On Boundary Recognition without Location Information in Wireless Sensor Networks

Transcription

1 Rheinische Friedrich-Wilhelms-Universität Bonn Institute for Computer cience IV Römerstraße 164 D Bonn On Boundary Recognition without Location Information in Wireless ensor Networks Olga aukh 1, Robert auter 1, Matthias Gauger 1, Pedro José Marrón 1 and Kurt Rothermel 2 1 Universität Bonn and Fraunhofer IAI, 2 Universität tuttgart, Germany {saukh, sauter, gauger, pjmarron}@cs.uni-bonn.de, rothermel@ipvs.uni-stuttgart.de

2 Overview Motivation Related Work Boundary Definition Patterns Patterns for UDGs Patterns for d-qudgs Pattern properties Boundary Recognition Algorithm Evaluation Conclusions 2

3 Motivation Recognition of communication and coverage holes Routing, localization, topology control Events entering and leaving the region Role assignment Topological map of the region 3

4 Related Work Three groups based on assumptions on: Node distribution in non-hole region. P. Fekete, A. Kröller, D. Pfisterer,. Fischer, and C. Buschmann. Neighborhood-based topology recognition in sensor networks. In Proc. of the 1st Int. Workshop on Algorithmic Aspects of Wireless ensor Networks, Node density (the length of the shortest path between two nodes provides a reasonable approximation of the geometric distance between the nodes). Funke. Topological hole detection in wireless sensor networks and its applications. In Proc. of the Joint Workshop on Foundations of Mobile Computing, Funke and C. Klein. Hole detection or: How much geometry hides in connectivity?. In Proc. of the 22nd ymp. on Computational Geometry, Y. Wang, J. Gao, and J.. Mitchell. Boundary recognition in sensor networks by topological methods. In Proc. of the 12th Int. Conf. on Mobile Computing and Networking, Radio model (UDG and d-qudg) A. Kröller,. P. Fekete, D. Pfisterer, and. Fischer. Deterministic boundary recognition and topology extraction for large sensor networks. In Proc. of the 17th ymp. on Discrete Algorithms,

5 Boundary Definition (1) What is a boundary? Geometry: ingle 2D embedding (realization) is considered and known Coordinates are known Boundaries are defined with respect to the fixed embedding: Example: convex hull Graph Theory: No embedding No coordinates, only edges No embedding, so... no boundary can be defined! 5

6 Boundary Definition (2) WN: Embedding in 2D is assumed, but not given No coordinates, only edges (communication links) How to define a boundary? We are interested in vertex-based boundary definition due to the nature of wireless links Problem 1: Boundary is not unique even for a fixed embedding Problem 2: Boundary changes with embedding F a) 2 b) F 6

7 Boundary Definition (3) Problem 3: If the boundary is vertex-based defined it might contain no vertex Problem 4: Boundary is not continuous (nodes 1 and 2 always lie inside of the network, as highlighted graph is a pattern) 1 F F F Problem 5: It is generally impossible to distinguish different hole boundaries without additional location information hortest cycles that contain the holes C 0 C 1 F 0 F 1 C 0 C 1 F * 1 F * 0 F * inf F inf 7

8 Boundary Definition (4) Our Definition of Boundary: If there is a d-qude of G, such that the node v belongs to the geometric boundary of G v is called exterior node If there is NO d-qude of G, such that the node v belongs to the geometric boundary of G v is called interior node The set of all exterior nodes of a graph is called the boundary of G Properties: Every node is either exterior or interior The boundary is unique if d is fixed and minimal The boundary is not continuous [A. Kröller et al.]: Boundary = Chordless cycle that follows the perimeter of the region. There is always one cycle for outer perimeter 8

9 Patterns: Terminology, Assumptions (1) Geometric Properties: d-quasi Unit Disk Embedding (d-qude) p of G is a realization of G in 2D such that ( i, j) E p( i) p( j) 2 p( i) d p( j) ( i, 2 1 j) E 1 d If such embedding exist, G is called d-quasi Unit Disk Graph (d-qudg) 1-QUDG = Unit Disk Graph (UDG) The smaller d in a d-qudg the more realistic is the model Assumptions: 2D embedding d-qude, for d

10 Patterns: Terminology, Assumptions (2) Graph Properties: Graph G(V, E) Vertex-induced subgraph - is a subset of the vertices of G together with any edges whose endpoints are both in this subset. Chordless cycle - vertex-induced subgraph of G, which is a cycle of the length at least 4 with no cycle chord. Independent set - is a maximum subset of nodes such that there is no edge between any two of them Independent set property (IP): I ( D) fit ( V ( C) ) C > then D is outside of C N fit 1 (N) fit 1 (N) fit 2/2 (N) d 10

11 Patterns: Terminology, Assumptions (3) Vertex-induced subgraph Constructed from selected vertices Chordless Cycle With allowed and forbidden edges (chords) 11

12 Patterns: Terminology, Assumptions (4) C C: I C D: Independent set Constructed from the whole subgraph Independent set property I ( D) fit ( V ( C) ) C > d 12

13 Pattern Concept No coordinate knowledge required Reliable recognition of interior nodes Exterior nodes recognition requires consideration of all possible embeddings d-qudg model is used for Every node individually decides if it is an internal node d Pattern with respect to d is a vertex-induced subgraph which guarantees that for any d-qude a certain subset of vertices lies inside 13

14 Patterns for UDGs (Examples) d=1 UDE a) b) c) d) Insufficient constructions for UDG e) f) g) h) Patterns for UDG 14

15 Patterns for d-qudgs (Examples) d 2/2 d-qude a) b) c) d) Insufficient constructions for d-qudg, if d < 1 e) f) g) Patterns for d-qudg (f) and g) are taken from A. Kröller et al.) 15

16 Weak and trong Patterns Pattern(,d) n is a vertex-induced subgraph of G composed of chordless cycles C i such that i, j = 0.. n 1, i j the following conditions hold: (1) (2) V Ci V C C i j N1( ) = C..., 0 C 1 2 n 8 1, j = 2, j 3, j ( i ± 1) mod n = ( i ± 1) mod n n = ( i ± 1) mod n n > = (3) For Ci,(Pattern(, d) \ N1( Ci )) the extended independent set property holds (4) For Ci, j ic j there exists no critical intersection (5) For each conjunction Ji Ci, C i neither V ( + 1) mod J i \ 2 or V Ji \ = 1 and an edge exists between N ( J \ ) and N J \ i C i 1 ( i ) C( i + 1) mod n 16

17 Chordless Cycle Combinations Relations between the think cycle and other cycles of the pattern: (1) It may contain them (b) (2) It may intersect them (c,d,e) (3) It might lie on the different side of them (a,d,e) (4) It might be reflected (b,c) 17

18 Extended Independent et Property (eip) N fit 1 (N) fit 1 (N) fit 2/2 (N)

19 Critical Intersection (1) 1. elect one chordless cycle 2. Find independent nodes in 3. Find independent nodes in C i C i ( x) N1( Ci If a chordless cycles appears to be partitioned in at least 2 connected parts in ( x) N1( Ci ) fail(ip) 4. Enumerate the nodes of each chordless cycle in (x) with letters: same letter in C j from node to the first independent node. otherwise - different letters 5. Nodes and edges connecting forbidden letters combinations compose critical intersection ) 19

20 Critical Intersection (2) Types of critical intersection (1) Vertex-based critical intersection (2) Edge-based critical intersection (one of the dotted edges must exist) 20

21 upporting Lemmas (1) Lemma 4.1. Let x, y, w, v be four different nodes in V, where xy E and wv E. Assume the straight-line d-qude (for d 2/2 ) of xy and wv intersect. Then at least one of the edges in F = {xv, vy, yw,wx} is also in E. [A. Kröller et al.] Lemma 4.2. Let x, y, w, v be four different nodes in V, where xy E, wv E and xw E. Assume the straight-line UDE of xy and wv intersect. Then at least one of the edges in F = {xv, yw} is also in E. v v This edge might exist y y x x w w 21

22 upporting Lemmas (2) Lemma 4.3. If node VG is the seed of a weak pattern P(, 1) G, then is an inner node for any UDE of G. Lemma 4.4. If node VG is the seed of a strong pattern P (, d) G, then is an inner node for any d-qude of G with d 2/2. trong Pattern Found by a node in its neighborhood graph 22

23 Pattern Properties trong patterns provide distance guarantees: d 2 1 that can be accumulated into nesting levels 4 Inclusion property: if Pattern(,d) then Pattern(,q), q d Maximum pattern cardinality: π / arcsin( d / 2) Discreteness: for UDGs both weak and strong patterns can be used, but for d-qudgs only strong patterns make sense oundness Incompleteness Time: O( n h+ 3 3 max h ), space: O( n max, h ), message: O( n max h ) 23

24 Qualitative Evaluation (1) Topo.: UDG, 180 nodes grid, 4.12 avg. degree. Params: UDG, h=6. Inner nodes: 62 (L1:29, L2:33) 24

25 Qualitative Evaluation (2) Topo.: UDG, 400 nodes random, 6.74 avg. degree. Params: UDG, h=6. Inner nodes: 130 (L1:41, L2:88, L3:1) 25

26 Qualitative Evaluation (3) Topo.:UDG, 1600 nodes grid, avg. degree. Params:UDG, h=3, Inner nodes:936 (L1:322,L2:479,L3:111,L4:24) 26

27 Qualitative Evaluation (4) Topo.: 0.75-QUDG, 250 nodes grid, 5.11 avg. degree. Params: d-qudg, h=6. Inner nodes: 96 (L2:96) 27

28 Qualitative Evaluation (5) Topo.: 0.75-QUDG, 500 nodes grid, 6.89 avg. degree. Params: d-qudg, h=5. Inner nodes: 283 (L2:163, L4:113, L6:7) 28

29 Qualitative Evaluation (6) Topo.: 0.75-QUDG, 500 nodes grid, avg. degree. Params: d-qudg, h=3, Inner nodes: 276 (L2:165, L4:104, L6:4) 29

30 Quantitative Evaluation (1) Chordless cycles found in the neighborhood with and without ϻ-filtering imilarity metric reduces the number of chordless cycles to store by avoiding storing similar ones. ϻ-filtering: Center node c (C) relative to is the node in C that has the greatest hop distance to. Two cycles are similar, if they share the same anchors and the center node of one cycle is also part of the other. The longer of similar cycles is kept to increase the probability of constructing a strong pattern. 30

31 Quantitative Evaluation (2) Chordless cycles found in the neighborhood with and without ϻ-filtering over the cycle length 31

32 Quantitative Evaluation (3) Required Pattern Cardinality 32

33 Adaptation In random topologies there usually are areas with a very high average node degree and areas with a very low average node degree in the same network. In this example, the same value of h was selected for the complete network. Our algorithm finds nearly all inner nodes in the left part where the value of h is sufficient for the average node degree of this area. However, the sparser area in the right part of the figure requires a greater value of h. This motivates the need for adapting the value of h to the local density of the region, Topo.: 0.75-QUDG, 330 nodes rnd., 7.67 avg. degree. Params: which we plan to investigate as QUDG, h=5, Inner nodes: 87 (L2:87) part of future work. 33