Shayok Chakraborty Ph.D. student, Department of Computer Science and Engineering

1. Shayok Chakraborty Ph.D. student, Department of Computer Science and Engineering Arizona State University CSE 535: Mobile Computing Paper Presentation

2. Paper selection: • Title : Localization from Mere Connectivity • Authors : Y. Shang, W. Ruml, Y. Zhang, and M. Fromherz • Published in : Proceedings of the 4th ACM international symposium on Mobile ad hoc networking & computing

3. Agenda • Background and Motivation • Algorithm used • MDS-MAP • Details of the paper • Results • Novelty of the paper • Drawbacks and ways to overcome them • Relevance of the paper • Conclusion • References • Questions

4. Background and Motivation • Localization in wireless sensor networks is of utmost importance • Indispensible for exchange of sensor data (temperature, sound) among different nodes • Attaching GPS / sophisticated sensors to each node is not a practical proposition – a cost effective solution is required

5. Algorithm Used • MDS-MAP • Takes only the connectivity information as input • Time complexity is O(n3) for a network of n nodes • Outputs a relative map with the same neighboring relationships as in the network • If anchor nodes are available, this can be transformed into an absolute map • Performs well when the number of anchor nodes is less and they are placed uniformly

6. MDS-MAP • Consists of three steps: • 1> Find the shortest path between every possible pair of nodes – use a graph theory algorithm • 2> Apply classical MDS (multidimensional scaling) to the proximity matrix to get a relative map in a lower dimensional space(2D or 3D) which fits the proximity measures • 3> Use the known positions of the beacons to derive an absolute map

7. MDS-MAP : STEP 1 • Dijkstra’s algorithm • Given : A connected graph G = (V,E), A weight associated with each edge Output : Shortest distance between every pair of vertices Procedure : Select a particular vertex (s) At every step, declare a vertex as known and find the shortest distance from s to all the known vertices Iterate this for every possible s.

8. Dijkstra’s algorithm : Illustration s N1 • Final distances: • S to N1 = 2; S to N2 = 7; S to N3 = 3 2 9 1 N2 N3 4

9. MDS Classification • Based on data type - non-metric(qualitative) and metric(quantitative) • Based on number of matrices used - classical (one matrix), replicated(many matrices) and weighted • Based on representation - deterministic and probabilistic • Classical MDS used here – linear relationship between the proximities and actual distances: D = a + b(P)

10. MDS-MAP : STEP 2 • Goal : To replicate the distance information obtained in Step 1 by means of co-ordinate assignments to points in a lower dimensional space • Mathematical tool used : classical MDS • Input : Proximity matrix P (n x n) • Actual co-ordinate matrix X is n x n • Apply double centering to proximity matrix P

11. MDS-MAP : STEP 2

12. MDS-MAP : STEP 2 Double center • P B (n x n) • Double centering ensures B(i,j) = -2XiXj that is, B = -2XXT Let C = -1/2 B that is, C = XXT Now, use SVD (singular value decomposition) on C

13. MDS-MAP : STEP 2 SVD • C VDVT , where V and VT are orthogonal and D is a diagonal matrix • Therefore we have, VDVT = XXT or, VD1/2D1/2 VT = XXT , as D is a diagonal matrix or, VD1/2 [VD1/2 ]T = XXT Hence, X = VD1/2 , which gives the co-ordinates in a higher dimensional space

14. MDS-MAP : STEP 2 • X = VD1/2 • D – a diagonal matrix giving square roots of the eigenvalues • V – gives the corresponding eigenvectors • Sort eigenvalues, take the r largest (for solution in r dimensional space), the corresponding eigenvectors give the point co-ordinates in the lower dimensional space

15. MDS-MAP : STEP 3 • Transform relative map into absolute map using known positions of the anchors • Linear transformations (reflection, scaling etc) used to minimize sum of squared errors between estimated positions and actual positions of the anchors

16. Results • Results are evaluated for two placements: • 1> Random Placement - only proximity information known - neighboring distances known (modeled as true distance blurred with noise) Also, error is analyzed with respect to connectivity for either case • 2> Grid Placement - square and hexagonal grids Both proximity and distance information used

17. Random placement : proximity info only Average error = 0.46r

18. Random placement : distance info known Average error = 0.24r

19. Connectivity vs. error Error reduces

20. More results Connectivity vs. number of nodes localized Range error vs. Estimation error

21. Square grid Proximity only Distance info used

22. Square grid : Average error Proximity only Distance info used

23. Hexagonal grid Proximity only Distance info used

24. Hexagonal grid : Average error Proximity only Distance info used

25. Order Analysis • STEP 1 : Each step of Dijkstra’s algorithm takes O(n2) (for one s). Complexity of step 1 is thus O(n3) • STEP 2 : Classical MDS has complexity O(n3) • STEP 3 : Computing transformation parameters takes O(m2) steps, for m anchors. Applying the transformation takes O(n) time • Thus overall complexity is O(n3)

26. Novelty and contributions • Applicable even in the absence of anchor nodes – produces a relative location map • Works with mere connectivity information • Many localization algorithms depend heavily on the number and positioning of anchor nodes to give good results

27. Drawbacks • Approach requires global knowledge about the network and centralized computation • Performance drops when the number of anchor nodes increases – positioning information is only used in the third step

28. Overcoming limitations • Divide the network into sub-networks, apply MDS-MAP to each sub-network independently in parallel and combine • Use more advanced MDS techniques like ordinal MDS, anchor point method • Use MDS-MAP together with other methods

29. Relevance • Localization is an important aspect of mobile computing – helps in exchange of data and tracing movement patterns of the users • Indoor localization in kids’ network – apply algorithm at discrete time points to trace out the movement patterns of the kids • We have a few beacons (uniformly placed) and thus the scheme can prove effective

30. Conclusion • MDS-MAP works with mere proximity information, can also incorporate the distance information • Gives the relative positions even without anchors • Works well when the number of anchors is small and they are uniformly distributed • Performance can be improved by using advanced MDS methods, applying the algorithm to sub-networks in parallel or by using MDS-MAP in conjunction with other refinement methods

31. References • [1] C. Savarese, J. Rabaey, and K. Langendoen. Robust positioning algorithm for distributed ad-hoc wireless sensor networks. In USENIX Technical Annual Conf., Monterey, CA, June 2002. • [2] Lance Doherty, Kristofer Pister and Laurent El Ghaoui. Convex Position Estimation in Wireless Sensor Networks. IEEE InfoCom 2001. April 2001. • [3] A. Buja, D. F. Swayne, M. Littman, N. Dean, and H. Hofmann. XGvis: Interactive data visualization with multidimensional scaling. Journal of Computational and Graphical Statistics • [4] Vijayanth Vivekanandan and Vincent W.S. Wong. Ordinal MDS-based Localization for Wireless Sensor Networks

32. Questions Questions are guaranteed in life, answers aren’t!!!! …???

33. THANK YOU !!!