Scalable and Fully Distributed Scalable and Fully Distributed Localization With Mere Localization With Mere ConnectivityConnectivity

Localization with Mere Localization with Mere ConnectivityConnectivityLocalization 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 MethodsPrevious 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 MethodOur Proposed MethodWe 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 Contributions of Our Proposed MethodMethod

Fully distributed◦ All the involved computations for each node only

require information from its direct neighborsScalable

◦ The computational cost and communication cost are both linear to the size of the network

◦ Limited error propagationTheoretically sound

◦ Provably optimalNumerically stable

◦ Free of the choice of initial values and local minima with theoretical guarantee

Talk OverviewTalk OverviewTheory of flat metricDistributed algorithm on sensor

networkDiscussions

◦Costs◦Error propagation

Experiments and Comparison

Talk OverviewTalk OverviewTheory of flat metricDistributed algorithm on sensor

networkDiscussions

◦Costs◦Error propagation

Experiments and Comparison

Discrete Metric and Discrete Discrete Metric and Discrete Gaussian CurvatureGaussian CurvatureDiscrete metric: edge length of

the triangulationDiscrete Gaussian curvature:

induced by metric, measured as angle deficit

Circle Packing Metric Circle Packing Metric

Flat MetricFlat 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 MetricOptimal Flat Metric

Tool to Compute Optimal Flat Tool to Compute Optimal Flat MetricMetric

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 Tool to Compute Optimal Flat MetricMetric

Talk OverviewTalk OverviewTheory of flat metricDistributed algorithm on sensor

networkDiscussions

◦Costs◦Error propagation

Experiments and Comparison

Distributed AlgorithmDistributed Algorithm

Preprocessing: build a triangulation from network graph◦The dual of the landmark-based

Voronoi diagram

Distributed AlgorithmDistributed AlgorithmComputing 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 AlgorithmDistributed Algorithm

Distributed AlgorithmDistributed AlgorithmIsometric embedding

◦ Start embedding from one vertex

◦ Continuously propagate to the whole triangulation

Talk OverviewTalk OverviewTheory of flat metricDistributed algorithm on sensor

networkDiscussions

◦Costs◦Error propagation

Experiments and Comparison

CostsCostsPreprocessing 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 Convergence Rate of Step IINumber of iterations =

Error PropagationError PropagationIf 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 Testing of Error PropagationPropagationOne scenario: a much longer distance

measurement around one vertex due to slower response of the node to its neighboring nodes’ signals.

Talk OverviewTalk OverviewTheory of flat metricDistributed algorithm on sensor

networkDiscussions

◦Costs◦Error propagation

Experiments and Comparison

Experiments and Experiments and ComparisonComparisonVariant Density

Experiments and Experiments and ComparisonComparisonDifferent transmission models

Trans. models

UDG Quasi-UDG

Log-Norm Probability

Localization error

0.25 0.34 0.42 0.43

Experiments and Experiments and ComparisonComparisonNon-uniform distribution

Non-uniform distribution

Density ranging from 11.3 to 18.7

Localization error 0.46

Experiments and Experiments and ComparisonComparisonComparison with other methods

on networks with different topologiesNetwork

sFlat

MetricMDS-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 Summary and Future WorkWorkMere 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