Transcript Title: Your Title - University of Louisiana at Lafayette

Scalable and Fully Distributed
Localization With Mere Connectivity
Localization with Mere Connectivity

Localization is imperative to a variety of
applications in wireless sensor networks.

Considering the cost of extra equipment,
localization from mere connectivity is
ideal for large-scale sensor networks.
Previous Methods

Previous mere connectivity based
localization methods:
◦ Multi-Dimensional Scaling (MDS) based
 low scalability
 centralized
◦ Neural Network based
 numerically unstable
◦ Graph Rigidity Theory based
 low localization accuracy
Our Proposed Method

We model a planar sensor network as a discrete
surface with its metric represented as edge lengths
(approximated by hop counts) and curvatures at each
node as the angle deficits. Due to the approximation
error, the surface is curved, not flat in plane. We
compute the provably optimal flat metric which
introduces the least distortion from estimated metric to
isometrically embed the surface to plane.
Contributions of Our Proposed
Method

Fully distributed
◦ All the involved computations for each node only
require information from its direct neighbors

Scalable
◦ The computational cost and communication cost are
both linear to the size of the network
◦ Limited error propagation

Theoretically sound
◦ Provably optimal

Numerically stable
◦ Free of the choice of initial values and local minima
with theoretical guarantee
Talk Overview
Theory of flat metric
 Distributed algorithm on sensor network
 Discussions

◦ Costs
◦ Error propagation

Experiments and Comparison
Talk Overview
Theory of flat metric
 Distributed algorithm on sensor network
 Discussions

◦ Costs
◦ Error propagation

Experiments and Comparison
Discrete Metric and Discrete
Gaussian Curvature
Discrete metric: edge length of the
triangulation
 Discrete Gaussian curvature: induced by
metric, measured as angle deficit

Circle Packing Metric

Flat Metric
Flat metric: a set of edge lengths which
induce zero Gaussian curvature for all
inner vertices such that the triangulation
can be isometrically embedded into plane.
 Infinite number of flat metrics for a given
connectivity triangulation.

Optimal Flat Metric

Tool to Compute Optimal Flat Metric

Discrete Ricci flow
◦ [Hamilton 1982]: Ricci flow on closed
surfaces of non-positive Euler characteristic
◦ [Chow 1991]: Ricci flow on closed surfaces of
positive Euler characteristic
◦ [Chow and Luo 2003]: Discrete Ricci flow
including existence of solutions, criteria, and
convergence.
◦ [Jin et al. 2008]: a unified framework of
computational algorithms for discrete Ricci
flow
Tool to Compute Optimal Flat Metric

Talk Overview
Theory of flat metric
 Distributed algorithm on sensor network
 Discussions

◦ Costs
◦ Error propagation

Experiments and Comparison
Distributed Algorithm

Preprocessing: build a triangulation from
network graph
◦ The dual of the landmark-based Voronoi
diagram
Distributed Algorithm

Computing optimal flat metric (edge
length)
◦ All edge lengths initially = unit
 The curvature of each vertex is not zero
 The initial triangulation surface can’t be embedded
in plane.
◦ Find the flat metric, such that
 The triangulation surface can be isometrically
embedded in plan
 The introduced localization error is minimal
Distributed Algorithm

Distributed Algorithm

Isometric embedding
◦ Start embedding from one vertex
◦ Continuously propagate to the whole triangulation
Talk Overview
Theory of flat metric
 Distributed algorithm on sensor network
 Discussions

◦ Costs
◦ Error propagation

Experiments and Comparison
Costs

Preprocessing step - building triangulation
◦ Time complexity:
◦ Communication cost:

Step 1 – computing flat metric
◦ Time complexity:
◦ Communication cost:

Step II – isometric embedding
◦ Time complexity:
◦ Communication cost:
Convergence Rate of Step I

Number of iterations =
Error Propagation

If error is introduced to the estimated or
measured metric in a small area, error will
propagate to the entire network for general
localization methods.

The impact to our computing optimal flat
metric based method can be modeled as a
discrete Green function:
Testing of Error Propagation

One scenario: a much longer distance measurement
around one vertex due to slower response of the node
to its neighboring nodes’ signals.
Talk Overview
Theory of flat metric
 Distributed algorithm on sensor network
 Discussions

◦ Costs
◦ Error propagation

Experiments and Comparison
Experiments and Comparison

Variant Density
Experiments and Comparison

Different transmission models
Trans. models
UDG
Quasi-UDG
Log-Norm
Probability
Localization error
0.25
0.34
0.42
0.43
Experiments and Comparison

Non-uniform distribution
Non-uniform distribution
Density ranging from 11.3 to 18.7
Localization error
0.46
Experiments and Comparison

Comparison with other methods on
networks with different topologies
Networks
Flat Metric
MDS-MAP
MDS-MAP(P)
C-CCA
D-CCA
0.29
2.52
0.89
2.10
0.88
0.32
0.56
0.68
0.71
0.69
0.48
0.62
0.75
0.72
0.64
0.55
1.18
0.61
0.78
0.70
0.63
1.27
0.99
1.17
0.8
Summary and Future Work

Mere connectivity based localization
method
◦ Theoretically guaranteed and numerically stable
◦ Fully distributed and highly scalable with linear
computation time and communication cost and
dramatically decreased error propagation

Future work: localization for sensor
nodes deployed on general 3D surface