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Coverage and Connectivity Issues in Sensor Networks

Coverage and Connectivity Issues in Sensor Networks. Ten-Hwang Lai Ohio State University. A Sensor Node. Memory (Application). Transmission range. Processor. Sensing range. Network Interface. Actuator. Sensor. Sensor Deployment. How to deploy sensors over a field?

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Coverage and Connectivity Issues in Sensor Networks

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  1. Coverage and Connectivity Issues in Sensor Networks Ten-Hwang Lai Ohio State University

  2. A Sensor Node Memory (Application) Transmission range Processor Sensing range Network Interface Actuator Sensor

  3. Sensor Deployment • How to deploy sensors over a field? • Deterministic, planned deployment • Random deployment • Desired properties of deployments? • Depends on applications • Connectivity • Coverage

  4. Coverage, Connectivity • Every point is covered by 1 or K sensors • 1-covered, K-covered • The sensor network is connected • K-connected • Others 1 8 R 2 7 6 3 4 5

  5. Coverage & Connectivity: not independent, not identical • If region is continuous & Rt> 2Rs • Region is covered sensors are connected Rt Rs

  6. Problem Tree coverage connectivity probabilistic algorithmic per-node homo homo heterogeneous barrier coverage k-connected blanket coverage

  7. Connectivity Issues

  8. Power Control for Connectivity • Adjust transmission range (power) • Resulting network is connected • Power consumption is minimum • Transmission range • Homogeneous • Node-based

  9. Power control for k-connectivity • For fault tolerance, k-connectivity is desirable. • k-connected graph: • K paths between every two nodes • with k-1 nodes removed, graph is still connected 1-connected 2-connected 3-connected

  10. Two Approaches • Probabilistic • How many neighbors are needed? • Algorithmic • Gmax connected • Construct a connected subgraph with desired properties

  11. Growing the Tree coverage connectivity probabilistic algorithmic

  12. Probabilistic Approach How many neighbors are necessary and/or sufficient to ensure connectivity?

  13. How many neighbors are needed? • Regular deployment of nodes – easy • Random deployment (Poisson distribution) • N: number of nodes in a unit square • Each node connects to its k nearest neighbors. • For what values of k, is network almost sure connected? P( network connected ) → 1, as N → ∞

  14. An Alternative View N • A square of area N. • Poisson distribution of a fixed density λ. • Each node connects to its k nearest neighbors. • For what values of k, is the network almost sure connected? P( network connected ) → 1, as N → ∞

  15. A Related Old Problem • Packet radio networks (1970s/80s) • Larger transmission radius • Good: more progress toward destination • Bad: more interference • Optimum transmission radius?

  16. Magic Number • Kleinrock and Silvester (1978) • Model: slotted Aloha & homogeneous radius R & Poisson distribution & maximize one hop progress toward destination. • Set R so that every station has 6 neighbors on average. • 6is the magic number.

  17. More Magic Numbers • Tobagi and Kleinrock (1984) • Eight is the magic number. • Other magic numbers for various protocols and models: • 5, 6, 7, 8

  18. Are Magic Numbers Magic? • Xue & Kumar (2002) • For the network to be almost sure connected, Θ(log n) neighbors are necessary and sufficient. • Heterogeneous radius 8, 7, 6, 5 (Magic numbers)

  19. Θ(log n) neighbors needed for connectivity • N: number of nodes (or area). K: number of neighbors. • Xue & Kumar (2002): • If K < 0.074 log N, almost sure disconnected. • If K > 5.1774 log N, almost sure connected. • 2004, improved to 0.3043 log N and 0.5139 log N 0.3043 0.5139 K 0.074 log n5.1774log n

  20. Penrose (1999): “On k-connectivity for a geometric random graph” • As n → infinity • Minimum transmission range required • R(n): for graph to be k-connected • R’(n): for graph to have degree k • Homogeneous radius • R(n) and R’(n) are almost sure equal • P( R(n) = R’(n) ) → 1, as n → infinity. • If every node has at leastk neighbors then network is almost sure k-connected.

  21. Any contradiction? • Xue & Kumar (improved by others): If every node connects to its • Log n nearest neighbors, almost sure connected. • 0.3 Log n nearest neighbors, almost sure disconnected. • Node-based radius • Penrose: • If every node has at least 1 neighbor, then almost sure 1-connected. • Homogeneous radius

  22. Applying Asymptotic Results • Applying Xue & Kumar’s result • “The K-Neigh Protocol for Symmetric Topology Control in Ad Hoc Networks” • Blough et al, MobiHoc’03. • Applying Penrose’s result • “On the Minimum Node Degree and Connectivity of a Wireless Multihop Network” • Bettstetter, MobiHoc’02.

  23. Applying Penrose’s result to power control (Bettstetter, MobiHoc’02) • Nodes deployed randomly. • Given: number of nodes n, node density λ, transmission range R. • P = Probability(every node has at least k neighbors) can be calculated. • Adjust R so that P ≈ 1. • With this transmission range, network is k-connected with high probability.

  24. Application 1 • N = 500 nodes • A = 1000m x 1000m • 3-connected required • R = ? • With R = 100 m, G has degree 3 with probability 0.99. • Thus, G is 3-connected with high probability. 500 nodes

  25. Application 2: How many sensors to deploy? • A = 1000m x 1000m • R = 50 m • 3-connected required • N = ? • Choose N such that P( G has degree 3) is sufficiently high.

  26. Growing the Tree coverage connectivity probabilistic algorithmic per-node homo radius radius Xue&Kumar Penrose

  27. Algorithmic Approach

  28. Gmax: network with maximum transmission range • Gmax: assumed to be connected • Construct a connected subgraph of Gmax • With certain desired properties • Distributed & localized algorithms • Use the subgraph for routing • Adjust power to reach just the desired neighbor • What subgraphs?

  29. What Subgraphs? • Gmax(V): Network with max trans range • RNG(V): Relative neighborhood graph • GG(V): Gabriel graph • YG(V): Yao graph • DG(V): Delaunay graph • LMST(V): Local minimum spanning tree graph GG(V):

  30. Desired Properties of Proximity Graphs • PG ∩ Gmax is connected (if Gmax is) • PG is sparse, having Θ(n) edges • Bounded degree • Degree RNG, GG, YG ≤ n – 1 (not bounded) • Degree of LMST ≤ 6 • Small stretch factor • Others • See “A Unified Energy-Efficient Topology for Unicast and Broadcast,” Mobicom 2005.

  31. Growing the Tree coverage connectivity probabilistic algorithmic per-node homo Homogeneous max trans. range various connected subgraphs

  32. Maximum transmission range • Homogeneous • Same max range for all nodes • PG ∩ Gmax is connected (if Gmax is) • Heterogeneous • Different max ranges • PG ∩ Gmax is not necessarily connected (even if Gmax is) • PG: existing PGs

  33. Growing the Tree coverage connectivity probabilistic algorithmic per-node homo max range homo heterogeneous k-connected

  34. Some references • N. Li and J. Hou, L Sha, “Design and analysis of an MST-based topology control algorithms,” INFOCOM 2003. • N. Li and J. Hou, “Topology control in heterogeneous wireless control networks,” INFOCOM 2004. • N. Li and J. Hou, “FLSS: a fault-tolerant topology control algorithm for wireless networks,” Mobicom 2004.

  35. Coverage Issues

  36. Simple Coverage Problem • Given an area and a sensor deployment • Question: Is the entire area covered? 1 8 R 2 7 6 3 4 5

  37. Is the perimeter covered?

  38. K-covered 1-covered 2-covered 3-covered

  39. K-Coverage Problem • Given: region, sensor deployment, integer k • Question: Is the entire region k-covered? 1 8 R 2 7 6 3 4 5

  40. Is the perimeter k-covered?

  41. Reference • C. Huang and Y. Tseng, “The coverage problem in a wireless sensor network,” • In WSNA, 2003. • Also MONET 2005.

  42. Density (or topology) Control • Given: an area and a sensor deployment • Problem: turn on/off sensors to maximize the sensor network’s life time

  43. PEAS and OGDC • PEAS: A robust energy conserving protocol for long-lived sensor networks • Fan Ye, et al (UCLA), ICNP 2002 • “Maintaining Sensing Coverage and Connectivity in Large Sensor Networks” • H. Zhang and J. Hou (UIUC), MobiCom 2003

  44. PEAS: basic ideas • How often to wake up? • How to determine whether to work or not? Wake-up rate? yes Wake up Sleep Go to Work? work no

  45. How often to wake up? • Desired: the total wake-up rate around a node equals some given value

  46. Inter Wake-up Time • f(t) = λ exp(- λt) • exponential distribution • λ = average # of wake-ups per unit time

  47. Wake-up rates A f(t) = λ exp(- λt) B f(t) = λ’ exp(- λ’t) A + B: f(t) = (λ + λ’) exp(- (λ + λ’) t)

  48. Adjust wake-up rates • Working node knows • Desired total wake-up rate λd • Measured total wake-up rate λm • When a node wakes up, adjusts its λ by λ := λ (λd / λm)

  49. Go to work or return to sleep? • Depends on whether there is a working node nearby. Rp Go back to sleep go to work

  50. Is the resulting network covered or connected? • If Rt ≥ (1 + √5) Rp and … then P(connected) → 1 • Simulation results show good coverage

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