Abstract
We analyze continuous and discrete symmetries of the maximum lifetime problem in two dimensional sensor networks.We show how a symmetry of the network and invariance of the problem under a given transformation group G can be utilized to simplify its solution. We prove that for a G-invariant maximum lifetime problem there exists a G-invariant solution.Constraints which follow from the G-invariance allows us to reduce the problem and its solution to the subset of the sensor network. The subset we call an optimal fundamental region of network with respect to the action of the symmetry group G. We analyze in detail solutions of the maximum network lifetime problem invariant under a group of isometry transformations of a two dimensional Euclidean plane.
Highlights
Let us denote by SKN a sensor network built of N sensors and K data collectors
Because a selection of a fundamental region F for given symmetry group G is not unique, and the problem cannot be reduced for every fundamental region, we show in this paper how to construct the optimal one and we investigate its properties
Proposition 5 Let q be a solution of RðMÞ-invariant problem (2) for SKN [ Cð0Þ network with Ei;j satisfying (15), the sensors pm:i from the region Vm can send their data to the elements pm0:i of the sensor network SKN [ Cð0Þ only when they lie in Vm [ VmÆÇ1, i.e., Fig. 3 The dashed arrows optimal data transmission path between elements of SKN which lie on different orbits
Summary
Let us denote by SKN a sensor network built of N sensors and K data collectors. We split the set SKN into two subsets, the set of data collectors CK and the set of sensors SN, such that SKN 1⁄4 CK [ SN. That the sensors can send their data to any element of the network SKN, the data collectors can receive any amount of data without costs, the initial energy of each sensor and the maximum amount of data which the sensor can send to other nodes of the network SKN is sufficiently large that at least one solution of the problem exists, i.e., there is no upper bound for qi;jðpÞ. The second formula defines the energy consumed by each sensor to send all of its data in a one cycle of the network lifetime. This equation is a matrix form of (1).
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