Jie Gao Joint work with Amitabh Basu*, Joseph Mitchell, Girishkumar Sabhnani* @ Stony Brook

1. Distributed Localization using Noisy Distance and Angle Information Jie Gao Joint work with Amitabh Basu*, Joseph Mitchell, Girishkumar Sabhnani* @ Stony Brook To appear in ACM MobiHoc 2006

2. Localization in sensor networks • Given local measurements • Connectivity • Distance measurements • Angle measurements • Find • Relative positions • Absolute positions

3. 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.

4. 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.

5. 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!

6. 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.

7. Our results • Finding a feasible solution with noisy distance + angle is NP-hard. • A distributed, iterative algorithm for a relaxation.

8. 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].

9. This paper • εnoise in distance, δnoise in angle, for arbitrarily smallε,δ, finding a feasible solution is NP-hard. • Accurate distance, relative angle, non-neighbors >1: NP-hard. • Reduction from 3SAT. or

10. 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.

11. 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.

12. 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.

13. Linear programming • We can also solve weak and strong deployment by LP. • Let’s look at weak deployment first.

14. 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).

15. 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.

16. 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.

17. 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.

18. Forward constraint propagation • Each node keeps a current feasible region Ri. • Region Ri shrinks region Rj. • Rj  Rj ∩ Ri  Fij. Rj Fij Ri Minkowski sum XY={p+q | p∊ X, q ∊ Y}

19. Rj Ri Backward constraint propagation • When Rj shrinks, then Ri can also shrink. • Ri  Ri ∩ Rj  (-Fij). -Fij

20. 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.

21. 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

22. 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

23. p p* Convergence • Recall p  Ri*. Thus either • One region Pj is empty. • 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. Pj

24. Convergence • Recall p  Ri*. Thus either • One region Pj is empty. • 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

25. Why the regions are O(m)? • All the operations are Minkowski sums and intersections. Minkowski sum XY: boundary comes from the boundaries of X and Y

26. 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.

27. Convergence rate • Nodes randomly deployed. • Communication graph: unit disk graph.

28. Robustness to link variation • Links switch on ↔ off with prob p: 0~1. • The deployment regions are stable.

29. 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.

30. Comparison to SDP [Biswas & Ye] • SDP only uses noisy distance measurements. • We use angle range /4. Less dependency on # anchors.

31. Comparison to SDP [Biswas & Ye] • SDP only uses noisy distance measurements. • We choose angle range /4. Two metrics: • Center • furthest point. WD: weak deployment SD: strong deployment

32. Strong deployment • Strong deployment • Inherent uncertainty. Lower bound. • Pick any point within each region independently  a feasible solution.

33. 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?

34. 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.

35. Strong deployment • Reduce to weak deployment. • Distributed algorithm. • Guess the size r. • Solve for center of the strong deployment region. • Binary search on r.

36. Conclusion • Localization with noisy distance + angle measurements. • Complete the hardness results. • Upper/lower bound: weak/strong deployment regions. • Linear programming and distributed implementation.

37. 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.

38. Questions? • Thank you!