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?