View
213
Download
0
Tags:
Embed Size (px)
Citation preview
Time-Variant Spatial Network Model
Vijay Gandhi, Betsy George(Group : G04)
Group ProjectOverview of Database Research
Fall 2006
Outline
Motivation Related Work Problem Statement Contributions Future Work
Motivation
A network is a system that allows its abstractrepresentation as a graph
• Most networks change with time
Problem Definition
Given : A set of frequent queries posed on a Network; Changes in the network with time.
Output : A model that supports efficient and correct algorithms for computing the query results.
Objective : Minimize storage and computation costs.
Constraints : Edge cost are positive.
Related Work: Time Expanded Graphs
N1 N2
N3 N4
1
1
2 2
N1 N2
N3 N4
1
2 2
N1 N2
N3 N44
2 2
t = 1 t = 3t = 2
N1
N2
N3
N4
Time ExpandedGraphRepresentation
Graph changingover time
A Conceptual Model
t = 1
N1 N2
N3 N4
1
1
2 2
N1 N2
N3 N4
1
2 2
N1 N2
N3 N44
2 2
t = 3t = 2
• Time series of snapshots
A Physical Model
Time Aggregated Graph
1,1,
2,2,3
N1 N2
N3 N41, ,4
2,2,2
88
• Attributes are aggregated over edges and nodes.
N1 N2
N3 N4
1
1
2 2
N1 N2
N3 N4
1
2 2
N1 N2
N3 N44
2 3
t = 3t = 2
Physical Model - Finding the shortest path
Challenge • Not all shortest paths display optimal substructure.
Lemma 1: If there is a path from source to destination in the time aggregated graph then there is at least one optimal path which satisfies the optimal substructure
property.
How do we address this?
11
22
2
[1,4]
N2
N3
N4N1 N5
Physical Model - Finding the shortest path
Uses a greedy algorithm to find the
shortest path
At every step, Picks the node with the least cost .Chooses the edge that is available at the
earliest.
Time Dependency of Shortest Paths
Start time
Shortest Path Travel time
t=1 N1-N2-N3 6
t=2 N1-N2-N3 5
t=3 N1-N2-N3 4
• Shortest Path Travel time is time-dependent
Best-Start-Time Shortest Path
Shortest Paths
N2N1 N31,2,2,2,2,2 2,--,--,--,2,2
Best Start Time Shortest Paths
Challenges Prefixes need not be optimal Optimal prefix can lead to longer waits
N2N1 N31,2,2,2,2,2 2,--,--,--,2,2
Start time
Shortest Path
Travel time
t=1 N1-N2-N3 6
t=3 N1-N2-N3 4
Best Start Time Shortest Paths
Strategies
Minimize waits by postponing the start of the journey
Reduce search space by forming path classes
• Journeys with • consecutive start times• Same travel times
Pruning
• Node count in a shortest path is bounded by n.• Eliminate irrelevant journeys
Experimental Design
Experimental Results
Shortest Path Algorithm(Length of time series=1500)
0.1
1
10
100
1000
10000
111 277 562 786
No: of nodes
Run
Tim
e (lo
g sc
ale) TAG
TEXP
Experimental Results
Least Travel time Algorithm(Length of time series = 150)
1
10
100
1000
111 277 562 786
No: of Nodes
Algo
rithm
Run
-tim
e (lo
g sc
ale)
TAG
Memory(Length of time series=150)
100
1100
2100
3100
4100
5100
111 277 562 786
No: of nodes
Stor
age u
nits
(KB) TAG
TEXP
A Logical Model
• Examples of Logical Operators
Future Work
Heuristics need to be explored for Best-start-time path algorithm to improve the performance.
Add spatial properties to nodes, edges
Extend spatial graph indexing methods to time-aggregated graphs
Incorporate capacities on edges.
Questions?