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Network Optimization

Network Optimization. Lecture 16 - 1. Critical Transmission Range (1/5). Definition 4.0.1 (CTR for connectivity) Suppose n nodes are placed in a certain region R = [0 , l ] d , with d = 1 , 2 , or 3 .

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Network Optimization

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  1. Network Optimization Lecture 16 - 1

  2. Critical Transmission Range (1/5) • Definition 4.0.1 (CTR for connectivity) • Suppose n nodes are placed in a certain region R = [0, l]d , with d = 1, 2, or 3. • Which is the minimum value of r such that the r-homogeneous range assignment is connecting? disconnected CTR connected CTR

  3. Critical Transmission Range (2/5) • Definition: Tree • Definition: Spanning tree • Definition: Minimum spanning tree (MST) • Definition: Euclidean MST (EMST)

  4. Critical Transmission Range (3/5) • Q: The CTR for connectivity rC of the network composed of nodes in N ? 4 4 1 3 5 10 9 9 7 9 18 8 2 9 2 4 6 8 9 3 9

  5. Critical Transmission Range (4/5) • e:longest edge in T • l(e):length of e • l(e)-homogeneous RA produces a graph containing T • By definition of EMST • T is connected • l(e)-homogeneous RA is connected • By definition of CTR • rC l(e) e EMST T

  6. Critical Transmission Range (5/5) • T1, T2: two connected component by removing e • By definition of EMST • e is the shortest edge connecting any pair (u, v), u ∈ T1 and v ∈ T2 • any node in T1 is at distance at least l(e) from any node in T2 • setting the transmitting range smaller than l(e) results in a disconnected graph • rC =l(e) u T1 e v T2 EMST T

  7. CTR in Probabilistic Perspective • Shortcomings of deterministic approach • Finding the longest edge in the EMST is not apt to distributed implementation • Requirement of knowing exact node positions is very strong • Probabilistic perspective • Given a number n of nodes to be deployed in a certain region R, and given distribution F, which is the minimum value rC(n,F)of the transmitting range that ensures connectivity with high probability?

  8. CTR in Probabilistic Perspective • Corollary 4.1.2 If R is the unit square and n nodes are distributed uniformly at random in R, then the CTR for connectivity is f (n) is an arbitrary function such that

  9. CTR in Probabilistic Perspective • Proof • Mn:random variable of the length of the longest MST edge built on the n nodes • (Penrose 1997) For any β ∈ R,

  10. CTR in Probabilistic Perspective • Proof (cont.) • For any arbitrary function f(n) such that limn→∞f (n) = +∞ • Gr:communication graph obtained when the transmitting range is set to r • Gris w.h.p. connected if and only if e.g. set f (n) = loglog n

  11. CTR in Probabilistic Perspective • How fast is the convergence of the actual CTR to the asymptotic value?

  12. CTR in Probabilistic Perspective -52.0% -28.7%

  13. CTR in Probabilistic Perspective • Giant component phenomenon • Relatively large connected component is formed quite soon when increasing the transmission range

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