Hierarchical model for routing query in spatio -temporal network

1. Hierarchical model for routing query in spatio-temporal network Project presentation

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

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 Question: Can we build a model which can support various spatio-temporal network routing queries? 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 I94 @ Hamline Ave at 8AM & 10AM *U. Demiryurek, F. B. Kashani, and C. Shahabi. Towards k-nearest neighbor search in time dependent spatial network databases. In Proceedings of DNIS, 2010

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.

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

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 [1,1,1,1] B [1,1,1,1] [1,1,3,1] A C E [2,2,2,2] [1,1,1,1] D

8. Related Work • 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.

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

11. Proposed Solution Output: hierarchical graph & pre-computed information (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 [m1,…..,mT] Partitioned sub network: Shortest path cost: mi- travel time at t=i

12. Proposed Solution Base level graph & pre-computed information Fragment 1 [m1,…..,mT] Shortest path cost: mi- travel time at t=i Partitioned sub network:

13. Proposed Solution Base level graph & pre-computed information Fragment 2 [m1,…..,mT] Shortest path cost: mi- travel time at t=i Partitioned sub network:

14. Proposed Solution Higher level graph & pre-computed information Boundary graph [m1,…..,mT] Shortest path cost: mi- travel time at t=i Partitioned sub network:

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

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 niϵBN(i)∧nj∈BN(j)∧(SPC(A,C)+SPC(C,E)+SPC(E,G))≤(SPC(A,ni)+SPC(ni, nj)+SPC(nj,G)) ----------------------------------------------------------------------------------------------- A ϵ Fragment i, G ϵ Fragment j, i ≠ j C ϵBN(Fragment i), E ϵBN(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 ----------------------------------------------------------------------------------------------- Find Shortest path from A to G SP(A,G)=SP(A,C)+SP(C,E)+SP(E,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. 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. 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(C,E,T1)=2 BG.SPC(C,E,T1)=2 G.SPC(A,C,T2)=2

19. Hierarchical routing theory in Spatio-temporal network 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. Find SP(A,G,t1) t1 t2 t3

20. Shortest path algorithm in spatio-temporal network

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

22. Find SP(A,E,T2) :SP(A,E,T2)=SP(A,C,T2)+SP(C,E,T5) T1 T2 T3 T4 T5 T6 T7 T8 A C E T2 initial T4 T5 T7

23. Summary • 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

24. References 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)