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Time-Aggregated Graphs- Modeling Spatio-temporal Networks

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Time-Aggregated Graphs-Modeling Spatio-temporal Networks

Prof. Shashi Shekhar

Department of Computer Science and Engineering

University of Minnesota

August 29, 2008

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.

- Introduction
- Motivation
- Problem Statement
- Related Work

- Contributions

- Representation

- Routing Algorithms

- Conclusion and Future Work

U.P.S. Embraces High-Tech Delivery Methods (July 12, 2007)By Claudia H. Deutsch

“The research at U.P.S. is paying off. ……..— savingroughly three million gallons of fuel in good part by mapping routes that minimize left turns.”

9 PM, November 19, 2007

Sensors on Minneapolis Highway Network periodically report time varying traffic

4 PM, November 19, 2007

7 PM, November 19, 2007

3) Knowledge discovery from Sensor data.

- Spreading Hotspots

1) Transportation network Routing

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

2) Crime Analysis

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

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.

Signal delays at left turns cancause non-FIFO travel times.

Pictures Courtesy: http://safety.transportation.org

- Input :
a) A Spatial Network

b) Temporal changes of the network topology and parameters.

- Output : A model that supports efficient correct algorithms for computing the query results.

- Objective : Minimize storage and computation costs.

- Constraints :
(i) Predictable future

(ii) Changes occur at discrete instants of time,

(iii) Logical & Physical independence,

- Key assumptions violated.

- Ex., Prefix optimality of shortest paths
- (greedy property behind Dijkstra’s algorithm..)

- Conflicting Requirements

- Expressive Power

- Storage Efficiency

- New and alternative semantics for common graph operations.

- Ex.,Shortest Paths are time dependent.

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Edge:

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Travel time

Holdover Edge

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Transfer Edges

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t=4

t=6

t=7

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t=5

t=2

(1) Snapshot Model

[Guting04]

(2) Time Expanded Graph (TEG)

[Kohler02, Ford65]

High Storage Overhead

Redundancy of nodes across time-frames

Additional edges across time frames in TEG.

- Computationally expensive Algorithms
- Increased Network size due to redundancy.

- Inadequate support for modeling non-flow parameters on edges in TEG.

- Lack of physical independence of data in TEG.

- Introduction
- Motivation
- Problem Statement
- Related Work

- Contributions

- Representation

- Time Aggregated Graph (TAG)

- Case Studies

- Routing Algorithms

- Conclusion and Future Work

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Node:

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N..

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Edge:

Travel time

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t=4

t=5

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

Time Aggregated Graph

- Attributes are aggregated over edges and nodes.

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Node

[,1,1,1,1]

[1,1,1,1,1]

[2,, , ,2]

N4

N5

N1

Edge

[m1,…..,(mT]

[2,2,2,2,2]

[2,2,2,2,2]

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mi- travel time at t=i

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.

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[,1,1,1,1]

On edge N4-N5

* [2,∞,∞,∞,2] is a time series of attribute;

[1,1,1,1,1]

[2,, , ,2]

N4

N5

N1

* At t=1, the edge has an attribute value of 2.

[2,2,2,2,2]

[2,2,2,2,2]

* At t=2, the ‘∞’ can indicate the absence of connectivity between the nodes at t=2.

N3

[ew1,..,ewT ] |

TAG = (N,E,T,

[nw1…nwT ],

nwi : N RT,

ewi : E RT

Minneapolis CBD [1/2, 1, 2, 3 miles radii]

- Road data
- Mn/DOT basemap for MPLS CBD.

(*) All edge and node parameters might not display time-dependence.

(**) D. Sawitski, Implicit Maximization of Flows over Time, Technical Report (R:01276),University of Dortmund, 2004.

- 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)

- 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.

- Introduction
- Motivation
- Problem Statement
- Related Work

- Contributions

- Representation

- Time Aggregated Graph (TAG)

- Routing Algorithms

- Conclusion and Future Work

- Violation of optimal prefix property

- Not all optimal paths show optimal prefix property.

- New and Alternate semantics

- Termination of the algorithm: an infinite non-negative cycle over time

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Find the shortest path travel time from N1 to N5 for start time t = 1.

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Solution: Reaches N5 at t=8.

Total time = 7

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∞

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∞

Optimal path: Reach N4 at t=3;

Wait for t=4;

Reach N5 at t=6

Total time = 5

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Dijkstra’s, A*….

Stationary

Predictable Future

Special case (FIFO)

[Kanoulas07]

Non-stationary

LP, Label-correcting Alg. on TEG

General Case

Unpredictable Future

[Orda91, Kohler02, Pallotino98]

SP-TAG, SP-TAG*,CapeCod

Limitations:

Label correcting algorithm over long time periods and large networks is computationally expensive.

LP algorithms are costly.

Time Aggregated Graph (TAG)

- Representation

- Routing Algorithms

- Shortest Path for a given start time

in general (FIFO & non-FIFO) Networks

- Analytical & Experimental Evaluation

Start time = 1; Start node : N1

Iteration 1: N1_1 selected

N1_2 = 2; N2_2 = 2; N3_3 = 3

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 selected

N4_5 = 5; N5_6 = 6

- Selection of node to expand is random.

- Algorithm terminates when no node gets updated.

N1

N2

N3

N4

N5

t=8

t=3

t=4

t=6

t=7

t=2

t=1

t=5

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

(*) Cherkassky 93,Zhan01, Ziliaskopoulos97

N2

N2

[1,1,1,1,1]

[1,1,1,1,1]

[2,3,4,5,6]

[2,3,4,5,6]

[1,2,5,2,2]

[2,4,8,6,7]

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[3,4,5,6,7]

[2,2,2,2,2]

[3,4,5,6,7]

[2,2,2,2,2]

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[2,3,4,5,6]

[2,3,4,5,6]

[2,4,6,6,7]

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[3,4,5,6,7]

[3,4,5,6,7]

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When start time is fixed, earliest arrival least travel time

(Shortest path)

Arrival Time Series Transformation (ATST) the network:

travel times arrival times at end node Min. arrival time series

Result is a Stationary TAG.

Greedy strategy (on cost of node, earliest arrival) works!!

N2

Select Minimum {Cost of edge ij }

[2,3,4,5,6]

[2,3,4,5,6]

t ≥ arrival at i

[2,4,8,6,6]

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N5

N1

[3,4,5,6,7]

[3,4,5,6,7]

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Greedy strategy on transformed TAG:

Cost of a node = Arrival time at the node

Expand the node with least cost.

Update costs of adjacent nodes.

Trace of NF-SP-TAG Algorithm

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- 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.

for t1 < t2

[aij(t)]

≤

[aij(t)]

Minimum

Minimum

t t1

t t2

NF-SP-TAG Algorithm is correct.

- 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.

Maintains admissibility since

- Computational Complexity

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

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

Length of Time Series

Real Dataset (without time series)

Time Series Generation

Road network with travel time series

Network Expansion

TAG Based Algorithms

Shortest Path Algorithms on Time Expanded Graph

Run-time

Run-time

Data Analysis

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

Time expanded network

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

Setup: Fixed length of time series = 100

Experiment 2: Effect of Length of time series.

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

Experiment 1

Experiment 2

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

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.

- 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.

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Node:

t=4

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t=5

Edge:

Travel time

Finding the shortest path from N1 to N5..

Start at t=3:

Start at t=1:

Shortest Path is N1-N2-N4-N5; Travel time is 4 units.

Shortest Path is N1-N3-N4-N5; Travel time is 6 units.

Fixed Start Time Shortest Path

Least Travel Time (Best Start Time)

Shortest Path is dependent on start time!!

- Time Aggregated Graph (TAG)

- Routing Algorithms

- Formulate new algorithms.

- Incorporate time-dependent turn restrictions in shortest path computation.
- Develop ‘frequent route discovery’ algorithms based on TAG framework.

Thank you.