Abstract
AbstractA fundamental problem for wireless ad hoc networks is the assignment of suitable transmission powers to the wireless devices such that the resulting communication graph is connected. The goal is to minimize the total transmit power in order to maximize the life‐time of the network. Our aim is a probabilistic analysis of this power assignment problem. We prove complete convergence for arbitrary combinations of the dimension d and the distance‐power gradient p. Furthermore, we prove that the expected approximation ratio of the simple spanning tree heuristic is strictly less than its worst‐case ratio of 2.Our main technical novelties are two‐fold: First, we find a way to deal with the unbounded degree that the communication network induced by the optimal power assignment can have. Minimum spanning trees and traveling salesman tours, for which strong concentration results are known in Euclidean space, have bounded degree, which is heavily exploited in their analysis. Second, we apply a recent generalization of Azuma‐Hoeffding's inequality to prove complete convergence for the case for both power assignments and minimum spanning trees (MSTs). As far as we are aware, complete convergence for p > d has not been proved yet for any Euclidean functional. © 2017 The Authors Random Structures & Algorithms Published by Wiley Periodicals, Inc., 51, 483–505, 2017
Highlights
Wireless ad hoc networks have received significant attention due to their many applications in, for instance, environmental monitoring or emergency disaster relief, where wiring is difficult
We provide a probabilistic analysis of the minimum spanning trees (MSTs) heuristic for the geometric case
We introduce the notion of typically smooth Euclidean functionals, prove convergence of such functionals, and show that minimum spanning trees and power assignments are typically smooth
Summary
Wireless ad hoc networks have received significant attention due to their many applications in, for instance, environmental monitoring or emergency disaster relief, where wiring is difficult. Wireless ad hoc networks lack a backbone infrastructure. We consider the case that each node can adjust its transmit power for the purpose of power conservation. In the assignment of transmit powers, two conflicting effects have to be taken into account: if the transmit powers are too low, the resulting network may be disconnected. If the transmit powers are too high, the nodes run out of energy quickly. The goal of the power assignment problem is to assign transmit powers to the transceivers such that the resulting network is connected and the sum of transmit powers is minimized [13]
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