Los Angeles September 27, 2006

1. MOBICOM 2006 Los Angeles September 27, 2006

2. MOBICOM 2006 Localization in Sparse Networks using Sweeps D. K. Goldenberg P. Bihler M. Cao J. Fang B. D. O. Anderson A. S. Morse Y. R. Yang Yale University Los Angeles September 27, 2006

3. MOBICOM 2006 Localization in Sparse Networks using Sweeps D. K. Goldenberg P. Bihler M. Cao J. Fang B. D. O. Anderson A. S. Morse Y. R. Yang Yale University Los Angeles September 27, 2006

4. MOBICOM 2006 Localization in Sparse Networks using Sweeps D. K. Goldenberg P. Bihler M. Cao J. Fang B. D. O. Anderson A. S. Morse Y. R. Yang Yale University Los Angeles September 27, 2006

5. Roadmap Motivation Problem Formulation Theoretical Foundation Related Work Our Contribution Experimental Evaluations Future Work

6. Roadmap Motivation Problem Formulation Theoretical Foundation Related Work Our Contribution Experimental Evaluations Future Work

7. Motivation Location necessary in order for sensed data to be meaningful: e.g., Forest fire detection. Location information is taken for granted in many network designs: e.g., Geographic routing. Equipping each node with GPS is not always feasible due to power constraints and other limitations inherent to sensor networks. Nodes can often measure their distances to nearby nodes: Acoustic ranging (e.g. L. Girod et al.), ultra-wideband ranging (e.g. Ubisense), radio interferometry (e.g. Vanderbilt). Localize using inter-node distances!

8. Roadmap Motivation Problem Formulation Theoretical Foundation Related Work Our Contribution Experimental Evaluations Future Work

9. Roadmap Motivation Problem Formulation Theoretical Foundation Related Work Our Contribution Experimental Evaluations Future Work

10. ? ? A network in the plane. Anchors are nodes whose positions are known. ? ? Anchor positions from GPS or manual configuration. The distances between some nodes are known. ? ? The network localization problem is to determine the positions of all the nodes. The network is localizable if there exists exactly one position in the plane corresponding to each non-anchor node so that all known inter-node distances are satisfied. A node is localizable if its position is uniquely determined by the known inter-node distances and anchor positions.

11. The localization problem issolvableif and only if the network is localizable. The network localization problem is NP-Hard. (Aspnes et al.) Even assuming exact distance measurements, there is currently no algorithm that can localize a large class of localizable networks without requiring high connectivity while giving correctness guarantees. Our contribution – An algorithm that provably and tractably localizes a class of localizable networks with average degree as low as three under the assumption of exact distance measurements. Techniques to adapt our algorithm to noisy measurements - No proven results.

12. Roadmap Motivation Problem Formulation Theoretical Foundation Related Work Our Contribution Experimental Evaluations Future Work

13. Roadmap Motivation Problem Formulation Theoretical Foundation Related Work Our Contribution Experimental Evaluations Future Work

14. Consider the network nodes as vertices in a graph. There is an edge between two vertices if the distance between the corresponding nodes are known. This is the graph of the network. A network in the plane whose node position coordinates are algebraically independent over the rationals is localizable if and only if it has at least three non-collinear anchors and its graph is generically globally rigid in the plane. (Eren et al.) There are polynomial time algorithms to determine if a graph is generically globally rigid in the plane. Can almost always efficiently check if a network in the plane is localizable by analyzing its graph! Assume the node position coordinates of the networks we consider are algebraically independent over the rationals.

15. Roadmap Motivation Problem Formulation TheoreticalFoundation Related Work Our Contribution Experimental Evaluations Future Work

16. Roadmap Motivation Problem Formulation TheoreticalFoundation Related Work Our Contribution Experimental Evaluations Future Work

17. Global Approach Nodes are localized by processing all nodes at once. Global optimization susceptible to local minimums. May not be effective for networks where average degree is low. Typically assume uniform deployment of nodes.

18. Sequential Approach Nodes are localized by sweeping through the network in some order and processing the nodes one by one. ? Trilateration based Multilateration (Savvides et al.) (e.g. Eren et al., Moore et al.) Experimental evaluations suggest trilateration based method is not effective for sparse networks where average degree is low even assuming exact distances. Our work extends the trilateration based methods in order to localize sparse networks where average degree is as low as three.

19. A graph has a trilateration ordering if its vertices can be relabeled as v1,...,vnso that (i) the subgraph induced by {v1,v2,v3}is complete. v1, v2, v3 are the seeds of the ordering. (ii) each vi, i>3, is adjacent to at least three distinct vertices vj, j< i.

20. A graph has a trilateration ordering if its vertices can be relabeled as v1,...,vnso that (i) the subgraph induced by {v1,v2,v3}is complete. v1, v2, v3 are the seeds of the ordering. (ii) each vi, i>3, is adjacent to at least three distinct vertices vj, j< i. v2 v3 v1

21. A graph has a trilateration ordering if its vertices can be relabeled as v1,...,vnso that (i) the subgraph induced by {v1,v2,v3}is complete. v1, v2, v3 are the seeds of the ordering. (ii) each vi, i>3, is adjacent to at least three distinct vertices vj, j< i. v2 v3 v1 v4

22. A graph has a trilateration ordering if its vertices can be relabeled as v1,...,vnso that (i) the subgraph induced by {v1,v2,v3}is complete. v1, v2, v3 are the seeds of the ordering. (ii) each vi, i>3, is adjacent to at least three distinct vertices vj, j< i. v2 v3 v1 v5 v4 Suppose this is the graph of a network.

23. A graph has a trilateration ordering if its vertices can be relabeled as v1,...,vnso that (i) the subgraph induced by {v1,v2,v3}is complete. v1, v2, v3 are the seeds of the ordering. (ii) each vi, i>3, is adjacent to at least three distinct vertices vj, j< i. q2 v2 q3 v3 q1 v1 v5 v4 Suppose this is the graph of a network.

24. A graph has a trilateration ordering if its vertices can be relabeled as v1,...,vnso that (i) the subgraph induced by {v1,v2,v3}is complete. v1, v2, v3 are the seeds of the ordering. (ii) each vi, i>3, is adjacent to at least three distinct vertices vj, j< i. q2 v2 q3 v3 q1 v1 v5 v4 Suppose this is the graph of a network.

25. A graph has a trilateration ordering if its vertices can be relabeled as v1,...,vnso that (i) the subgraph induced by {v1,v2,v3}is complete. v1, v2, v3 are the seeds of the ordering. (ii) each vi, i>3, is adjacent to at least three distinct vertices vj, j< i. q2 v2 q3 v3 q1 v1 v5 v4 q4 Suppose this is the graph of a network.

26. A graph has a trilateration ordering if its vertices can be relabeled as v1,...,vnso that (i) the subgraph induced by {v1,v2,v3}is complete. v1, v2, v3 are the seeds of the ordering. (ii) each vi, i>3, is adjacent to at least three distinct vertices vj, j< i. q2 v2 q3 v3 q1 v1 q5 v5 v4 q4 Suppose this is the graph of a network.

27. A graph has a trilateration ordering if its vertices can be relabeled as v1,...,vnso that (i) the subgraph induced by {v1,v2,v3}is complete. v1, v2, v3 are the seeds of the ordering. (ii) each vi, i>3, is adjacent to at least three distinct vertices vj, j< i. q2 v2 q3 v3 q1 v1 q5 v5 v4 q4 Suppose this is the graph of a network. A network with three anchors can be localized using just trilaterations followed by an Euclidean transformation if and only if its graph has a trilateration ordering.

28. Roadmap Motivation Problem Formulation Theoretical Foundation Related Work Our Contribution Experimental Evaluations Future Work

29. Roadmap Motivation Problem Formulation Theoretical Foundation Related Work Our Contribution Experimental Evaluations Future Work

30. Our work is an extension of the trilateration based localization method for networks whose graphs may not have a trilateration ordering. Sweeps is a sequential localization algorithm that provably and tractably localizes a class of sparse localizable networks with average degree as low as three assuming exact distance measurements. Experimental evaluations suggest Sweeps is feasible and consistently localizes 90% or more of the nodes in sparse networks of 1000 nodes with average degree five. Sweeps also identifies all localizable nodes. We can efficiently check if Sweeps will successfully localize a network by just analyzing the network’s graph. We propose techniques to deal with noisy measurements, which experimental evaluations suggest is promising, but no proven results.

31. A graph has a bilateration ordering if its vertices can be relabeled as v1,...,vn so that • the subgraph induced by {v1,v2,v3}is complete. v1, v2, v3 are the seeds of the ordering. (ii) each vi, i>3, is adjacent to at least two distinct vertices vj , j < i. v4 v2 No trilateration ordering. v3 v5 v1 Sweeps is for localizable networks whose graphs have a bilateration ordering, but not necessarily a trilateration ordering. Experimental evaluations suggest such networks occur with high probability even in networks with average degree as low as three.

32. The algorithm Sweeps consists of performing a sequence of bilaterations and set reductions, combined with consistency checking. Candidate position set of a node is a set of points in the plane which contains the node's position. Bilateration - Determining a finite candidate positions set for a node using its distances to two or more nodes with finite candidate positions sets. {pa p'a} {pb} A B Bilateration at node c C {pc p'c1 p'c2 p'c3} If nodes A and B are positioned at pa and pb, then that determines at most two positions for node C, one of which must be the position of node C. If nodes A and B are positioned at p'a and pb, then that determines at most two positions for node C. Bilateration with consistency checkingis where only a subset of a finite candidate positions set is chosen to use in a bilateration operation.

33. Set reduction - Removing points from a node's finite candidate positions set using its distances to one or more nodes with finite candidate positions sets. Set reduction at node 2 2 {p2, p'2} ║p'2 - p1║  d ║p'2 - p'1║  d d 1 {p1, p'1} Remove point p'2 from the candidate positions set of node 2 because the true position of node 2 must be distance d to at least one point in the candidate positions set of node 1. Set reduction with consistency checkingreduces a node’s candidate positions set even further using an additional criteria. (See paper for details)

34. Sweeps Localizable network whose graph has a bilateration ordering. pv1 pv2 v1 v2 This graph does not have a trilateration ordering, so cannot be localized by a trilateration based method! Assign positions to seed vertices so their inter-node distances are satisfied. v3 pv3 This determines a unique position for each vertex relative to the seed vertices. v4 P v5 Sweep through the network according to the bilateration ordering, and perform a bilateration operation at each unlocalized node. Q v6 pv6

35. Sweeps Localizable network whose graph has a bilateration ordering. p1 p2 v1 v2 This graph does not have a trilateration ordering, so cannot be localized by a trilateration based method! Assign positions to seed vertices so their inter-sensor distances are satisfied. v3 p3 This determines a unique position for each vertex relative to the seed vertices. v4 P v5 Sweep through the network according to the bilateration ordering, and perform a bilateration operation at each unlocalized node. Q v6 pv6 Sweep through the network in a different order performing set reduction at each unlocalized node.

36. Sweeps Localizable network whose graph has a bilateration ordering. p1 p2 w1 w2 This graph does not have a trilateration ordering, so cannot be localized by a trilateration based method! Assign positions to seed vertices so their inter-sensor distances are satisfied. w3 p3 This determines a unique position for each vertex relative to the seed vertices. pw6 w5 pw5 P w6 Sweep through the network according to the bilateration ordering, and perform a bilateration operation at each unlocalized node. Q w4 pv6 Sweep through the network in a different order performing set reduction at each unlocalized node. Localizable in two sweeps plus Euclidean transformation.

37. Theorem: A localizable network whose graph has a bilateration ordering can be localized with two sweeps followed by a Euclidean transformation. We say such networks are sweepable. A network’s graph must have a bilateration ordering for the network to be sweepable. There are localizable networks whose graphs do not have bilateration orderings, and so cannot be sweepable. However, experimental evaluations suggest sweepable networks occur with high probability even in networks with average degree as low as three.

38. Roadmap Motivation Problem Formulation Theoretical Foundation Related Work Our Contribution Experimental Evaluations Future Work

39. Roadmap Motivation Problem Formulation Theoretical Foundation Related Work Our Contribution Experimental Evaluations Future Work

40. Uniformly random 250 node network. Percentage of localizable nodes localized by Trilateration. Percentage of localizable nodes localized by Sweeps. Ratio Average Degree

41. Sweeps localizes more nodes than trilateration at the expense of computational complexity. Worst case complexity of Sweeps is exponential, but experimental evaluations suggest Sweeps is practically feasible. At average degree 6, the maximum size of the candidate positions sets is at most 8 for just 75% of the nodes. Cumulative Proportion of Nodes At average degrees 3 and 9.5, the maximum size of the candidate positions sets is at most 8 for 95% of the nodes. Maximum Size of Candidate Positions Sets

42. Uniformly random deployment of 100 nodes, 5 anchors, average degree 8. Zero-mean Gaussian noise with std 5% of sensing range added to distance measurements. Proportion of nodes with less than given position error 90% of nodes localized by Sweeps have position error less than 50%. Cumulative proportion of nodes Position Error (% of Sensing Range)

43. Roadmap Motivation Problem Formulation Theoretical Foundation Related Work Our Contribution Experimental Evaluations Future Work

44. Roadmap Motivation Problem Formulation Theoretical Foundation Related Work Our Contribution Experimental Evaluations Future Work

45. Future Work Obtain theoretical results relating the probability of a network’s graph having bilateration ordering with the average degree of the network. Obtain theoretical results on the effectiveness of Sweeps in the presence of noisy distance measurements. Extending sweeps to 3-D.