Time-Aggregated Graphs- Modeling Spatio-temporal Networks

1

Time-Aggregated Graphs-Modeling Spatio-temporal Networks

September 7, 2007

Betsy George

Department of Computer Science and Engineering University of Minnesota

Advisor : Prof. Shashi Shekhar

2

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, Accepted for presentation at the 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 (In second review) , Special issue of Selected papers from ER 2006.

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, Accepted for presentation at 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.

3

Outline

Introduction Motivation Problem Statement Related Work

Contributions

Conclusion and Future Work

Representation Case Studies

Routing Algorithms

Sensor Data Representation

4

Motivation

accurate computation of frequent routing queries.

Varying Congestion Levels and turn restrictions travel time changes.

Examples: Transportation network Routing, Crime pattern analysis, knowledge discovery from Sensor data.

Many Applications…

I94 @ Hamline Ave at 8AM & 10AM

Traffic sensors on Twin-Cities, MN Road Network monitor traffic levels/travel time on the road network. (Courtesy: MN-DoT (www.dot.state.mn.us) )

Identification of frequent routes

Crime Analysis

Identification of congested routes

Network Planning

5

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) Changes occur at discrete instants of time, (ii) Logical & Physical independence

6

Challenges

Conflicting Requirements

Expressive Power

Storage Efficiency

New and alternative semantics for common graph operations.

Ex., Shortest Paths are time dependent.

Key assumptions violated.

Ex., Prefix optimality of shortest paths

7

Related Work

Graph-based Models

Operations Research

Databases

Spatial Graphs Spatio-temporal Graphs(Time Aggregated

Graphs)

Flow networks ( Time Expanded Graphs)

Spatial Graphs [Erwig’94, Guting’96, Mouratidis’06, Shekhar’97] Does not model temporal variations in the network topology, parameters Supports operations such as shortest path computation on static graphs Maintains connectivity of link-node networks

Flow Networks (Time expanded Graphs)[Ford’58, Kaufman’93, Kohler’02,Dean’04]

Models time-dependent flow networks Maintains a copy of the graph for each time instant. Cannot model scenarios where edge parameter does not represent a

“flow”.

8

Related Work

t=1

N2

N1

N3

N4 N5

1

2

2

2

t=2

N2

N1

N3

N4 N5

1

22

1

t=3

N2

N1

N3

N4 N5

1

22

1

t=4

N2

N1

N3

N4 N5

1

22

1

t=5

N2

N1

N3

N4 N5

12

22

1N..

Travel time

Node:

Edge:

Time Expanded Graph

t=1

N1

N2

N3

N4

N5

t=2

N1

N2

N3

N4

N5t=3

N1

N2

N3

N4

N5t=4

N1

N2

N3

N4

N5

N1

N2

N3

N4

N5t=5

N1

N2

N3

N4

N5t=6

N1

N2

N3

N4

N5t=7

Holdover Edge

Transfer Edges

Snapshots at t=1,2,3,4,5

9

Related Work

Shortest Paths in Time Expanded Graphs

LP solvers (NETFLO, RELAX IV) provide support for Shortest Path Computation.

Models the time-expanded graph as an Uncapacitated flow network.

E : set of edges in the TEG

C(e) : Edge Cost

x(e) =1 if edge e is taken

= 0, otherwise

10

Limitations of Related Work

High Storage Overhead Redundancy of nodes across time-frames Additional edges across time frames.

Inadequate support for modeling non-flow parameters and uncertainty on edges.

Time Expanded Graph

Lack of physical independence of data.

Computationally expensive Algorithms Increased Network size.

12

Our Contributions

Graph Time Expanded Graph (TEG) & Time Aggregated Graph (TAG)

LP Solver (flow networks)

Flow algorithms based on LP


Label Correcting Algorithms

Two-Q Algorithm,..

BEST-TAG Algorithm

Label Setting Algorithms

Dijkstra’s Algorithm,..

SP-TAG Algorithm

Lack of optimal prefix

Shortest Path Shortest Path (Fixed Start

Time)

Shortest Path (Best Start Time)

Static Networks

Time-variant Networks

13

Our Contributions

Time Aggregated Graph (TAG)

Shortest Path for the ‘best’ start time

Shortest Path for a given start time

Analytical & Experimental Evaluation

Representation Case Studies

Routing Algorithms


14

Time Aggregated Graph

t=1

N2

N1

N3

N4 N5

1

2

22

t=2

N2

N1

N3

N4 N5

1

22

1

t=3

N2

N1

N3

N4 N5

1

22

1

t=4

N2

N1

N3

N4 N5

1

22

1

t=5

N2

N1

N3

N4 N5

1

2

22

1N..

Travel time

Node:

Edge:

Snapshots of a Network at t=1,2,3,4,5


N1

[,1,1,1,1]

[2,2,2,2,2]

[1,1,1,1,1]

[2,2,2,2,2]

[2,, , ,2]

N2

N3

N4 N5

[m1,…..,(mT]

mi- travel time at t=i

Edge

N..

Node

Attributes are aggregated over edges and nodes.

15


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.

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, [nw1…nwT ],

[ew1,..,ewT ] |nwi : N RT, ewi : E RT

N1

[,1,1,1,1]

[2,2,2,2,2]

[1,1,1,1,1]

[2,2,2,2,2]

[2,, , ,2]

N2

N3

N4 N5

16

Case Study -Routing Algorithms

t=1

N2

N1

N3

N4 N5

1

2

22

t=2

N2

N1

N3

N4 N5

1

22

1

t=3

N2

N1

N3

N4 N5

1

22

1

t=4

N2

N1

N3

N4 N5

1

22

1

t=5

N2

N1

N3

N4 N5

1

2

22

1N..

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..

17

Shortest Path Algorithm for Given Start Time

Challenges(1) Not all shortest paths

show optimal substructure.

[1,1,1,1,1]

[2,2,2,2,2]

[1,1,1,1,1]

[2,2,2,2,2]

[2,, , ,2]

N2

N3

N4 N5N1For start time t=1N1- N2- N4- N5t=1

t=2

t=3

wait till t=5 !!

t=7

(1)

N1-N3-N4 -N5 has non-optimal prefix

N1-N3-N4 -N5 N1-N2-N4 -N5 & are optimal (6 units).

N1- N3- N4- N5t=1

t=3

t=5

t=7

(2)

Lemma: At least one optimal path satisfies the optimal substructure property. N1-N2-N4-N5 in the example has optimal prefixes.

18


Challenge-1

Lemma: At least one optimal path satisfies the optimal substructure property.

Proof:

• For a given start time, the non-optimal substructure is due to waits at intermediate nodes.

• For the path from ‘s’ to ‘d’, let ‘u’ be an intermediate, wait node. • Append the optimal path from ‘s’ to ‘u’ to the path from ‘u’ to ‘d’ allowing wait at ‘u’.

• This path is optimal. (by Contradiction)

(1) Not all shortest paths show optimal substructure.

Greedy algorithm can be used to find the shortest path.

19


Challenges

Assume FIFO travel times.

(2) Correctness : Determining when to traverse an edge.

N11,1,1,1 N2 N3

1,3,1,2

When to traverse the edge N2-N3 for start time t=1 at N1?Traversing N2-N3 as soon as N2 is reached, would give sub optimal solution.

(3) Termination of the algorithm : An infinite non-negative cycle over time

Finite time windows are assumed.

20


Algorithm

Every node has a cost ( arrival time at the node). Greedy strategy:

Select the node with the lowest cost to expand.

Traverse every edge at the earliest available time.

N1

[,1,1,1,1]

[2,2,2,2,2]

[1,1,1,1,1]

[2,2,2,2,2]

[2,, , ,2]

N2

N3

N4 N5

Source: N1; Destination: N5; time: t=1;

1 ∞ ∞ ∞ ∞

1 3 3 ∞ ∞

1 3 3 4 ∞

1 3 3 4 ∞

1 3 3 4 7

N1 N2 N3 N4 N5

(1)

(∞)

(∞)

(∞) (∞)

(3)

(3)

(4) (7)

21


Initialize c[s] = 0; v ( s), c[v] = ∞. Insert s in the priority queue Q. while Q is not empty do u = extract_min(Q); close u (C = C {u}) for each node v adjacent to u do { t = min_t((u,v), c[u]); // min_t finds the earliest departure time for (u,v) If t + u,v(t) < c[v] c[v] = t + u,v(t) parent[v] = u insert v in Q if it is not in Q; } Update Q.

22


Correctness of the Algorithm (Optimality of the result)

The SP-TAG is correct under the assumption of FIFO travel times and finite time windows.

Lack of optimal substructure of some shortest paths is due to a potential wait at an intermediate node. Algorithm picks the path that shows optimal substructure and allows waits.

Lemma: When a node is closed, the cost associated with the node is the shortest path cost.

Based on proof for Dijkstra’s algorithm. Difference - Earliest availability of edge

- Admissible guarantees optimality

23

Analytical Evaluation

* B. C. Dean, Algorithms for Minimum Cost Paths in Time-dependent Networks, Networks 44(1), August 2004.

Computational Complexity

Dijkstra’s Cost Model extended to include the dynamic nature of edge presence.

Each edge traversal Binary search to find the earliest departure O(log T )

Complexity of shortest path algorithm is O(m( log T+ log n))

[n: Number of nodes, m – Number of edges, T – length of the time series]

For every node extracted, Earliest edge lookup – O(log T) Priority queue update – O(log n) Overall Complexity = O(degree(v). (log T + log n)) = O(m( log T+ log n))

24

Analytical Evaluation

Complexity of Shortest Path algorithm based on TAG is O(m( log T+ log n))

Complexity of Shortest Path Algorithm based on Time Expanded Graph is O(nT log T+mT) (*)

Lemma : Time-aggregated graph performs asymptotically better than time expanded graphs when log (n) < T log (T).

* B. C. Dean, Algorithms for Minimum Cost Paths in Time-dependent Networks, Networks 44(1), August 2004.

25

Best Start Time Shortest Path Algorithm

Finds a start time and a path such that the time spent in the network is minimized.

t=1

N2

N1

N3

N4 N5

1

2

22

t=2

N2

N1

N3

N4 N5

1

22

1

t=3

N2

N1

N3

N4 N5

1

22

1

t=4

N2

N1

N3

N4 N5

1

22

1

t=5

N2

N1

N3

N4 N5

1

2

22

1N..

Travel time

Node:

Edge:

Start Time:

Path : N1 – N2 – N4 – N5

Arrival Time:

Time Spent:

1 2 3 4 5

7 7 7 8 9

6 5 4 4 4

A Best Start Time!!

26


Challenges(1) Best Start Time shortest paths

need not have optimal prefixes.

N1[1,2,2,2,2,2] [2,∞, ∞,

∞,2,2]N2 N3

Optimal solution for the shortest path from N1 to N3 is suboptimal for N1 to N2 due to the wait at N2.

(2) Correctness: Lack of FIFO property.

Use Label-correcting approach instead Greedy methods.

Use node cost series instead of a scalar node cost.

(3) Termination of the algorithm : An infinite non-negative cycle over time

Finite time windows are assumed.

Costs assumed constant after T.

27

Best Start Time Shortest Path

Label Correcting Vs. Label Setting Algorithms



Node expanded

Least cost node Random

Termination Destination expanded No cost updates

# Expansions Once Many

Complexity O(n log m) O(n2m) *

(*) Two-Q Algorithm

Data Structure used – Pair of queues Q1, Q2 Q1 – Set of nodes scanned (expanded) before (repeated expansion)

Q2 – Set of nodes not scanned before (first expansion) Nodes from Q1 are given preference

* S. Pallottino, Shortest Path Methods: Complexity, Interrelations and New Propositions, Networks, 14:257-267, 1984.

28


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).

N1

[0,0,0,0,0]

N2

N3

N4 N5

N5 is selected;

Iteration 1: t=1:CN4(1) > (N4N5(1) + CN5(1+ N4N5(1)))

∞ > (4 + CN1(1+4))

Cu(t) = min(Cu(t), uv(t) + Cv(t+ uv(t) ) (,, , ,)

[4,4,3,3,3]

29


Key Ideas

Label correcting Algorithm for every time instant

Handles non-FIFO travel times

Finds the minimum travel time from all shortest paths

30

Performance Evaluation: Experiment Design

Network Expansion

TAG Based Algorithms Shortest Path Algorithms on Time

Expanded Graph

Data Analysis

Length of Time Series

Real Dataset (without time

series) Road network with travel time series

Run-time Run-time

Time Series Generation

Time expanded network

Goals

1. Compare TAG based algorithms with algorithms based on time expanded graphs (e.g. NETFLO):

- Performance: Run-time

2. Test effect of independent parameters on performance: - Number of nodes, Length of time series

Experiment Platform: CPU: 1.77GHz, RAM: 1GB, OS: UNIX.

Experimental Setup

31

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 dataMn/DOT basemap for MPLS CBD.

32

Comparison of Storage Cost

Memory(Length of time series=150)

100

1100

2100

3100

4100

5100

111 277 562 786

No: of nodes

Sto

rag

e u

nit

s (K

B)

TAG

TEXP

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)

(**) 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

33

Performance Evaluation :Experiment Results 1

Experiment 1: Effect of Number of Nodes

Setup: Fixed length of time series = 100

• TAG based algorithms are faster than time-expanded graph based algorithms.

Shortest Path – Given Start Time Shortest Path – Best Start Time

34

Performance Evaluation : Experiment Results 2

Experiment 2: Effect of Length of time series.

Setup: fixed number of nodes = 786, number of edges = 2106.

Shortest Path – Given Start Time Shortest Path – Best Start Time

• TAG based algorithms run faster than time-expanded graph based algorithms.

35

Comparison of Algorithm Complexity

For a network of n nodes and m edges and a time interval of length T

Algorithm Time Expanded Graph


Best Start TimeShortest Path

O(nT2+mT)T) (*) O(n2mT)(**)

Fixed Start TimeShortest Path

O(nT log T+mT) (*) O(m log T+log n) (**)

(*) B.C. Dean, Algorithms for Minimum Cost Paths in Time-Dependent Networks, Networks, 44(1) pages (41-46), 2004.

(**) 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 (SSTD’07), July 2007.

39

Conclusion

Graph Time Expanded Graph (TEG) & Time Aggregated Graph (TAG)

LP Solver (flow networks)




Two-Q Algorithm,..

BEST-TAG Algorithm


Dijkstra’s Algorithm,..

SP-TAG Algorithm

Lack of optimal prefix

Shortest Path Shortest Path (Fixed Start

Time)

Shortest Path (Best Start Time)

Static Networks

Time-variant Networks

Key Insights Fixed Start time shortest paths – Greedy strategy gives optimal solutions. Flexible Start time – Greedy strategy need not give optimal solution.

(Label correcting method)

40

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

Shortest Path for Fixed Start Time

Shortest Path for Fixed Start Time

Transportation Network Routing Algorithms


41

Future Work

Algorithms Performance Tuning of Best Start Time Algorithm Incorporate capacities on nodes/edges and

develop optimal algorithms for Evacuation Planning.

Incorporate time-dependent turn restrictions in shortest path computation.

Develop ‘frequent route discovery’ algorithms based on TAG framework.

42

Future Work

BEST-TAG Algorithm

Performance Tuning

Current Complexity – O(n2mT) Real datasets Heuristics

Proof of Optimality (all cases)

43

Future Work - Algorithms

Evacuation Planning

Given: (i) A transportation network, a directed graph G = ( N, E ) (ii) Capacity constraints for each edge and node, (iii) Time-dependent travel time for each edge, (iv) Number of evacuees and source nodes (v) Evacuation destinations.

Find : Evacuation plan consisting of a set of origin-destination routes & scheduling of evacuees on each route.

Objective: Minimize evacuation egress time, Computational cost. Optimize evacuation time subject to time-dependent travel times & Capacity constraints.

Problem Statement

44


Frequent route discovery Algorithm Motivation: Crime Analysis Effective patrolling

Routes are time-dependent

Time-dependent schedule of Public transportation

Route discovery on Spatio-temporal networks (Journey-to-crime)*

Explore TAG as a model for Spatio-temporal network data Spatio-temporal data mining.

Crime data is Spatio-temporal

* CrimeStat 3.0, Ned Levine & Associates

45


Shortest Path with time-dependent turn restrictions

Given: (i) A transportation network, a directed graph G = ( N, E ) (ii) Time-dependent travel time for each edge, (iii) Time-dependent turn costs (iv) Source node, Destination nodeFind : Shortest Path from the source to destination

Objective: Minimize Computational cost.

Problem Statement

For each node v, (degee (u) +1)T costs are maintained.

u v

Travel time series

w1w2

u v

w1

w2

tc(u,v,w1)

tc(u,v,w2)

47

Future Work

Spatio-temporal Network Databases

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.

Persistence Shared Interrelated

Three-Schema Architecture

49

References

ESRI, ArcGIS Network Analyst, 2006. Oracle, Oracle Spatial 10g, August 2005. M. Erwig, R.H. Guting, Explicit Graphs in a Functional Model for Spatial

Databases, IEEE Transactions on Knowledge and Data Engineering, 6(5), 1994.

S. Shekhar, D. Liu, Connectivity Clustered Access Method for Networks and Network Analysis, IEEE Transactions on Knowledge and Data Engineering, January, 1997.

L.R. Ford, D.R. Fulkerson, Constructing maximal dynamic flows from static flows, Operations Research, 6:419-433, 1958.

E. Kohler, K. Langtau and M. Skutella, Time expanded graphs for time-dependent travel times, Proc. 10th Annual European Symposium on Algorithms, 2002.

D.E. Kaufman, R.L. Smith, Fastest Path in Time-dependent Networks for Intelligent Vehicle Highway Systems Applications, IVHS Journal, 1(1), 1993.

K. Mouratidis, M. Yiu, D. Papadias, N. Mamoulis. Continuous Nearest Neighbor Monitoring in Road Networks. Proceedings of the Very Large Data Bases Conference (VLDB), pp. 43-54, Seoul, Korea, Sept. 12 - Sept. 15, 2006.

B.C. Dean, Algorithms for Minimum Cost Paths in Time-Dependent Networks, Networks, 44(1) pages (41-46), 2004.

50

Thank you.

Questions and Comments ?

Time-Aggregated Graphs- Modeling Spatio-temporal Networks

Documents

Transcript of Time-Aggregated Graphs- Modeling Spatio-temporal Networks