Time-Aggregated Graphs- Modeling Spatio-temporal Networks

1
Time-Aggregated Graphs-Modeling Spatio-temporal
Networks
Betsy George
Advisor Prof. Shashi Shekhar
Department of Computer Science and Engineering
University of Minnesota
September 7, 2007
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

4
Motivation
Many Applications
Examples Transportation network Routing, Crime
pattern analysis, knowledge discovery from Sensor
data.

• Varying Congestion Levels and turn restrictions
  ? travel time changes.

• ? accurate computation of frequent routing
  queries.

I94 _at_ Hamline Ave at 8AM 10AM

• Identification of frequent routes
• Crime Analysis

• Identification of congested routes
• Network Planning

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)
)
5
Problem Definition

• 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) 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.

7
Related Work

• Spatial Graphs Erwig94, Guting96,
  Mouratidis06, Shekhar97
• 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)Ford58,
  Kaufman93, Kohler02,Dean04
• 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
Snapshots at t1,2,3,4,5
Time Expanded Graph
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.

10
Limitations of Related Work

• Time Expanded Graph

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

• Computationally expensive Algorithms
• Increased Network size.

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

• Lack of physical independence of data.

11
Limitations of Related Work
Time Expanded Graphs cannot model this
dynamic relationship since it does not involve
a flow.
12
Our Contributions
Static Networks Time-variant Networks
Graph Time Expanded Graph (TEG) Time Aggregated Graph (TAG) Time Expanded Graph (TEG) Time Aggregated Graph (TAG)
LP Solver (flow networks) Flow algorithms based on LP Flow algorithms based on LP
Label Correcting Algorithms Two-Q Algorithm,.. BEST-TAG Algorithm
Label Setting Algorithms Dijkstras Algorithm,.. SP-TAG Algorithm Lack of optimal prefix
Shortest Path Shortest Path (Fixed Start Time) Shortest Path (Best Start Time)
13
Our Contributions
14
Time Aggregated Graph
Snapshots of a Network at t1,2,3,4,5
Time Aggregated Graph

• Attributes are aggregated over edges and nodes.

N2
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
N3
mi- travel time at ti
15
Time Aggregated Graph
ew1,..,ewT
TAG (N,E,T,
nw1nwT ,
nwi N? RT,
ewi E? RT
16
Case Study -Routing Algorithms
Finding the shortest path from N1 to N5..
Start at t3
Start at t1
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!!
17
Shortest Path Algorithm for Given Start Time
Challenges
(1) Not all shortest paths show optimal
substructure.
For start time t1
N1- N2- N4- N5
(1)
t1
t2
t7
t3
wait till t5 !!
Lemma At least one optimal path satisfies the
optimal substructure property. N1-N2-N4-N5 in the
example has optimal prefixes.
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Shortest Path Algorithm for Given Start Time
Challenge-1
(1) Not all shortest paths show optimal
substructure.
Lemma At least one optimal path satisfies the
optimal substructure property.
? Greedy algorithm can be used to find the
shortest path.
19
Shortest Path Algorithm for Given Start Time
Challenges
(2) Correctness Determining when to traverse an
edge.
When to traverse the edge N2-N3 for start time
t1 at N1? Traversing N2-N3 as soon as N2 is
reached, would give sub optimal solution.
Assume FIFO travel times.
(3) Termination of the algorithm An infinite
non-negative cycle over time
Finite time windows are assumed.
20
Shortest Path Algorithm for Given Start Time
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.

Source N1 Destination N5 time t1
(3)
(8)
8
1
8
8
8
N2
?,1,1,1,1
1,1,1,1,1
8
8
1
3
3
2,?, ?, ?,2
N4
N5
(1)
N1
8
1
3
4
3
(8)
(4)
(8)
(7)
2,2,2,2,2
2,2,2,2,2
8
1
3
4
3
N3
(8)
(3)
1
3
3
4
7
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Shortest Path Algorithm for Given Start Time

• Initialize
• cs 0 ?v (? s), cv 8.
• 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), cu)
• // min_t finds the
  earliest departure time for (u,v)
• If t ?u,v(t) lt cv
• cv t ?u,v(t)
• parentv u
• insert v in Q if it
  is not in Q
• Update Q.

22
Shortest Path Algorithm for Given Start Time

• 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 Dijkstras algorithm.
• Difference - Earliest availability of edge
• - Admissible guarantees optimality

23
Analytical Evaluation

• Computational Complexity

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))

• Dijkstras 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))

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

• Lemma Time-aggregated graph performs
  asymptotically better than time expanded graphs
  when log (n) lt 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.

A Best Start Time!!
Path N1 N2 N4 N5
Start Time
1
2
3
4
5
Arrival Time
7
7
7
8
9
Time Spent
6
5
4
4
4
26
Best Start Time Shortest Path Algorithm
Challenges
(1) Best Start Time shortest paths need not have
optimal prefixes.
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.
(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

Label Setting Algorithms Label Correcting 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, 14257-267, 1984.
28
Best Start Time Shortest Path Algorithm
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).

Cu(t) min(Cu(t), ?uv(t) Cv(t ?uv(t) )
N2
N5 is selected
0,0,0,0,0
N4
N5
N1
Iteration 1 t1 CN4(1) gt (?N4N5(1) CN5(1
?N4N5(1)))
4,4,3,3,3
N3
8 gt (4 CN1(14))
29
Best Start Time Shortest Path Algorithm

• 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
Experimental Setup
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.
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 data Mn/DOT basemap for MPLS CBD.
32
Comparison of Storage Cost

• 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(nmkT)

• For a Time Expanded Graph,

• Storage requirement is O(nT) O(nm)T ()

• Experimental Evaluation

33
Performance Evaluation Experiment Results 1
Experiment 1 Effect of Number of Nodes Setup
Fixed length of time series 100
Shortest Path Best Start Time
Shortest Path Given Start Time

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

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 Best Start Time
Shortest Path Given 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 Time Aggregated Graph
Best Start Time Shortest Path O(nT2mT)T) () O(n2mT)()
Fixed Start Time Shortest Path O(nT log TmT) () O(m log Tlog 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 (SSTD07), July 2007.
36
Case Study - Sensor Data Representation
37
Case Studies on Sensor TAG ()

• Anomaly Detection

• Sensors that display measured values different
  from expected values are identified.

• Basic Hotspot Detection

• Sensor nodes that show anomalies (hotspots) are
  detected through a search of the Sensor TAG

• Growing Hotspot Detection

• Increase in the number of sensors that report an
  anomaly is predicted based on the physical
  attributes modeled on the edges of Sensor TAG.

B. George, J. Kang, S. Shekhar, STSG A Data
Model for Representation of Sand Knowledge
Discovery in Sensor Data, Proceedings of ACM
SIGKDD Workshop on Knowledge Discovery in Sensor
Data, August, 2007.
38
Representation of a Dynamic Relationship
tt9
Relationship between Two Objects at Various
Instants
Dynamic Relationship expressed in TAG
39
Conclusion
Static Networks Time-variant Networks
Graph Time Expanded Graph (TEG) Time Aggregated Graph (TAG) Time Expanded Graph (TEG) Time Aggregated Graph (TAG)
LP Solver (flow networks) Flow algorithms based on LP Flow algorithms based on LP
Label Correcting Algorithms Two-Q Algorithm,.. BEST-TAG Algorithm
Label Setting Algorithms Dijkstras Algorithm,.. SP-TAG Algorithm Lack of optimal prefix
Shortest Path Shortest Path (Fixed Start Time) Shortest Path (Best Start Time)

• 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

• Transportation Network Routing Algorithms

• Shortest Path for Fixed Start Time

• Shortest Path for Fixed Start Time

• Sensor Data Representation

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

• Problem Statement

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.

44
Future Work - Algorithms
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)

• Crime data is Spatio-temporal

? Explore TAG as a model for Spatio-temporal
network data ? Spatio-temporal data mining.
CrimeStat 3.0, Ned Levine Associates
45
Future Work - Algorithms
Shortest Path with time-dependent turn
restrictions

• Problem Statement

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 node Find
Shortest Path from the source to
destination Objective Minimize Computational
cost.
For each node v, (degee (u) 1)T costs are
maintained.
46
Future Work - Algorithms
Turn restriction Node Expansion
47
Future Work

• Spatio-temporal Network Databases

• Three-Schema Architecture

48
Future Work
Logical Model
A Sample Set of Logical Operators
Operator Snapshot Time-Aggregate
get get(node,time) getNodeSeries(node)
getEdge getEdge(node1,node2,time) getEdgeSeries(node1,node2)
get_node_Presence getNodePresence(node,time) getNodeSeries(node)
get_edge_Presence getEdgePresence(node1,node2,time) getEdgeSeries(node1,node2)
get_Graph get_Graph(time) get_Graph()
49
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Thank you. Questions and Comments ?