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

Localization in sensor networks

• Given local measurements– Connectivity– Distance measurements– Angle measurements

• Find – Relative positions– Absolute positions

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.

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.

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!

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.

Our results

• Finding a feasible solution with noisy distance + angle is NP-hard.

• A distributed, iterative algorithm for a relaxation.

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].

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

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.

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.

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.

Linear programming

• We can also solve weak and strong deployment by LP.

• Let’s look at weak deployment first.

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).

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.

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.

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.

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

Backward constraint propagation

Ri

Rj• When Rj shrinks,

then Ri can also shrink.

• Ri Ri ∩ Rj (-Fij).

-Fij

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.

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

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

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

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

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

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.

Convergence rate

• Nodes randomly deployed.• Communication graph: unit disk

graph.

Robustness to link variation

• Links switch on ↔ off with prob p: 0~1.• The deployment regions are stable.

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.

Comparison to SDP [Biswas & Ye]

• SDP only uses noisy distance measurements.• We use angle range /4.

Less dependency on # anchors.

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

Strong deployment

• Strong deployment– Inherent uncertainty. Lower bound.– Pick any point within each region

independently a feasible solution.

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?

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.

Strong deployment

• Reduce to weak deployment.

• Distributed algorithm.– Guess the size r.– Solve for center of

the strong deployment region.

– Binary search on r.

Conclusion

• Localization with noisy distance + angle measurements.

• Complete the hardness results.

• Upper/lower bound: weak/strong deployment regions.

• Linear programming and distributed implementation.

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.

Questions?

• Thank you!