Chen Wang and Hongbo Jiang Huazhong University of Science and Technology, China

1. SURF: A Connectivity-based Space Filling Curve Construction Algorithm in High Genus 3D Surface WSNs ChenWang and Hongbo Jiang Huazhong University of Science and Technology, China {chenwang, hongbojiang}@hust.edu.cn Hong Kong, April 29th, 2015 IEEE INFOCOM 2015

2. Introduction Outline Preliminary SURF Algorithm Performance Evaluation Conclusion

3. Introduction Outline Preliminary SURF Algorithm Performance Evaluation Conclusion

4. High Genus 3D Surface WSNs The three networks in (a-c) are homotopically equivalent to (d).

5. Space Filling Curve (SFC) Almostall existing SFCs are constructed in squares or hyper-cubes. In mathematical analysis, the space filling curve refers to a curve whose range contains the entire 2D unit square (or more generally an N-D hypercube).

6. SFC Applications in WSNs Linearization. • Serial Data Fusion [1, 2] The query is successively (i.e., serially) update from node to node until all nodes in the network are visited. The last node holds the right value of the query. • Path Planning of Mobile Nodes • Localization and coverage [3] • Sensor battery recharge [4] • Data collection by the data mules near the sink [5]

7. SFC Construction in WSNs Both these algorithms are designed for 2D networks. • X. Ban, M. Goswami, W. Zeng, X. Gu, and J. Gao, "Topology Dependent Space Filling Curves for Sensor Networks and Applications," in 32nd IEEE INFOCOM, 2013, pp. 2166-2174. • A. Mostefaoui, A. Boukerche, M. A. Merzoug, and M. Melkemi, "A Scalable Approach for Serial Data Fusion in Wireless Sensor Networks," Computer Networks, vol. 79, pp. 103-119, 2015.

8. Our Approach Iso-contours of a closed spherical surface naturally form an embryonic form of the SFC. Directly connecting the iso-contours forms a SFC. Intuition of SURF

9. Our Approach Constructing SFCsin regions before connecting them. Cuttingoff genus to form regions. Genus incurs twoiso-contours.

10. Introduction Outline Preliminary SURF Algorithm Performance Evaluation Conclusion

11. Cut and Genus How can we know wherethe genus is? genus=1 genus=2 genus=3 A cut is referred to as a disjoint closed simple curve on a connected and orientable surface M. The genus of M is defined as the maximum number of cuts without rendering M disconnected.

12. Iso-contour and Reeb Graph A iso-contour is a connected component of a level set, i.e. a curve whose points have a constant value. The Reeb graph reveals the evolution of its level set.

13. Cut Identification Theorem 1. The Reeb graph of a closed orientable genus-n surface has exactly n loops [6]. An arc of the Reeb graph of M is a loop-end arc, if it is merged from two different arcs. Corollary 2. Each loop in the Reeb graph of M corresponds to one loop-end arc.

14. Introduction Outline Preliminary SURF Algorithm Performance Evaluation Conclusion

15. Proprocessing Triangulation of the Network [7] The triangular structure is still denoted by M, with its vertex (node) set V={vi} , and edge set E = { e = (vi, vj) | vj is called the neighbor of vi }.

16. Step 1: Contour Construction Hop count distance → iso-distance contour. Proposition 3: An iso-distance contour is a connected and closed cycle.

17. Step 2: Cut Identification (1) Assigning each node with an iso-contour ID. (2) Constructing regions (arcs). (3) Notifing loop-end regions (arcs). (4) Bisecting loop end regions.

18. Step 3: Serial Traversal Scheme The SFC construction follows several cases.

19. Introduction Outline Preliminary SURF Algorithm Performance Evaluation Conclusion

20. Visual Results of SURF SURF is robust to general topologies.

21. Network Coverage SURF has a faster traversal speed.

22. Coverage v.s. Path Length SURF guarantees a 100% coverage.

23. Introduction Outline Preliminary SURF Algorithm Performance Evaluation Conclusion

24. Conclusion • We proposed SURF, the first solution for the SFC construction in high genus 3D surface WSNs. • It requires connectivity information only, without the reliance on the location or distance measurement. • It does not rely on any particular communication model. • It is fully distributed and scalable, with a nearly constant storage and communication cost of every node. • A proportional coverage of the generated SFC, with an adaptive density for a given traversal budget or delay constrain will be an interesting direction for the future.

25. References • S. Patil, S. R. Das, and A. Nasipuri, "Serial Data Fusion Using Space-Filling Curves in Wireless Sensor Networks," in Proceedings of IEEE SECON, 2004, pp. 182-190. • A. Mostefaoui, A. Boukerche, M. A. Merzoug, et al., "A Scalable Approach for Serial Data Fusion in Wireless Sensor Networks," Computer Networks, vol. 79, pp. 103-119, 2015. • J. M. Bahi, A. Makhoul, and A. Mostefaoui, "Localization and Coverage for High Density Sensor Networks," Computer Communications, vol. 31, pp. 770-781, 2008. • L. Xie, Y. Shi, Y. T. Hou, et al., "Making Sensor Networks Immortal: An Energy-Renewal Approach with Wireless Power Transfer," IEEE/ACM Transactions on Networking, vol. 20, pp. 1748-1761, 2012. • R. Sugihara and R. K. Gupta, "Path Planning of Data Mules in Sensor Networks," ACM Transactions on Sensor Networks, vol. 8, pp. 1:1-1:27, 2011. • K. Cole-McLaughlin, H. Edelsbrunner, J. Harer, et al., "Loops in Reeb Graphs of 2-Manifolds," in Proceedings of ACM SoCG, 2003, pp. 344-350. • H. Zhou, H. Wu, S. Xia, et al., "A Distributed Triangulation Algorithm for Wireless Sensor Networks on 2d and 3d Surface," in Proceedings of IEEE INFOCOM, 2011, pp. 1053-1061.

26. Thanks for your attentions ! IEEE INFOCOM 2015 Networked and Communication Systems Research Group (NEST) http://ei.hust.edu.cn/teacher/hongbo/NEST/index.html