scalable and fully distributed localization with mere connectivity
TRANSCRIPT
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