A Distributed Triangulation Algorithm for Wireless Sensor Networks on 2D and 3D Surface

Transcription

1 A Distributed Triangulation Algorithm for Wireless Sensor Networks on 2D and 3D Surface Hongyu Zhou, Hongyi Wu, Su Xia, Miao Jin, Ning Ding The Center for Advanced Computer Studies(CACS) University of Louisiana at Lafayette

2 OUTLINE Introduction Triangulation Algorithm Extension to 3D Surface Application and Simulation Results Conclusion

3 INTRODUCTION Wireless sensor network exhibits randomness Wireless sensor network topology is a graph Triangulation is a subgraph of the topology Triangulation mesh is very important for many applications of sensor networks: geometry-based routing, localization, coverage, segmentation, data segmentation

4 INTRODUCTION Related work # # # (a) (a) (a) A (a) 2D A A2D network 2D network network graph. graph. graph. (b) (b) (b) Triangulation (b) Triangulation under under under ideal ideal CDG. ideal CDG. CDG. (c) (c) (c) CDG CDG (c) CDG (k=1): not (k=1): not not planar. not planar. H. Zhou, S. Xia, M. Jin, and H. Wu, Localized Algorithm for Precise Boundary Detection in 3D Wireless Networks, in Proc. of ICDCS, # # # # ## # (d) (d) CDM (k=1): planar but butnot triangulated. (e) (e) Failed triangulation. (f) CDG (k=2): triangulated but coarse. (d) (d) CDM (d) CDM CDM (k=1): (k=1): (k=1): planar planar planar but but not but not triangulated. not triangulated. (e) (e) Failed (e) Failed triangulation. triangulation. (f) (f) CDG (f) CDG (k=2): (k=2): triangulated triangulated but coarse. but coarse. Fig Illustration of of challenges in in triangulation, where the circles indicate sensor nodes and the dashed lines are communication links. The solid black line Fig. [2] Fig. illustrate 2. R. Fig. Sarkar, 2. Illustration 2. virtual Illustration X. Yin, edges, ofj. challenges of Gao, which of challenges F. are Luo, are in realized triangulation, in and inx. triangulation, by D. by corresponding Gu, where Greedy where where the the Routing circles paths the circles circles shown indicate with Guaranteed indicate as as solid sensor grey sensor nodes Delivery lines. nodes and and In In the Using subfigure and the dashed the Ricci dashed (b), lines Flows, the are lines are red communication Proc. lines are depict of IPSN, Voronoi links links. cells. The solid In The subfigure black solid bl illustrate illustrate (c)-(f), virtual virtual the the nodes virtual edges, edges, with edges, which which the the same which are are realized color are realized realized belong by by corresponding toby the corresponding same paths Voronoi paths paths shown shown cell, shown as while as solid solid the as grey large solid grey lines. circles(i.e., grey lines. lines. In Insubfigure A-E) subfigure (b), stand (b), the the for (b), red red landmarks. thelines red depict lines depict Voronoi Voronoi cells. cells. In Insubfig s [3](c)-(f), H. (c)-(f), the Zhou, the nodes the S. nodes Xia, nodes with M. with the with Jin, the same and the same same H. color Wu, color color belong Localized belong belong to to thealgorithm tosame same same Voronoi Voronoi for Voronoi Precise cell, cell, while Boundary cell, while while the the large Detection the large large circles(i.e., in circles(i.e., 3D Wireless A-E) A-E) A-E) stand Networks, stand for forlandmarks. in Proc. landmarks. of ICDCS, subset of of nodes as as landmarks where any two landmarks Landmark j), it it sends a connection packet to the latter. The

5 INTRODUCTION We proposed a distributed triangulation algorithm works for any arbitrary 2D sensor networks without communication model constraint, and we prove the correctness of the algorithm in 2D sensor networks tolerates some measurement errors also works for 3D open or closed surfaces

6 TRIANGULATION ALGORITHM Basic idea Each triangulation mesh edge will associate with two triangles, except boundary edge If we add extra edges into triangulation mesh, they will change this association The association can be used to identify B these extra edges. C D B C D A E A E H G F H G F

7 TRIANGULATION ALGORITHM Definitions node neighbor set (NNS), Nv(i)={h,k,l,m} edge neighbor set (ENS), Ne(eij)=Nv(i) Nv(j)={k,l,m} refined edge neighbor set (RNS), Re(eij)={l,m} edge weight, W(eij)=2, W(ehk)=2 associated edge neighbor set # (AENS), A(eij)={eil, eim, ejl, ejm} ' equivalent edges, eij, elm critical edge, ekj

8 TRIANGULATION ALGORITHM Definitions node neighbor set (NNS), Nv(i)={h,k,l,m} edge neighbor set (ENS), Ne(eij)=Nv(i) Nv(j)={k,l,m} refined edge neighbor set (RNS), Re(eij)={l,m} edge weight, W(eij)=2, W(ehk)=2 associated edge neighbor set # (AENS), A(eij)={eil, eim, ejl, ejm} ' equivalent edges, eij, elm critical edge, ekj

9 TRIANGULATION ALGORITHM Definitions node neighbor set (NNS), Nv(i)={h,k,l,m} edge neighbor set (ENS), Ne(eij)=Nv(i) Nv(j)={k,l,m} refined edge neighbor set (RNS), Re(eij)={l,m} edge weight, W(eij)=2, W(ehk)=2 associated edge neighbor set # (AENS), A(eij)={eil, eim, ejl, ejm} ' equivalent edges, eij, elm critical edge, ekj

10 TRIANGULATION ALGORITHM Definitions node neighbor set (NNS), Nv(i)={h,k,l,m} edge neighbor set (ENS), Ne(eij)=Nv(i) Nv(j)={k,l,m} refined edge neighbor set (RNS), Re(eij)={l,m} edge weight, W(eij)=2, W(ehk)=2 associated edge neighbor set # (AENS), A(eij)={eil, eim, ejl, ejm} ' equivalent edges, eij, elm critical edge, ekj

11 TRIANGULATION ALGORITHM Definitions node neighbor set (NNS), Nv(i)={h,k,l,m} edge neighbor set (ENS), Ne(eij)=Nv(i) Nv(j)={k,l,m} refined edge neighbor set (RNS), Re(eij)={l,m} edge weight, W(eij)=2, W(ehk)=2 associated edge neighbor set (AENS), A(eij)={eil, eim, ejl, ejm} ' # equivalent edges, eij, elm critical edge, ekj

12 TRIANGULATION ALGORITHM Definitions node neighbor set (NNS), Nv(i)={h,k,l,m} edge neighbor set (ENS), Ne(eij)=Nv(i) Nv(j)={k,l,m} refined edge neighbor set (RNS), Re(eij)={l,m} edge weight, W(eij)=2, W(ehk)=2 associated edge neighbor set # (AENS), A(eij)={eil, eim, ejl, ejm} ' equivalent edges, eij, elm critical edge, ekj

13 TRIANGULATION ALGORITHM Definitions node neighbor set (NNS), Nv(i)={h,k,l,m} edge neighbor set (ENS), Ne(eij)=Nv(i) Nv(j)={k,l,m} refined edge neighbor set (RNS), Re(eij)={l,m} edge weight, W(eij)=2, W(ehk)=2 associated edge neighbor set # (AENS), A(eij)={eil, eim, ejl, ejm} ' equivalent edges, eij, elm critical edge, ekj

14 TRIANGULATION ALGORITHM Lemma 1: A subgraph of G is triangulated if and only if every edge of the subgraph has a weight of two. Given a graph G and a triangulation subgraph T, extra edges are G-T Objective: remove all extra edges There are three type of extra edges, e0, e1, e2 # # # (a) e 0 (b) e 1 (c) e 2

15 TRIANGULATION ALGORITHM Theorem 1: An edge with a weight less than two must be removed in order to produce a triangulated subgraph. # # (a) e 0 (b) e 1

16 TRIANGULATION ALGORITHM Theorem 2:An non-critical edge can be recognized as an e2 edge and safely removed if all edges in its AENS have their weight greater than two. # (c) e

17 TRIANGULATION ALGORITHM Lemma 2: An e0 extra edge can exist in G, only if it depends on at least two other extra edges. Lemma 3: An e1 extra edge can exist in G, only if it at least depends on another extra edge. # # # ' ' (a) Loop chain. (b) Non-loop chain. (c) Same head and tail.

18 TRIANGULATION ALGORITHM Step1: Initialization. each edge calculate NNS, ENS, RENS, AENS, and edge weight; Step2: Iterative edge removal according to Theorems 1 and 2; Step3: Removal of e0 and e1 chains.

19 EXTENSION TO 3D SURFACE Similar algorithm can be applied in a sensor network on 3D surface with minor modification in determining RNS and edge weight ' # l i k j l'

20 APPLICATION AND SIMULATION RESULTS 2D network (a) Sensor network topology (b) Finest triangulation by [2] (c) Proposed triangulation

21 APPLICATION AND SIMULATION RESULTS 3D network with open surface (a) Sensor network topology (b) Finest triangulation by [3] (c) Proposed triangulation

22 APPLICATION AND SIMULATION RESULTS 3D network with closed surface (a) Sensor network topology (b) Finest triangulation by [3] (c) Proposed triangulation

23 APPLICATION AND SIMULATION RESULTS 3D network with closed surface (a) Sensor network topology (b) Finest triangulation by [3] (c) Proposed triangulation

24 (b) Triangulation under 20 distance errors. (c) Triangulation under 30 distance errors. APPLICATION AND SIMULATION RESULTS 2D network with distance errors (a) Triangulation under 10 distance errors.

25 APPLICATION AND SIMULATION RESULTS Difference Communication Models (a) UDG (b) Triangulation under Quasi-UDG (α = 0.4). (c) Triangulation under Quasi-UDG (α = 0.6) (d) Triangulation under Log-normal.

26 APPLICATION AND SIMULATION RESULTS Greedy routing based on Ricci flow in 2D networks (a) Finest triangulation by [2] (b) Ricci embedding (c) Proposed triangulation (d) Ricci embedding

27 APPLICATION AND SIMULATION RESULTS Greedy routing based on Ricci flow in 3D networks (a) Finest triangulation by [3] (b) Ricci embedding (c) Proposed triangulation (d) Ricci embedding

28 APPLICATION AND SIMULATION RESULTS COMPARISON OF STRETCH FACTOR IN GREEDY ROUTING. 2D networks 3D networks Triangulation by [2], [3] Proposed Triangulation Distribution of stretch factor in greedy routing 9 9 UTING. tworks For gud in and 9. For ), its triangue depicted in onding Ricci 0 1.0~ ~ ~ ~ ~ ~1.4 Triangulation Triangulation by [2] by [2] Proposed Triangulation Proposed Triangulation ~1.6 2~3 1.6~1.8 3~4 Stretching Factor ~5 2~3 (a) 2D networks (a) 2D networks. 5~6 3~4 Stretching Factor 6~ 4~5 (a) 2D networks. Fig ~ ~ ~2 0 2~3 1~2 3~4 2~3 4~5 3~4 Triangulation Triangulation by [3] by [3] Proposed Proposed Triangulation Triangulation 5~6 4~5 6~7 7~8 8~9 9~10 (b) Stretching 3DFactor networks. (b) 3D networks (b) 3D networks. 5~6 6~7 7~8 Stretching Factor Distribution of stretch factor in greedy routing ~9 9~10 10~ 10~

29 CONCLUSION In summary, we have proposed a distributed algorithm that can triangulate an arbitrary sensor network without position information. The algorithm can achieve the finest triangulation and tolerate distance measurement errors. We have proven its correctness in 2D and extended it to 3D surface. A range of geometry-based network algorithms can benefit from the proposed triangulation.