Optimal deployment of large wireless sensor networks

Toumpis, Stavros; Tassiulas, Leandros

doi:10.1109/tit.2006.876256

Cited by 118 publications

(60 citation statements)

References 17 publications

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“…The p = 2 case can be viewed as a trade-off between packet delay and optimal load balancing. Furthermore, in [4] it is proved that while maintaining connectivity of network, minimizing (5) minimizes the number of nodes required to be deployed after the optimizer is discretized. Also, it is shown in [2] that the vector field that minimizes (5), D * (z), has the following property:…”

Section: Background: Continuous Information Flow Modelmentioning

confidence: 99%

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Filling Gap between Discrete and Continuous Space Flow Models in Dense Wireless Networks

Anaraki

Kalantari

2011

2011 IEEE International Conference on Communications (ICC)

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Abstract-Optimizing information flow in a dense wireless network using discrete methods can be computationally prohibitive. Instead of treating the nodes as discrete entities, these networks can be modeled as continuum of nodes providing a medium for information transport. To model information routes in continuous space, information flow vector field is defined over the geographical domain of the network. At each point of the network, the orientation of the vector field shows the direction of the flow of information, and its magnitude shows the density of information flow. Using multivariate calculus techniques in continuous domain, an information flow vector field can be found such that it minimizes a suitable cost function. Then the solution is discretized. Conventionally, a centralized method of calculating the optimal information flow in the network is suggested; however, using a centralized method to optimize information flow in a dynamic network is prohibitive. Additionally, the value of information flow vector field is needed only at the locations of nodes in the network. This poses a gap between the continuous space and discrete space models of information flow in dense wireless networks. This gap is how to calculate and apply the optimum information flow derived in continuous domain in a network with finite number of nodes. As a first step to fill this gap, a specific quadratic cost function is considered. It is proved that the the vector field that minimizes this cost function is irrotational, thus it is written as the gradient of a potential function. This potential function satisfies a Poisson Partial Differential Equation (PDE) which in conjunction with Neumann boundary condition has a unique solution up to a constant. The PDE resulted by optimization in continuous domain is first discretized and then solved in a distributed fashion. The solution requires only neighboring nodes to communicate with each other. The gradient of the resulting potential defines the routes that the traffic should be forwarded.

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Section: Background: Continuous Information Flow Modelmentioning

confidence: 99%

“…Solution of this PDE leads to a simple mechanism for routing the traffic: each node forwards the traffic to a neighbor with least potential (steepest potential decent). In [4], it is shown that this routing method, minimizes the number of nodes required to carry a specified information density in a network area.…”

Section: Introductionmentioning

confidence: 99%

Filling Gap between Discrete and Continuous Space Flow Models in Dense Wireless Networks

Anaraki

Kalantari

2011

2011 IEEE International Conference on Communications (ICC)

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“…In this case the optimal paths are obtained by solving a set of partial differential equations similar to Maxwell's equations. Using a similar analogy with electrostatics, Toumpis and Tassiulas [2] consider a related problem of optimal placement of the nodes in a dense sensor network. It is assumed that, at any point in the network, the information flows exactly in one direction.…”

Section: Related Workmentioning

confidence: 99%

“…In practice, this means that, locally, a packet is forwarded to the furthest reachable node in that direction. 1 At the macroscopic level, one is concerned with the end-to-end paths, and the assumption of a strong separation between the different scales justifies describing the underlying network as a continuous medium and the paths as smooth continuous curves [2][3][4][5][6][7][8][9][10]. In the present paper, we focus on studying the optimal paths at the macroscopic level.…”

Section: Introductionmentioning

confidence: 99%

On the optimality of field-line routing in massively dense wireless multi-hop networks

Hyytiä

Virtamo²

2009

Performance Evaluation

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A B S T R A C TWe consider the load balancing problem in large wireless multi-hop networks by applying the continuum approximation. The task is to find routes, geometric curves, such that the maximal traffic load in the network is minimized. In finite fixed networks, multi-path routes generally yield a lower congestion and thus allow higher throughput. In contrast, we show that in dense wireless multi-hop networks, the optimal load balancing can be achieved by a destination based single-path routing referred to as field-line routing. This is because any routing can be transformed to the corresponding field-line routing with the same or better performance, by using as paths, the field lines of the so-called destination flow associated with the original routing. The concepts are illustrated with two examples. In the case of a unit disk with unit traffic, the maximal load of 0.389 of a multi-path routing system is reduced to 0.343 by using the field-line routing. Similar improvements are also demonstrated for the unit square.

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“…This work was followed by [3,4], where the author showed that the methodology leads to an optimal deployment of sensor nodes. In this methodology a traffic density vector field models transportation of traffic in a wireless network.…”

Section: Introductionmentioning

confidence: 99%

A Steepest Descent Relocation Algorithm for Placement of Sinks in a Sensor Network

Kalantari

Shayman

2007

IEEE GLOBECOM 2007-2007 IEEE Global Telecommunications Conference

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Abstract-In this paper we use an information flow model for placement of traffic sinks in a wireless sensor network. Our mathematical model translates a communication network composed of countably many sensors into a continuum model described by a continuous vector field. This vector field models flow of information in a sensor network, and its magnitude is the intensity of the communication activity, and its orientation is the direction in which the traffic is forwarded. We use this formulation for design of a sensor network in which the traffic generated by the wireless sensors needs to be routed to one of the multiple traffic sinks in different locations of the network. We show that the optimal vector field satisfies a set of PDEs similar to Maxwell's equations in electrostatics, with Neumann boundary conditions. We prove that the orientation of the vector field at the location of each sink is the direction of the steepest descent of the total communication cost function of the network. This direction corresponds to the direction of force in an analogous electrostatics problem. We use this result in order to introduce iterations that relocate sinks and show that in the optimal placement of sinks, the value of the vector field is zero at the location of each sinks.

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Optimal deployment of large wireless sensor networks

Cited by 118 publications

References 17 publications

Filling Gap between Discrete and Continuous Space Flow Models in Dense Wireless Networks

Filling Gap between Discrete and Continuous Space Flow Models in Dense Wireless Networks

On the optimality of field-line routing in massively dense wireless multi-hop networks

A Steepest Descent Relocation Algorithm for Placement of Sinks in a Sensor Network

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