1

Time-Aggregated Graphs-Modeling Spatio-temporal Networks

August 29, 2008

Department of Computer Science and Engineering University of Minnesota

Prof. Shashi Shekhar

2

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.

3

Outline

Introduction Motivation Problem Statement Related Work

Contributions

Conclusion and Future Work

Representation

Routing Algorithms

4

Motivation

Delays at signals, turns, Varying Congestion Levels travel time changes.

1) Transportation network Routing

U.P.S. Embraces High-Tech Delivery Methods (July 12, 2007) By Claudia H. Deutsch“The research at U.P.S. is paying off. ……..— saving roughly three million gallons of fuel in good part by mapping routes that minimize left turns.”

Identification of frequent routes (i.e.) Journey to Crime

2) Crime Analysis

3) Knowledge discovery from Sensor data. Spreading Hotspots

9 PM, November 19, 2007

4 PM, November 19, 2007Sensors on Minneapolis Highway

Network periodically report time varying traffic

7 PM, November 19, 2007

5

Motivation

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

6

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,

8

Challenges in Representation

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 (greedy property behind Dijkstra’s algorithm..)

9

Related Work in Representation

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:

(2) Time Expanded Graph (TEG)

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

(1) Snapshot Model

[Guting04]

[Kohler02, Ford65]

10

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.

11

Outline

Introduction Motivation Problem Statement Related Work

Contributions

Conclusion and Future Work

Representation

Case Studies Routing Algorithms

Time Aggregated Graph (TAG)

12

Proposed Approach

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

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

N4 N5

[m1,…..,(mT]

mi- travel time at t=i

Edge

N..

Node

Attributes are aggregated over edges and nodes.

13

Time Aggregated Graph

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

14

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.

15

TAG: Storage Cost Comparison

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 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.

16

Outline

Introduction Motivation Problem Statement Related Work

Contributions

Conclusion and Future Work

Representation

Routing Algorithms

Time Aggregated Graph (TAG)

17

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.

18

Routing Algorithms- Challenges

t=1

N2

N1

N3

N4 N5

1

1

22

t=2

N2

N1

N3

N4 N5

1

22

1

t=3

N2

N1

N3

N4 N5

1

22

1

t=5

N2

N1

N3

N4 N5

1

1

22

1

12 5

t=4

N2

N1

N3

N4 N5

1

22

1

2

N1

1 ∞

2

1

3

3

3

N2 N5 N3 N4

1

1

2

2

∞ ∞ ∞

3

∞∞

∞

4 31 2 3 ∞

5 31 2 3 8

Solution: Reaches N5 at t=8.

Total time = 7Optimal 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.

19

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.

32

Our Contributions

Time Aggregated Graph (TAG)

Shortest Path for a given start time

Analytical & Experimental Evaluation

Representation

Routing Algorithms

in general (FIFO & non-FIFO) Networks

33

Related Work – Label Correcting Approach(*)

t=1 t=2 t=3 t=4 t=5 t=6 t=7

N1

N2

N3

N4

N5t=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 selectedN4_5 = 5; N5_6 = 6

Algorithm terminates when no node gets updated.

(*) Cherkassky 93,Zhan01, Ziliaskopoulos97

Implementation used the Two-Q version [O(n2T 3(n+m)]

34

Proposed Approach – Key Idea

Arrival Time Series Transformation (ATST) the network:

N2

N1

N3

N4 N5

[1,1,1,1,1] [1,1,1,1,1]

[2,2,2,2,2] [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)

35

SP Algorithm in Non-FIFO Networks (NF-SP-TAG)

Greedy strategy on transformed TAG:

Cost of a node = Arrival time at the nodeExpand the node with least cost.

Update costs of adjacent nodes.

Select Minimum {Cost of edge ij } t ≥ arrival at

i

N2

N1

N3

N4 N5

[2,3,4,5,6]

[3,4,5,6,7]

[2,3,4,5,6]

[2,4,8,6,6]

[3,4,5,6,7]

Trace of NF-SP-TAG Algorithm

N1

1 ∞

2

1

3

3

3

N2 N5 N3 N4

1

1

2

2

∞ ∞ ∞

3

∞∞

∞

4 31 2 3 ∞

5 31 2 3 6

36

NF-SP-TAG Algorithm- Pseudocode

Pre-process the network. Initialize c[s] = t_start; 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_arrival((u,v), c[u]); 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.

37

NF-SP-TAG Algorithm - Correctness

Earliest arrival for a given start time Shortest path

If it is not, it contradicts “the earliest arrival”.

Algorithm picks the node with the least cost

Ensures admissibility.

Algorithm updates the nodes based on the minimum arrival time.

NF-SP-TAG Algorithm is correct.

Maintains admissibility sinceMinimum

t t1[aij(t)] Minimum

t t2[aij(t)]≤ for t1 < t2

38

NF-SP-TAG: Analytical Evaluation

Computational Complexity

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

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

For every node extracted, Earliest arrival lookup – O(T)

Priority queue update – O(log n)

Overall Complexity = O(degree(v). (T + log n)) = O(m( T+ log n))

Complexity of label correcting algorithm is O(n2T3(n+m)]

39

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, average node degree.

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

Experimental Setup

40

Performance Evaluation - Results

Experiment 1: Effect of Number of Nodes (Fixed Start Time)

Setup: Fixed length of time series = 100

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

Experiment 2: Effect of Length of time series.

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

Experiment 1 Experiment 2

41

Performance Evaluation - Results

Experiment 3: Effect of Average Degree of Network.

Setup: Length of time series= 240.

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

42

Conclusions

Time Aggregated Graph (TAG) Time series representation of edge/node properties Non-redundant representation Often less storage, less computation time

Routing Algorithms

Faster shortest path for fixed start time in general (FIFO & non-FIFO networks.

43

Routing Algorithms – Alternate Semantics

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..

44

Contributions (Broader Picture)

Time Aggregated Graph (TAG)

Routing Algorithms

FIFO Non-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

49

Future Work

Formulate new algorithms.

Incorporate time-dependent turn restrictions in shortest path computation.

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

50

Thank you.