We are given a set of sensors at given locations, a set of potential locations for placing base stations (BSs, or sinks), and another set of potential locations for placing wireless relay nodes. There is a cost for placing a BS and a cost for placing a relay. The problem we consider is to select a set of BS locations, a set of relay locations, and an association of sensor nodes with the selected BS locations, so that number of hops in the path from each sensor to its BS is bounded by hmax, and among all such feasible networks, the cost of the selected network is the minimum. The hop count bound suffices to ensure a certain probability of the data being delivered to the BS within a given maximum delay under a light traffic model. We observe that the problem is NP-Hard, and is hard to even approximate within a constant factor. For this problem, we propose a polynomial time approximation algorithm (SmartSelect) based on a relay placement algorithm proposed in our earlier work, along with a modification of the greedy algorithm for weighted set cover. We have analyzed the worst case approximation guarantee for this algorithm. We have also proposed a polynomial time heuristic to improve upon the solution provided by SmartSelect. Our numerical results demonstrate that the algorithms provide good quality solutions using very little computation time in various randomly generated network scenarios.
Several variations of the optimal node placement problem have been studied in the literature. @cite_8 @cite_9 @cite_19 studied variations of the problem of , where a minimum number of relays have to be placed in a two-dimensional region (they can be placed anywhere; no locations are given) to obtain a tree spanning a given set of vertices (sources and BS). No QoS constraint was imposed in their formulations. They showed the problems to be NP-Hard, and proposed approximation algorithms. @cite_7 studied the problem of optimal relay placement for @math connectivity. They proposed an @math approximation algorithm for the problem with any @math . However, they also did not impose any QoS constraint.
@cite_19 @cite_9 @cite_7 @cite_8
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A wireless sensor network consists of many low-cost, low-power sensor nodes, which can perform sensing, simple computation, and transmission of sensed information. Long distance transmission by sensor nodes is not energy efficient since energy consumption is a superlinear function of the transmission distance. One approach to prolonging network lifetime while preserving network connectivity is to deploy a small number of costly, but more powerful, relay nodes whose main task is communication with other sensor or relay nodes. In this paper, we assume that sensor nodes have communication range r>0, while relay nodes have communication range Rgesr, and we study two versions of relay node placement problems. In the first version, we want to deploy the minimum number of relay nodes so that, between each pair of sensor nodes, there is a connecting path consisting of relay and or sensor nodes. In the second version, we want to deploy the minimum number of relay nodes so that, between each pair of sensor nodes, there is a connecting path consisting solely of relay nodes. We present a polynomial time 7-approximation algorithm for the first problem and a polynomial time (5+epsi)-approximation algorithm for the second problem, where epsi>0 can be any given constant This paper addresses the following relay sensor placement problem: given the set of duty sensors in the plane and the upper bound of the transmission range, compute the minimum number of relay sensors such that the induced topology by all sensors is globally connected. This problem is motivated by practically considering the tradeoff among performance, lifetime, and cost when designing sensor networks. In our study, this problem is modelled by a NP-hard network optimization problem named Steiner Minimum Tree with Minimum number of Steiner Points and bounded edge length (SMT-MSP). In this paper, we propose two approximate algorithms, and conduct detailed performance analysis. The first algorithm has a performance ratio of 3 and the second has a performance ratio of 2.5. We consider the problem of deploying or repairing a sensor network to guarantee a specified level of multi-path connectivity (k-connectivity) between all nodes. Such a guarantee simultaneously provides fault tolerance against node failures and high capacity through multi-path routing. We design and analyze the first algorithms that place an almost-minimum number of additional sensors to augment an existing network into a k-connected network, for any desired parameter k. Our algorithms have provable guarantees on the quality of the solution. Specifically, we prove that the number of additional sensors is within a constant factor of the absolute minimum, for any fixed k. We have implemented greedy and distributed versions of this algorithm, and demonstrate in simulation that they produce high-quality placements for the additional sensors. We are also in the process of using our algorithms to deploy nodes in a physical sensor network using a mobile robot. Abstract In this paper, we study the Steiner tree problem with minimum number of Steiner points and bounded edge-length (STPMSPBEL), which asks for a tree interconnecting a given set of n terminal points and a minimum number of Steiner points such that the Euclidean length of each edge is no more than a given positive constant. This problem has applications in VLSI design, WDM optimal networks and wireless communications. We prove that this problem is NP-complete and present a polynomial time approximation algorithm whose worst-case performance ratio is 5.