Sentry Selection in Wireless Networks

Abstract

Let P be a Poisson process of intensity one in the infinite plane R 2 . We surround each point x of P by the open disc of radius r centred at x. Now let S n be a fixed disc of area n, and let C r (S n ) be the set of discs which intersect S n . Write E k r for the event that C r (S n ) is a k-cover of S n , and F k r for the event that C r (S n ) may be partitioned into k disjoint single covers of S n . We prove thatlog n , and that this result is best possible. We also give improved estimates for P(E k r ). Finally, we study the obstructions to k-partitionability in more detail. As part of this study, we prove a classification theorem for (deterministic) covers of R 2 with half-planes that cannot be partitioned into two single covers.

Sentry Selection in Sensor Networks: Theory and Algorithms

“…The proof of Theorem 2, which is rather complicated, is given in [5]. The same paper also contains a short proof of Theorem 1.…”

“…These facts point us towards the investigation of non-2-partitionable 2-covers with half-planes. In [5], the following theorem is proved. C 3 s and C 5 s are the configurations which occur with probability Θ(1/ log n) and make Theorem 2 best possible for k = 2.…”

Secrecy coverage in two dimensions

“…If the distance of i from pq is c i /t then the heights of D p and D q above m 0 are asymptotically t 3 Figure 1) is entirely contained in R ∪ R . Therefore, s will not be covered by A λ , since, by construction, V ⊂ R ∪ R is free of black points; all black discs will be stopped by p, q, r, or another red point before they cover s .…”

“…For instance, writing πr 2 = log n + log log n + t, Svante Janson proved in 1986 [7] that as n → ∞, coverage occurs with probability asymptotically exp(−e −t ). One approach to this result [2], [3] uses the fact that the obstructions to coverage are small uncovered regions, which essentially form their own Poisson process, of intensity e −t /n. Although these uncovered regions may be of different shapes, they are all roughly the same size.…”

“…This model, based on the secrecy graph [5], was inspired by the issue of security in wireless networks, and was studied in [11], where it was proved that if λ 3 n → ∞ then p λ (n) → 0, while if λ 3 n(log n) 3 → 0 then p λ (n) → 1. In this paper we prove that the correct indicator of coverage is f (λ, n) = λ 3 n log n. Specifically, if λ 3 n log n = y then, for sufficiently large n, e −8π 2 y ≤ p λ (n) ≤ e −2π 2 y/3 .…”