrival time Path cost: given a path with start time, path cost is the arrival time minus start time P(p,q,t0): a path from p to q start at time t0 SP(p,q,t0): shortest path from node p to node q start at time t0 PC(p,q,t0): path cost from node p to node q start at time t0 SPC(p,q,t0): shortest path cost from p to q start at time t0 ∆t: wait time G: original graph BG: boundary graph BN(Gi): boundary nodes of fragment Gi

18. 18 Hierarchical routing theory in Spatio-temporal network Theorem 1 G.PC(p,q,t0)=G.SPC(p,q,t0)+ ∆t, ∆t is wait time at node q 2 G.SPC(p,q,t0)=BG.SPC(p,q,t0), p, q ∈ BG G.PC(A,C,T2)=4 G.SPC(A,C,T2)=2 G.SPC(C,E,T1)=2BG.SPC(C,E,T1)=2

19. 19 Theorem: Let P = {G1,G2,…,Gp} be a partition of original graph G, BG be the boundary graph. For node s ∈ Node set of Gu, node d ∈ Node set of Gv, where 1≤u,v≤p, and u≠v. Start time is fixed at t1. Then SP(s,d,t1)=Fragment(Gu).SP(s,ni,t1)+BG.SP(ni,nj,t2)+Fragment(Gv).SP(nj,d,t3) ni ∈ BN(Gu), nj ∈ BN(Gv) t2=t1+SPC(s,ni,t1)+ ∆t1, t3=t2+SPC(ni,nj,t2)+ ∆t2 ∆t1 is wait time at ni, ∆t2 is wait time at nj. t1t2t3 Find SP(A,G,t1) Hierarchical routing theory in Spatio-temporal network

20. Shortest path algorithm in spatio-temporal network

21. 21 Shortest path algorithm in spatio-temporal network C E T1 T2 T3 T4 T5 T6 T7 T8 A Time expended graph

22. 22 C E T1 T2 T3 T4 T5 T6 T7 T8 A Find SP(A,E,T2):SP(A,E,T2)=SP(A,C,T2)+SP(C,E,T5) Node IDArrival timeParent nodeParent start timeStatus AT2-- Explored CInfinite-- Toexplore EInfinite-- Toexplore Node IDArrival timeParent nodeParent start timeStatus AT2-- closed CT4AT2Toexplore EInfinite-- Toexplore T2initial Node IDArrival timeParent nodeParent start timeStatus AT2-- closed CT4AT2Explored EInfinite-- Toexplore T4 Node IDArrival timeParent nodeParent start timeStatus AT2-- closed CT4AT2Explored ET7CT5Toexplore T5 Node IDArrival timeParent nodeParent start timeStatus AT2-- closed CT4AT2closed ET7CT5explored T7

23. 23 What we have done Hierarchical model Hierarchical routing theory in spatio-temporal network Future work Study data structure support the hierarchical model Optimize algorithm and storage cost Study impact of network update Experiments Summary

24. 24 1.Materialization Trade-Offs in Hierarchical Shortest Path Algorithms, S. Shekhar, A. Fetterer, and B. Goyal, Proc. Intl. Symp. on Large Spatial Databases, Springer Verlag. 1997 2. Betsy George, Shashi Shekhar, Time Aggregated Graphs for Modeling Spatio-temporal Network, Journal on Semantics of Data (Editors: J.F. Roddick, S. Spaccapietra), Vol XI, December, 2007 3. Fast object search on road networks, C.K. Lee, A. Wang-Chien Lee, and Beihua Zheng, Proceedings of the 12th International Conference on Extending Database Technology: Advances in Database Technology. Vol 360. 2009 4. Ugur Demiryurek, Farnoush Banaei-Kashani, and Cyrus Shahabi. Efficient K-Nearest Neighbor Search in Time-Dependent Spatial Networks. 2010 5. Hierarchical Encoded Path Views for Path Query Processing: An Optimal Model and Its Performance Evaluation, Ning Jing, Yun-Wu Huang, Elke A. Rundensteiner, IEEE Transactions on Knowledge and Data Eng., May/June 1998 (Vol. 10, No. 3) References