Download - Jie Gao Joint work with Amitabh Basu*, Joseph Mitchell, Girishkumar Sabhnani* @ Stony Brook
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Jie Gao Joint work with
Amitabh Basu*, Joseph Mitchell, Girishkumar Sabhnani* @ Stony
Brook
Distributed Localization Distributed Localization using Noisy Distance and using Noisy Distance and
Angle InformationAngle Information
To appear in ACM MobiHoc 2006
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Localization in sensor networks
• Given local measurements– Connectivity– Distance measurements– Angle measurements
• Find – Relative positions– Absolute positions
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Localization in sensor networks
• Location info is important for– Integrity of sensor readings– Many basic network functions
• Topology control• Geographical routing• Clustering and self-organization.
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Localization problem
• Extensively studied.• Anchor-based methods
– Anchors know positions, e.g., via GPS.– Triangulation-type of methods, e.g.,
[Savvides et al.]• Anchor-free methods
– Local measurements global layout.– We use this approach.
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Anchor-free localization
• Distance information only– Global optimization
• MDS [Shang 03], SDP [Biswas & Ye 04]
– Localized, distributed algorithm• Mass-spring optimization, robust
quadrilateral [Moore 04], etc.
• Graph rigidity!
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Our approach
• Distance + angle information• Measurements are noisy.
Assume a global north.
Upper/lower bound on distance and direction of neighbors.Goal: find an embedding that satisfies all the constraints.
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Our results
• Finding a feasible solution with noisy distance + angle is NP-hard.
• A distributed, iterative algorithm for a relaxation.
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Hardness results
• Accurate distance + angle: trivial.
• Infinite noise, non-neighbors >1 = Unit disk graph embedding: NP-hard [Breu & Kirkpatrick].
• Accurate angle, infinite noise in distance, non-neighbors >1: NP-hard [Bruck05].
• Accurate distance, infinite noise in angle, non-neighbors >1: NP-hard [Aspnes et. al. 04].
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This paper
1. εnoise in distance, δnoise in angle, for arbitrarily smallε,δ, finding a feasible solution is NP-hard.
2. Accurate distance, relative angle, non-neighbors >1: NP-hard.
• Reduction from 3SAT. or
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Solve a relaxation
• Use a convex approximation to the non-convex frustum, e.g, a trapezoid.
All the constraints are linear.
Use linear programming to solve for an embedding.
Solution not unique. Compute all of them.
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Weak deployment regions
• We solve for Regions of Deployment
• Weak deployment– All feasible solutions. Upper bound.– Fix a sensor, a feasible solution for
the other sensors.
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Strong deployment regions
• We solve for Regions of Deployment
• Strong deployment– Inherent uncertainty. Lower bound.– Pick any point within each region
independently a feasible solution.
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Linear programming
• We can also solve weak and strong deployment by LP.
• Let’s look at weak deployment first.
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Weak deployment and LP
• LP for feasibility of embedding.• n sensors, m edges.• Variables: (xi, yi) for each sensor i.• # variables 2n, # constraints: 8m.• A valid embedding is a point in R2n.• The feasible polytope P in R2n :
collection of all feasible solutions.
Weak deployment region for sensor i = projection of P onto plane (xi, yi).
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Theory of convex polytope
• The feasible polytope P has 8m faces.
• In general, the complexity of P (# vertices) and its projection, can be exponential in 8m.
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Solve for weak deployment
Our problem has special structures:• The weak deployment region has
O(m) complexity in the worst case.• We can solve it in polynomial time
by linear programming.• There is a distributed algorithm
that finds the same solution as the global LP.
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What next?
• A distributed, iterative algorithm for the weak deployment problem.
• Show why the complexity of weak deployment region is O(m).
• Simulation results.
• Strong deployment.
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Ri
Rj
Forward constraint propagation
• Each node keeps a current feasible region Ri.
• Region Ri shrinks region Rj.
• Rj Rj ∩ Ri Fij.
Minkowski sumXY={p+q | p ∊ X, q ∊ Y}
Fij
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Backward constraint propagation
Ri
Rj• When Rj shrinks,
then Ri can also shrink.
• Ri Ri ∩ Rj (-Fij).
-Fij
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Iterative algorithm
• Pin down one node at the origin.• Initialize all other regions as R2.• Until all regions stabilize
– For each sensor, compute new regions from all neighbors’ regions• Both forward & backward propagation.
– Shrink its current region to the common intersection.
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Iterative algorithm correctness
• The iterative algorithm computes the weak deployment regions.
• Proof sketch: – Regions always shrink.– It converges to weak deployment
region when shrinking stops.– The algorithm stops after a finite
number of steps
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Convergence
• Prove by contradiction. Assume a point p Ri* for sensor i.
• For every sensor j, propagate the constraints from i to j along all possible paths.
• Take the common intersection of these regions, say Pi.
p
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Convergence
• Recall p Ri*. Thus either1. One region Pj is empty.
2. The origin k is outside Pk.
• 1 is not possible. – The shape of Pj doesn’t depend on p.
– Start from a point in Ri*, the LP is infeasible.
p
p*
Pj
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Convergence
• Recall p Ri*. Thus either1. One region Pj is empty.
2. The origin k is outside Pk.
• If 2 happens. – Reverse the paths from k to i.– The point p will be eliminated.– The algorithm hasn’t converged.
p k=origin
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Why the regions are O(m)?
• All the operations are Minkowski sums and intersections.
Minkowski sum XY: boundary comes from the boundaries of X and Y
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Why the regions are O(m)?
• All the operations are Minkowski sums and intersections.
• Slopes of the region boundary come from the original constraints.
• There are only 8m different slopes.
• If we use rectangle constraints, then all the deployment regions are rectangles.
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Convergence rate
• Nodes randomly deployed.• Communication graph: unit disk
graph.
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Robustness to link variation
• Links switch on ↔ off with prob p: 0~1.• The deployment regions are stable.
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Robustness to link variation
• Links switch on ↔ off with prob p: 0~1.
Due to network disconnection. When p is small, it is slow to get re-connected.
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Comparison to SDP [Biswas & Ye]
• SDP only uses noisy distance measurements.• We use angle range /4.
Less dependency on # anchors.
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Comparison to SDP [Biswas & Ye]
• SDP only uses noisy distance measurements.• We choose angle range /4.
Two metrics:• Center • furthest point.
WD: weak deploymentSD: strong deployment
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Strong deployment
• Strong deployment– Inherent uncertainty. Lower bound.– Pick any point within each region
independently a feasible solution.
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Strong deployment
• More subtle!• One can shrink the region for one to get
a larger region for the others.
• We propose to find the same shaped region for every node, e.g., square, as large as possoble.
• Formulate as LP? Infinite # constraints?
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Strong deployment
• By convexity, if the constraints are satisfied for every pair of corners of the deployment regions, then the constraints are satisfied for every pair of internal points.
• Formulate a LP w/ constraints on all pairs of corners.
• Maximize the size of the region r.
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Strong deployment
• Reduce to weak deployment.
• Distributed algorithm.– Guess the size r.– Solve for center of
the strong deployment region.
– Binary search on r.
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Conclusion
• Localization with noisy distance + angle measurements.
• Complete the hardness results.
• Upper/lower bound: weak/strong deployment regions.
• Linear programming and distributed implementation.
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Future work
• Convergence rate of the distributed iterative algorithm.
• Bound the approximation through the relaxation of non-convex constraints.
• Generalize the noise model to probabilistic distributions.
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Questions?
• Thank you!