Rigidity, Computation, and Randomization in Network Localization (original) (raw)

A Theory of Network Localization

IEEE Transactions on Mobile Computing, 2006

In this paper we provide a theoretical foundation for the problem of network localization in which some nodes know their locations and other nodes determine their locations by measuring the distances to their neighbors. We construct grounded graphs to model network localization and apply graph rigidity theory to test the conditions for unique localizability and to construct uniquely localizable networks. We further study the computational complexity of network localization and investigate a subclass of grounded graphs where localization can be computed efficiently. We conclude with a discussion of localization in sensor networks where the sensors are placed randomly.

A theoretical analysis of the conditions for unambiguous node localization in sensor networks

2004

In this paper we provide a theoretical foundation for the problem of network localization in which some nodes know their locations and other nodes determine their locations by measuring distances or bearings to their neighbors. Distance information is the separation between two nodes connected by a sensing/communication link. Bearing is the angle between a sensing/communication link and the x-axis of a node's local coordinate system. We construct grounded graphs to model network localization and apply graph rigidity theory and parallel drawings to test the conditions for unique localizability and to construct uniquely localizable networks. We further investigate partially localizable networks.

On the Computational Complexity of Sensor Network Localization

Algorithmic Aspects of Wireless Sensor Networks, 2004

Determining the positions of the sensor nodes in a network is essential to many network functionalities such as routing, coverage and tracking, and event detection. The localization problem for sensor networks is to reconstruct the positions of all of the sensors in a network, given the distances between all pairs of sensors that are within some radius r of each other. In the past few years, many algorithms for solving the localization problem were proposed, without knowing the computational complexity of the problem. In this paper, we show that no polynomial-time algorithm can solve this problem in the worst case, even for sets of distance pairs for which a unique solution exists, unless RP = NP. We also discuss the consequences of our result and present open problems.

Further results on sensor network localization using rigidity

Proceeedings of the Second European Workshop on Wireless Sensor Networks, 2005., 2005

Two further results, which extend the previous work on the use of rigidity in sensor network localization, are given, The previous work provided the conditions for the localization of an entire network in which some nodes know their locations and other nodes determine their locations by measuring the distances to their neighbors, First, the paper gives the conditions for partial localization of a subnetwork when an entire network i s not localizable. Second, the paper gives the conditions for localization in which some nodes know their locations and other nodes determine their locations by measuring the bearings (angle of arrivals) to their neighbors rather than the distances. 0-7803-880 1-1/05/$20.00 (c)2005 IEEE.

EFFECT OF RIGIDITY ON TRILATERATION TECHNIQUE FOR LOCALIZATION IN WIRELESS SENSOR NETWORKS

The localization of wireless sensor networks is an important problem where the location of wireless sensors is determined using the distance between sensors. Trilateration is a geometric technique used to find location of points in 2D using distances. Using geometry, one can find the location of a point uniquely in 2D given its distance to three other points in 2D. The problem of finding the trilateration order of vertices even if the network of sensors is a uniquely localizabe is NP-Complete. The 2D localization problem is closely related to the problem of graph rigidity. A graph can be uniquely realized in 2D if and only if the underlying network graph is globally rigid. Therefore by examining the structure of the underlying graph for rigidity and localization guided by rigidity is another technique used in localization. We study the performance of trilateration which is based on geometry and local information to see if it is effected by graph rigidity which is a global property. In particular, we compare the performance of the trilateration on connected non-rigid networks and connected rigid networks. We focus on sparse networks graphs of lower radius.

Graphical properties of easily localizable sensor networks

Wireless Networks, 2009

The sensor network localization problem is one of determining the Euclidean positions of all sensors in a network given knowledge of the Euclidean positions of some, and knowledge of a number of inter-sensor distances. This paper identifies graphical properties which can ensure unique localizability, and further sets of properties which can ensure not only unique localizability but also provide guarantees on the associated computational complexity, which can even be linear in the number of sensors on occasions. Sensor networks with minimal connectedness properties in which sensor transmit powers can be increased to increase the sensing radius lend themselves to the acquiring of the needed graphical properties. Results are presented for networks in both two and three dimensions.

Network localization in partially localizable networks

2005

Knowing the positions of the nodes in a network is essential to many next generation pervasive and sensor network functionalities. Although many network localization systems have recently been proposed and evaluated, there has been no systematic study of partially localizable networks, i.e., networks in which there exist nodes whose positions cannot be uniquely determined. There is no existing study which correctly identifies precisely which nodes in a network are uniquely localizable and which are not. This absence of a sufficient uniqueness condition permits the computation of erroneous positions that may in turn lead applications to produce flawed results. In this paper, in addition to demonstrating the relevance of networks that may not be fully localizable, we design the first framework for two dimensional network localization with an efficient component to correctly determine which nodes are localizable and which are not. Implementing this system, we conduct comprehensive evaluations of network localizability, providing guidelines for both network design and deployment. Furthermore, we study an integration of traditional geographic routing with geographic routing over virtual coordinates in the partially localizable network setting. We show that this novel cross-layer integration yields good performance, and argue that such optimizations will be likely be necessary to ensure acceptable application performance in partially localizable networks.

Localization in wireless sensor networks

2000

A fundamental problem in wireless sensor networks is localization -the determination of the geographical locations of sensors. Most existing localization algorithms were designed to work well either in networks of static sensors or networks in which all sensors are mobile. In this paper, we propose two localization algorithms, MSL and MSL*, that work well when any number of sensors are static or mobile. MSL and MSL* are range-free algorithms -they do not require that sensors are equipped with hardware to measure signal strengths, angles of arrival of signals or distances to other sensors. We present simulation results to demonstrate that MSL and MSL* outperform existing algorithms in terms of localization error in very different mobility conditions. MSL* outperforms MSL in most scenarios, but incurs a higher communication cost. MSL outperforms MSL* when there is significant irregularity in the radio range. We also point out some problems with a well known lower bound for the error in any range-free localization algorithm in static sensor networks.

Analysis of Node Localization in Wireless Sensor Networks

2012

1. Abstract Sensor networks are dense wireless networks of small, low-cost sensors, which collect and disseminate environmental data. Sensor nodes are very small, lightweight, and unobtrusive. The problem of localization, that is, “determining where a given node is physically located in a network”, can be mainly divided into two parts range-based (fine-grained) or range-free (coarse-grained) schemes. This Paper presents the analysis of range based algorithms on the basis of few network parameters (Network Size, Anchor node Density, Array node density) and tried to find out the best range based algorithms by doing simulation on matlab. The metric selected for the analysis is Standard Deviation of localization error.

Localizability of Wireless Sensor Networks: Beyond Wheel Extension

Lecture Notes in Computer Science, 2013

A network is called localizable if the positions of all the nodes of the network can be computed uniquely. If a network is localizable and embedded in plane with generic configuration, the positions of the nodes may be computed uniquely in finite time. Therefore, identifying localizable networks is an important function. If the complete information about the network is available at a single place, localizability can be tested in polynomial time. In a distributed environment, networks with trilateration orderings (popular in real applications) and wheel extensions (a specific class of localizable networks) embedded in plane can be identified by existing techniques. We propose a distributed technique which efficiently identifies a larger class of localizable networks. This class covers both trilateration and wheel extensions. In reality, exact distance is almost impossible or costly. The proposed algorithm based only on connectivity information. It requires no distance information.