Time-Aggregated Graphs- Modeling Spatio-temporal Networks

1
Time-Aggregated Graphs-Modeling Spatio-temporal
Networks
Prof. Shashi Shekhar
Department of Computer Science and Engineering
University of Minnesota
August 29, 2008
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

• Representation

• Routing Algorithms

• Conclusion and Future Work

4
Motivation
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

5
Motivation
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 can cause non-FIFO
travel times.
Pictures Courtesy http//safety.transportation.or
g
6
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) Predictable future
• (ii) Changes occur at discrete instants of
  time,
• (iii) Logical Physical independence,

7
Problem Definition (contd.)

• Predictable Future
• Values of edge attributes largely predictable

• Assumption not unreasonable in planning scenarios

• Operational scenarios reasonable in the absence
  of random events (ex., public transportation
  scheduling)

8
Challenges in Representation

• Conflicting Requirements

• Expressive Power

• Storage Efficiency

• New and alternative semantics for common graph
  operations.

• Ex., Shortest Paths are time dependent.

9
Related Work in Representation
(1) Snapshot Model
Guting04
(2) Time Expanded Graph (TEG)
Kohler02, Ford65
10
Limitations of Related Work

• 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.

11
Outline

• Introduction
• Motivation
• Problem Statement
• Related Work

• Contributions

• Representation

• Time Aggregated Graph (TAG)

• Case Studies

• Routing Algorithms

• Conclusion and Future Work

12
Proposed Approach
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
13
Time Aggregated Graph
ew1,..,ewT
TAG (N,E,T,
nw1nwT ,
nwi N? RT,
ewi E? RT
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 data Mn/DOT basemap for MPLS CBD.
15
TAG Storage Cost Comparison

• 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(nmkT)

• For a Time Expanded Graph,

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

• Experimental Evaluation

• Storage cost of TAG is less than that of TEG if k
  ltlt m.

• TAG can benefit from time series compression.

16
Outline

• Introduction
• Motivation
• Problem Statement
• Related Work

• Contributions

• Representation

• Time Aggregated Graph (TAG)

• Routing Algorithms

• Conclusion and Future Work

17
Routing Algorithms- Challenges

• 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

18
Routing Algorithms- Challenges
Find the shortest path travel time from N1 to N5
for start time t 1.
N1
N2
N5
N3
N4
Solution Reaches N5 at t8. Total
time 7
1
8
1
8
8
8
Optimal path Reach N4 at t3
Wait for t4
Reach N5 at t6 Total
time 5
2
1
8
3
2
8
3
3
3
2
1
8
4
3
2
3
1
8
5
3
3
1
2
8
19
Routing Algorithms Related Work
SP-TAG, SP-TAG,CapeCod
Limitations
Label correcting algorithm over long time periods
and large networks is computationally expensive.
LP algorithms are costly.
20
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.
21
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.
FIFO travel times ? Greedy algorithms, A search
Non-FIFO travel times ? ATST transformation ?
Greedy Algorithm
(3) Termination of the algorithm An infinite
non-negative cycle over time
Finite time windows are assumed.
22
SP-TAG Algorithm for Given Start Time
Greedy Algorithm (SP-TAG) for FIFO

• 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
23
SP-TAG 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.

24
SP-TAG Algorithm for a 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

• 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))

25
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))

• 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.
26
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.
27
SP-TAG (A based Algorithm) for FIFO
Cost function f(n) g(n) h(n)
g(n) Actual cost from the source node to node n
h(n) Estimated cost node n to the destination
node n
Actual Cost g(n) Arrival time at node n
Heuristic function h(n)
Lemma 1 Heuristic function h(n) is admissible.
? A search will result in an optimal solution.
Lemma 2 Heuristic function h(n) is monotone.
? Search is optimal ? Closed nodes are not
reopened.
28
SP-TAG
Lemma Heuristic function h(n) is admissible.
Proof
S_TAG Static network derived from TAG with
minimum travel time on each edge. Let P be the
shortest path from node i to the destination
d. Shortest path travel time SP min ? dpq
min
pq ? P
Let P(t) be the shortest path in TAG that starts
at i at time t.
P(t) is a feasible path in S_TAG.
29
SP-TAG
Lemma Heuristic function h(n) is monotone.
Proof
A heuristic is monotone if h(i) ? dij h(j),
?ij ? E
min
min
Since dij ? dij (t),
30
SP-TAG
min
SPi-d for every node i ? d

• Preprocess to find the
• Initialize
• As tstart ?u (? s), Au
  8, fu 8.
• fs SPs C ? S s .
• while u ? d do
• u extract_min(Q) close u (C C
  ? u) S S u
• for each node v adjacent to u do
• if fv gt Au duv(Au)
  SPvd(Au)
• Av Ai
  du,v(Au)
• fv Au
  duv(Au) SPvd(Au)
• S S ? v if it is
  not in Q

31
SP-TAG Execution Trace
To find the shortest path from N1 to N5 for start
time t 1
Heuristic h SPminN1,N2,N3,N4 4,3,4,2
f(N1) g(N1) h(N1) 1 4 5 f 8
32
Our Contributions
Time Aggregated Graph (TAG)

• Representation

• Routing Algorithms

• Shortest Path for a given start time

in general (FIFO non-FIFO) Networks

• Analytical Experimental Evaluation

33
Related Work Label Correcting Approach()

• Selection of node to expand is random.

• Algorithm terminates when no node gets updated.

N1
N2
N3
N4
N5
t8
t3
t4
t6
t7
t1
t2
t5

• Implementation used the Two-Q version O(n2T
  3(nm)

() Cherkassky 93,Zhan01, Ziliaskopoulos97
34
Proposed Approach Key Idea
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!!
35
SP Algorithm in Non-FIFO Networks (NF-SP-TAG)
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
N1
N2
N5
N3
N4
1
8
8
1
8
8
2
1
8
3
2
8
3
3
3
1
2
8
4
3
3
2
1
8
5
3
2
3
1
6
36
NF-SP-TAG Algorithm- Pseudocode

• Pre-process the network.
• Initialize
• cs t_start ?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_arrival((u,v), cu)
  
• 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.

37
NF-SP-TAG Algorithm - Correctness
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
38
NF-SP-TAG Analytical Evaluation

• 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(nm)

39
Performance Evaluation Experiment Design
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
40
Performance Evaluation - Results
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.

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
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!!
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
45
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.
46
CP-NF-BEST (Best Start Time)

• Key Ideas
• NF-SP-TAG for each start time
• Handles non-FIFO travel times
• Maintains copies of nodes ? logical concurrency
• Terminates when a copy of destination is expanded.

• Algorithm is correct
• NF-SP-TAG correctly computes shortest path for
  every instant.
• Shortest path is a function of start time and
  network parameters.
• Since the algorithm computes the SP for every
  start time,
• it finds the least travel time.

47
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

48
Best Start Time Shortest Path Algorithm
Time Iterated SP-TAG Algorithm for FIFO Networks
(TI-SP-TAG)

• Key Ideas
• SP-TAG (A based) iterated for every start time.
• Handles FIFO travel times
• Finds the minimum travel time from all shortest
  paths
• Performance optimization Re-use heuristic costs
  from previous iterations.

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.