Project presentation

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Agenda Motivation Problem Statement Related Work Proposed Solution Hierarchical routing theory

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MotivationMany 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

MotivationExisting work on transportation network routingBased on constant edge value.In real worldTravel 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?

Problem StatementInput:

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.

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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

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ChallengesNew 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

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AA

BB

CC

[1,1,1,1]

[1,1,1,1]

DD[2,2,2,2]

[1,1,1,1]

EE[1,1,3,1]

Related Work

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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

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.

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Proposed Solution

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Input: spatio-temporal network snapshots at t=1,2,3,4,5

AANode:

travel timeEdge:

t=1 t=2

t=3 t=4

t=5

Proposed Solution

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Output: hierarchical graph & pre-computed information

[m1,…..,mT]

mi- travel time at t=iShortest 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:

Proposed Solution

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Base level graph & pre-computed information

[m1,…..,mT]

mi- travel time at t=i

Partitioned sub network:

Shortest path cost:

Fragment 1

Proposed Solution

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Base level graph & pre-computed information

[m1,…..,mT]

mi- travel time at t=i

Partitioned sub network:

Shortest path cost:

Fragment 2

Proposed Solution

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Higher level graph & pre-computed information

[m1,…..,mT]

mi- travel time at t=i

Partitioned sub network:

Shortest path cost:

Boundary graph

Agenda Motivation Problem Statement Related Work Proposed Solution Hierarchical routing theory

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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 ≠ jC ϵ BN(Fragment i), E ϵ BN(Fragment j)BN(Fragment i): boundary nodes set of Fragment iSP(A,G): shortest path from A to GSPC(A,G): shortest path cost from A to GBG: 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).

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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 t0SP(p,q,t0): shortest path from node p to node q start at time t0PC(p,q,t0): path cost from node p to node q start at time t0SPC(p,q,t0): shortest path cost from p to q start at time t0∆t: wait timeG: original graphBG: boundary graphBN(Gi): boundary nodes of fragment Gi

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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

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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.

t1 t2 t3

Find SP(A,G,t1)

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C

E

T1 T2 T3 T4 T5 T6 T7 T8 A

Time expended graph

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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 ID Arrival time Parent node Parent start time StatusA T2 -- -- ExploredC Infinite -- -- ToexploreE Infinite -- -- Toexplore

Node ID Arrival time Parent node Parent start time StatusA T2 -- -- closedC T4 A T2 ToexploreE Infinite -- -- Toexplore

T2initial Node ID Arrival time Parent node Parent start time StatusA T2 -- -- closedC T4 A T2 ExploredE Infinite -- -- Toexplore

T4 Node ID Arrival time Parent node Parent start time StatusA T2 -- -- closedC T4 A T2 ExploredE T7 C T5 Toexplore

T5 Node ID Arrival time Parent node Parent start time StatusA T2 -- -- closedC T4 A T2 closedE T7 C T5 explored

T7

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What we have doneHierarchical modelHierarchical routing theory in spatio-temporal network

Future workStudy data structure support the hierarchical modelOptimize algorithm and storage costStudy impact of network updateExperiments

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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)