In this paper, we propose solutions for the minimum-energy broadcasting problem for cross networks, where N nodes are located on two perpendicular lines. Our solutions consist of an algorithm which finds the optimal assignment in polynomial time, a near-optimal algorithm with less complexity (O(N)), and a distributed algorithm with complexity O(1) that gives acceptable results. To the best of our knowledge, this is the first study presenting an optimal solution for the minimum-energy broadcasting problem for a 2-D network (with cross configuration). We compare our algorithms with the broadcast incremental power (BIP) algorithm, one of the most commonly used methods for solving this problem with complexity O(N2). The results show that while the proposed optimal algorithm finds the best solution, our near-optimal algorithm performs better than BIP in all cases, and the distributed algorithm performs close to it. The performance of our non-optimal algorithms tend to be closer to the optimal solution for larger networks. We prove that all the algorithms perform the same in the asymptotic regime.

In this paper, we study the hop-count properties of one-dimensional wireless ad-hoc networks, where the nodes are placed independently and identically according to a Poisson distribution with an arbitrary density function. We derive exact equations to calculate the probability mass function of the number of hops needed for a node located at an arbitrary location in the network to receive a message from the source (located at one end of the linear network). Based on the derived formulas, we then propose localization methods. Through simulations, we show that our best proposed localization method not only has a competitive performance for a range-free method, but also outperforms range-based methods with a local distance measurement error of 10% or more. An important feature of our methods is that they are applicable to arbitrary densities. This is unlike the existing methods that are limited only to the case of uniform node densities. Moreover, the hop-count equations derived in this work can be used in analyzing other aspects of broadcasting protocols such as location verification, quality of service, and delay.

In this paper, we study the hop-count properties of one-dimensional wireless ad-hoc networks, where the nodes are placed independently and identically according to a Poisson distribution with an arbitrary density function. We derive exact equations to calculate the probability mass function of two hop-count random variables: the number of hops needed for a node located at an arbitrary location in the network to receive a message from a node located at one end of the linear network, and the number of hops needed for a node located at one end of the network to receive a message from a node at an arbitrary location. Based on the derived formulas, we then propose localization and location verification methods. Through simulations, we show that our proposed localization method not only has a competitive performance for a range-free method, but also outperforms range-based methods with a local distance measurement error of 10% or more. Furthermore, the proposed location verification protocol is shown to have better results compared to the existing verification systems that also use the hop-count information. An important feature of our methods is that they are applicable to arbitrary densities. This is unlike the existing methods that are limited only to the case of uniform node densities. Using simulations, we also evaluate the proposed schemes in the presence of Rician fading and show that their performance is rather robust with respect to the change in the fading parameter. Moreover, the hop-count equations derived in this work can be used in analyzing other aspects of broadcasting protocols such as quality of service and delay.