Date post: | 31-Dec-2015 |
Category: |
Documents |
Upload: | imogene-dixon |
View: | 30 times |
Download: | 3 times |
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.