Fast Estimation of Area-Coverage for Wireless Sensor Networks Based on Digital Geometry

In this paper, we propose a fast and nearly-accurate method of estimating the area covered by a set of <inline-formula><tex-math notation="LaTeX">$n$</tex-math><alternatives> <inline-graphic xlink:href="saha-ieq1-2598737.gif"/></alternatives></inline-formula> identical and active sensor nodes that are randomly scattered over a 2D region. Since the estimation of collective coverage-area turns out to be computationally complex in the euclidean geometry, we represent each Euclidean circle in <inline-formula> <tex-math notation="LaTeX">$\mathbb {R}^2$</tex-math><alternatives><inline-graphic xlink:href="saha-ieq2-2598737.gif"/> </alternatives></inline-formula> with a digital circle in <inline-formula><tex-math notation="LaTeX">$\mathbb {Z}^2$ </tex-math><alternatives><inline-graphic xlink:href="saha-ieq3-2598737.gif"/></alternatives></inline-formula> for enabling faster computation. Based on the underlying geometric properties of digital circles, we present a novel <inline-formula><tex-math notation="LaTeX">$O(n \log n)$</tex-math><alternatives> <inline-graphic xlink:href="saha-ieq4-2598737.gif"/></alternatives></inline-formula> centralized algorithm and <inline-formula><tex-math notation="LaTeX">$O(d^2\log d)$</tex-math><alternatives> <inline-graphic xlink:href="saha-ieq5-2598737.gif"/></alternatives></inline-formula> distributed algorithm for coverage estimation, where <inline-formula><tex-math notation="LaTeX">$d$</tex-math><alternatives> <inline-graphic xlink:href="saha-ieq6-2598737.gif"/></alternatives></inline-formula> denotes the maximum degree of a node. In order to further expedite the estimation procedure, we approximate each digital circle by the tightest square that encloses it as well as by the largest square inscribed within it. Such approximation allows us to estimate the coverage-area, in a much simpler way, based on the intersection-geometry of a set of axis-parallel rectangles. Our experiments with random deployment of nodes demonstrate that the proposed algorithms estimate the area coverage with a maximum error of only 1.5 percent, while reducing the computational effort significantly compared to earlier work. The technique needs only simple data structures and requires a few primitive integer operations in contrast to classical methods, which need extensive floating-point computations for exact estimation. Furthermore, for an over-deployed network, the estimation provides an almost-exact measure of the covered area.

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