## Rigidity, Computation, and Randomization in Network Localization (2004)

### Cached

### Download Links

- [www.ieee-infocom.org]
- [www.cs.columbia.edu]
- [www.cs.columbia.edu]
- [www1.cs.columbia.edu]
- [www1.cs.columbia.edu]
- [www.cs.columbia.edu]
- [erente.googlepages.com]
- [www.cs.sunysb.edu]
- [athos.rutgers.edu]
- [www.cs.iastate.edu]
- [www.cs.sunysb.edu]
- [www.cs.yale.edu]
- [www.cs.rutgers.edu]
- [www-net.cs.yale.edu]
- [courses.csail.mit.edu]
- DBLP

### Other Repositories/Bibliography

Venue: | In Proceedings of IEEE INFOCOM ’04, Hong Kong |

Citations: | 81 - 14 self |

### BibTeX

@INPROCEEDINGS{Eren04rigidity,computation,,

author = {T. Eren and D. K. Goldenberg and W. Whiteley and Y. R Yang and A. S. Morse and B. D. O. Anderson and P. N. Belhumeur},

title = {Rigidity, Computation, and Randomization in Network Localization},

booktitle = {In Proceedings of IEEE INFOCOM ’04, Hong Kong},

year = {2004},

pages = {2673--2684}

}

### Years of Citing Articles

### OpenURL

### Abstract

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.

### Citations

1622 | GPSR: greedy perimeter stateless routing for wireless networks
- Karp, Kung
(Show Context)
Citation Context ...a printing job to the nearest printer. In sensor networks, the sensor nodes need to know their locations in order to detect and record events, or to route packets using geometric-aware routing, e.g., =-=[3]-=-. One method to determine the location of a node is manual configuration. However, this is unlikely to be feasible for any large-scale deployment or when nodes move often. Another possibility is GPS [... |

1310 | an in-building RF-based user location and tracking system
- Bahl, Padmanabhan
(Show Context)
Citation Context ... in connectedness above the phase transition at which the radius is minimal for total localizability. VII. RELATED WORK Network localization is an active research field, e.g., [28], [29], [30], [31], =-=[32]-=-, [33], [34], [35], [36], [37], [38], [39], [40], [5], [41], [6], [42]. The previous approaches can be classified into two types: coarse-grained and fine-grained. percentage localized nodes 1 0.9 0.8 ... |

1136 |
Extremal graph theory
- Bollobas
- 1978
(Show Context)
Citation Context ...803-8356-7/04/$20.00 (C) 2004 IEEE IEEE INFOCOM 2004sasymptotic and that because of this, we neglect collinearity as a low probability phenomenon. As in the case of the Erdös-Rényi random graph mode=-=l [24]-=-, there is a phase transition in the random geometric graph model at which the graph becomes connected with high probability [25]. Penrose [26] generalizes this to kconnectivity with the result that i... |

1102 | The active badge location system
- Want, Hopper, et al.
- 1992
(Show Context)
Citation Context ...due to a sudden increase in connectedness above the phase transition at which the radius is minimal for total localizability. VII. RELATED WORK Network localization is an active research field, e.g., =-=[28]-=-, [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [5], [41], [6], [42]. The previous approaches can be classified into two types: coarse-grained and fine-grained. percentage lo... |

896 | Next century challenges: Scalable coordination in sensor networks
- Estrin, Govindan, et al.
- 1999
(Show Context)
Citation Context ...crease in connectedness above the phase transition at which the radius is minimal for total localizability. VII. RELATED WORK Network localization is an active research field, e.g., [28], [29], [30], =-=[31]-=-, [32], [33], [34], [35], [36], [37], [38], [39], [40], [5], [41], [6], [42]. The previous approaches can be classified into two types: coarse-grained and fine-grained. percentage localized nodes 1 0.... |

753 | GPS-less low cost outdoor localization for very small devices
- Bulusu, Heidemann, et al.
- 2000
(Show Context)
Citation Context ...nnectedness above the phase transition at which the radius is minimal for total localizability. VII. RELATED WORK Network localization is an active research field, e.g., [28], [29], [30], [31], [32], =-=[33]-=-, [34], [35], [36], [37], [38], [39], [40], [5], [41], [6], [42]. The previous approaches can be classified into two types: coarse-grained and fine-grained. percentage localized nodes 1 0.9 0.8 0.7 0.... |

714 | Location systems for ubiquitous computing
- Hightower, Borriello
- 2001
(Show Context)
Citation Context ...the phase transition at which the radius is minimal for total localizability. VII. RELATED WORK Network localization is an active research field, e.g., [28], [29], [30], [31], [32], [33], [34], [35], =-=[36]-=-, [37], [38], [39], [40], [5], [41], [6], [42]. The previous approaches can be classified into two types: coarse-grained and fine-grained. percentage localized nodes 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 ... |

618 | Dynamic fine-grained localization in ad-hoc networks of sensors
- Savvides, Han, et al.
- 2001
(Show Context)
Citation Context ... view to the GPS satellites. Recently, novel schemes have been proposed to determine the locations of the nodes in a network where only some special nodes (called beacons) know their locations, e.g., =-=[5]-=-, [6]. In these schemes, network nodes measure the distances to their neighbors and then try to determine their locations. T. Eren and P.N. Belhumeur are supported by NSF grants ITR IIS-0085864, EIA-0... |

393 | Convex position estimation in wireless sensor networks
- Doherty, Pister, et al.
- 2001
(Show Context)
Citation Context ...on at which the radius is minimal for total localizability. VII. RELATED WORK Network localization is an active research field, e.g., [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], =-=[39]-=-, [40], [5], [41], [6], [42]. The previous approaches can be classified into two types: coarse-grained and fine-grained. percentage localized nodes 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 5 10 15 20... |

391 | A New Location Technique for the Active Office
- Ward, Jones, et al.
- 1997
(Show Context)
Citation Context ... a sudden increase in connectedness above the phase transition at which the radius is minimal for total localizability. VII. RELATED WORK Network localization is an active research field, e.g., [28], =-=[29]-=-, [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [5], [41], [6], [42]. The previous approaches can be classified into two types: coarse-grained and fine-grained. percentage localize... |

373 | GPS-Free positioning in mobile ad-hoc networks
- Hamdi, Hubaux
(Show Context)
Citation Context ...above the phase transition at which the radius is minimal for total localizability. VII. RELATED WORK Network localization is an active research field, e.g., [28], [29], [30], [31], [32], [33], [34], =-=[35]-=-, [36], [37], [38], [39], [40], [5], [41], [6], [42]. The previous approaches can be classified into two types: coarse-grained and fine-grained. percentage localized nodes 1 0.9 0.8 0.7 0.6 0.5 0.4 0.... |

372 | The challenges of mobile computing - Forman, Zahorjan - 1994 |

358 | Range-free localization schemes for large scale sensor networks
- He, Huang, et al.
- 2003
(Show Context)
Citation Context ... to the GPS satellites. Recently, novel schemes have been proposed to determine the locations of the nodes in a network where only some special nodes (called beacons) know their locations, e.g., [5], =-=[6]-=-. In these schemes, network nodes measure the distances to their neighbors and then try to determine their locations. T. Eren and P.N. Belhumeur are supported by NSF grants ITR IIS-0085864, EIA-02-244... |

334 | Ad hoc positioning system (APS) using AoA
- Niculescu, Nath
- 2003
(Show Context)
Citation Context ...nimal for total localizability. VII. RELATED WORK Network localization is an active research field, e.g., [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [5], [41], [6], =-=[42]-=-. The previous approaches can be classified into two types: coarse-grained and fine-grained. percentage localized nodes 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 5 10 15 20 step r = 0.1 r = 0.13 r = 0... |

303 | Robust positioning algorithms for Distributed ad hoc wireless sensor networks[C
- Savarese, Rabaey, et al.
(Show Context)
Citation Context ...adius is minimal for total localizability. VII. RELATED WORK Network localization is an active research field, e.g., [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [5], =-=[41]-=-, [6], [42]. The previous approaches can be classified into two types: coarse-grained and fine-grained. percentage localized nodes 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 5 10 15 20 step r = 0.1 r =... |

291 |
Global Positioning System Theory and Practice
- Hofmann-Wellenhoff
- 1997
(Show Context)
Citation Context ...]. One method to determine the location of a node is manual configuration. However, this is unlikely to be feasible for any large-scale deployment or when nodes move often. Another possibility is GPS =-=[4]-=-. However, GPS is costly both in terms of hardware and power requirements. Furthermore, since GPS requires line-of-sight between the receiver and the satellites, it may not work well in buildings or i... |

195 |
On graphs and rigidity of plane skeletal structures
- Laman
- 1970
(Show Context)
Citation Context ...or the plane we have a strong combinatorial characterization of the generically rigid graphs. We note that this leads to afast(O(|V | 2 ) algorithm for generic rigidity testing [13]. Theorem 4 (Laman =-=[14]): A g-=-raph G = (V,L) with n vertices is generically rigid in IR 2 if and only if L contains a subset E consisting of 2n − 3 edges with the property that for any nonempty subset E ′ ⊂ E, the number of ... |

179 |
Ad Hoc Positioning System
- Niculescu, Nath
- 2001
(Show Context)
Citation Context ...which the radius is minimal for total localizability. VII. RELATED WORK Network localization is an active research field, e.g., [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], =-=[40]-=-, [5], [41], [6], [42]. The previous approaches can be classified into two types: coarse-grained and fine-grained. percentage localized nodes 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 5 10 15 20 step ... |

178 |
Distance geometry and molecular conformation
- Crippen, Havel
- 1988
(Show Context)
Citation Context ...er vectors of points {x1,x2,...,x |V |} of the following form, minimize � (i,j)∈E (δ(i, j)− �xi − xj �) 2 . This formulation of the problem has been used by biologists studying molecular =-=conformation [20]-=-. Because such optimization is computationally expensive, strategies such as divide-and-conquer [21] and objective function smoothing [22] have been proposed. In the context of network localization, d... |

170 |
Dividing a Graph into Triconnected Components
- Hopcroft, Tarjan
- 1973
(Show Context)
Citation Context ...n graph G is generically globally rigid in IR 2 , one must establish that it is both 3-connected and redundantly rigid in IR 2 . Various tests for 3-connectivity are known, and we refer the reader to =-=[16], -=-[17] for details including measures of the complexity of the tests involved. Tests for redundant rigidity in IR 2 have been derived [13] based on variants of Laman’s theorem [14]. Since these proper... |

154 | Robust range estimation using acoustic and multimodal sensing
- Girod, Estrin
- 2001
(Show Context)
Citation Context ...ase transition at which the radius is minimal for total localizability. VII. RELATED WORK Network localization is an active research field, e.g., [28], [29], [30], [31], [32], [33], [34], [35], [36], =-=[37]-=-, [38], [39], [40], [5], [41], [6], [42]. The previous approaches can be classified into two types: coarse-grained and fine-grained. percentage localized nodes 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 ... |

151 |
Anit Chakraborty, and Hari Balakrishnan. The cricket location-support system
- Priyantha
- 2000
(Show Context)
Citation Context ...dness above the phase transition at which the radius is minimal for total localizability. VII. RELATED WORK Network localization is an active research field, e.g., [28], [29], [30], [31], [32], [33], =-=[34]-=-, [35], [36], [37], [38], [39], [40], [5], [41], [6], [42]. The previous approaches can be classified into two types: coarse-grained and fine-grained. percentage localized nodes 1 0.9 0.8 0.7 0.6 0.5 ... |

141 |
Embeddability of weighted graphs in k-space is strongly NP-hard
- Saxe
- 1979
(Show Context)
Citation Context ...lem is solvable are not known. 2) What is the computational complexity of network localization? Even though the computational complexity of general graph embeddability problem has been studied before =-=[7]-=-, the computational complexity of determining the locations of the nodes in a uniquely localizable network has not been studied. Furthermore, constraints on the communication and computational resourc... |

122 |
Some Computer Science Problems in Ubiquitous Computing
- Weiser
- 1993
(Show Context)
Citation Context ...tion in sensor networks where the sensors are placed randomly. I. INTRODUCTION Location service is a basic service of many emerging computing/networking paradigms. For example, in pervasive computing =-=[1]-=-, [2], knowing the locations of the computers and the printers in a building will allow a computer to send a printing job to the nearest printer. In sensor networks, the sensor nodes need to know thei... |

112 | Conditions for unique graph realizations
- Hendrickson
- 1992
(Show Context)
Citation Context ...rem 3 hold [9], [10]. For the plane we have a strong combinatorial characterization of the generically rigid graphs. We note that this leads to afast(O(|V | 2 ) algorithm for generic rigidity testing =-=[13]. -=-Theorem 4 (Laman [14]): A graph G = (V,L) with n vertices is generically rigid in IR 2 if and only if L contains a subset E consisting of 2n − 3 edges with the property that for any nonempty subset ... |

100 |
Designing a positioning system for finding things and people indoors
- Werb, Lanzl
- 1998
(Show Context)
Citation Context ...den increase in connectedness above the phase transition at which the radius is minimal for total localizability. VII. RELATED WORK Network localization is an active research field, e.g., [28], [29], =-=[30]-=-, [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [5], [41], [6], [42]. The previous approaches can be classified into two types: coarse-grained and fine-grained. percentage localized node... |

90 |
On k-connectivity for a geometric random graph. Random Structures Algorithms
- Penrose
- 1999
(Show Context)
Citation Context ...on. As in the case of the Erdös-Rényi random graph model [24], there is a phase transition in the random geometric graph model at which the graph becomes connected with high probability [25]. Penros=-=e [26] gen-=-eralizes this to kconnectivity with the result that if Gn(r) has a minimum vertex degree of k then with high probability Gn(r) is kconnected. � log n Since it is was proved in [27] for some r ∈ O(... |

71 |
Some matroids from discrete applied geometry. Matroid theory
- Whiteley
- 1995
(Show Context)
Citation Context ...s the number of vertices of G which are endpoints of edges in E ′ . There is no comparable complete result for 3-space, and no known polynomial time algorithm, though there are useful partial result=-=s [10]-=-, [11]. B. Conditions for Global Rigidity We are interested in the stronger concept of generic global rigidity. This concept is intimately related with first-order rigidity. If the formation Fp is not... |

69 | Global continuation for distance geometry problems
- Moré, Wu
- 1997
(Show Context)
Citation Context ...m has been used by biologists studying molecular conformation [20]. Because such optimization is computationally expensive, strategies such as divide-and-conquer [21] and objective function smoothing =-=[22]-=- have been proposed. In the context of network localization, distributed optimization algorithms may be desirable. In this case, algorithms such as [21] may be applied by dividing the global network i... |

67 | Solving Euclidean distance matrix completion problems via semidefinite programming
- ALFAKIH, KHANDANI, et al.
- 1999
(Show Context)
Citation Context ...ristics, and this paper provides the first theoretical analysis of network localization. A related problem called molecular conformation has been studied in the chemistry community, e.g., [21], [22], =-=[43]-=-. However, the focus of these studies is on 3D. Also, since the structure of a molecule is given, they do not consider the network construction process. One major building block of our analysis is rig... |

65 | An algorithm for two-dimensional rigidity percolation: the pebble game
- Jacobs, Hendrickson
- 1968
(Show Context)
Citation Context ...etwork localization, distributed optimization algorithms may be desirable. In this case, algorithms such as [21] may be applied by dividing the global network into small globally rigid sub-components =-=[23]-=- (clusters) to reduce overall complexity. Each cluster computes its relative localization using some optimization technique. Then the global localization can be achieved by merging the localizations o... |

62 | Connected rigidity matroids and unique realization graphs
- Jackson, Jordán
(Show Context)
Citation Context ...se set of points p ∈ IR dn at which Fp is a globally rigid formation with link set L. In the plane, a recent result gives a complete characterization of generically globally rigid graphs. Theorem 6 =-=( [15]):-=- A graph G with n ≥ 4 vertices is generically globally rigid in IR 2 if and only if it is 3-connected and redundantly rigid in IR 2 . Notice that to actually carry out a test to decide whether or no... |

59 |
Recursive position estimation in sensor networks
- Albowicz, Chen, et al.
- 2001
(Show Context)
Citation Context ...ansition at which the radius is minimal for total localizability. VII. RELATED WORK Network localization is an active research field, e.g., [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], =-=[38]-=-, [39], [40], [5], [41], [6], [42]. The previous approaches can be classified into two types: coarse-grained and fine-grained. percentage localized nodes 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 5 10... |

57 | The molecule problem: Exploiting structure in global optimization
- HENDRICKSON
- 1995
(Show Context)
Citation Context ... �) 2 . This formulation of the problem has been used by biologists studying molecular conformation [20]. Because such optimization is computationally expensive, strategies such as divide-and-conque=-=r [21]-=- and objective function smoothing [22] have been proposed. In the context of network localization, distributed optimization algorithms may be desirable. In this case, algorithms such as [21] may be ap... |

51 | Fault tolerant deployment and topology control in wireless ad hoc networks
- LI, WAN, et al.
- 2004
(Show Context)
Citation Context ...y [25]. Penrose [26] generalizes this to kconnectivity with the result that if Gn(r) has a minimum vertex degree of k then with high probability Gn(r) is kconnected. � log n Since it is was proved i=-=n [27] for som-=-e r ∈ O( n ), Gn(r) has a minimum vertex �degree of k for k ∈ O(1) log n with high probability, r ∈ O( n ) can also ensure kconnectivity. B. Global Rigidity of Random Geometric Graphs Recallin... |

49 |
Die Graphische Statik der starren Systeme
- Henneberg
- 1911
(Show Context)
Citation Context ... process. One major building block of our analysis is rigidity theory and computational geometry. Rigidity has been long studied in mathematics and structural engineering (see for example [14], [10], =-=[44]-=-, [9], [11]) and has a surprising number of applications in many areas. We formally analyzed the performance of network localization in networks of randomly placed nodes. Even though some researchers ... |

47 |
Rigid and flexible frameworks
- Roth
- 1981
(Show Context)
Citation Context ... {1, 2,...,n}; thelength of link (i, j) is the Euclidean distance between point pi and pj. The idea of a point formation is essentially the same as the concept of a “framework” studied in mathemat=-=ics [9]-=-, [10], [11] as well as within the theory of structures in mechanical and civil engineering. For our purposes, a point formation Fp =({p1,p2,...,pn}, L) provides a natural high-level model for an n-no... |

43 | Phase Transition phenomena in wireless ad hoc networks. Global Telecommunication Conference 5(2001
- chari, Wicker, et al.
(Show Context)
Citation Context ...ons in many areas. We formally analyzed the performance of network localization in networks of randomly placed nodes. Even though some researchers have studied random graphs in sensor networks, e.g., =-=[45],-=- [46], [47], [48], the focus is mainly on routing but not on localization. In [47], Díaz, Petit and Serna analyzed the performance of localization for optical sensor networks. However, their analysis... |

35 | A proof of Connelly’s conjecture on 3-connected circuits of the rigidity matroid
- Berg, Jordán
- 2003
(Show Context)
Citation Context ... remove the hub and one rim vertex, the graph remains a connected path on the remaining vertices. Therefore removing two vertices does not disconnect the graph, and it is 3-connected. As Lemma 2.1 of =-=[19]-=- observes, a wheel is a minimally redundantly rigid graph for the plane. By Theorem 6, it is generically globally rigid. We now analyze the complexity of realization of globally rigid graphs. A precis... |

30 |
Rigidity and scene analysis, Handbook of discrete and computational geometry, (pp. 893–916) Boca
- Whiteley
- 1997
(Show Context)
Citation Context ...n}; thelength of link (i, j) is the Euclidean distance between point pi and pj. The idea of a point formation is essentially the same as the concept of a “framework” studied in mathematics [9], [1=-=0], [11]-=- as well as within the theory of structures in mechanical and civil engineering. For our purposes, a point formation Fp =({p1,p2,...,pn}, L) provides a natural high-level model for an n-node network i... |

28 |
A random graph model for optical networks of sensors
- Díaz, Petit, et al.
(Show Context)
Citation Context ...areas. We formally analyzed the performance of network localization in networks of randomly placed nodes. Even though some researchers have studied random graphs in sensor networks, e.g., [45], [46], =-=[47],-=- [48], the focus is mainly on routing but not on localization. In [47], Díaz, Petit and Serna analyzed the performance of localization for optical sensor networks. However, their analysis is for the ... |

26 | A framework for maintaining formations based on rigidity
- Eren, Belhumeur, et al.
- 2002
(Show Context)
Citation Context ...of ˆ GN rather than GN which proves to be central to solvability of the localization problem under consideration. We begin by reviewing the point formation concept. By a d-dimensional point formation=-= [8] a-=-t p ∆ = column {p1,p2,...,pn}, written Fp, is meant a set of n points {p1,p2,...,pn} in IR d together with a set L of k links, labelled (i, j), where i and j are distinct integers in {1, 2,...,n}; t... |

25 | A new graph triconnectivity algorithm and its parallelization, in: STOC ’87
- Miller, Ramachandran
- 1987
(Show Context)
Citation Context ...h G is generically globally rigid in IR 2 , one must establish that it is both 3-connected and redundantly rigid in IR 2 . Various tests for 3-connectivity are known, and we refer the reader to [16], =-=[17] f-=-or details including measures of the complexity of the tests involved. Tests for redundant rigidity in IR 2 have been derived [13] based on variants of Laman’s theorem [14]. Since these properties a... |

23 |
The maximum vertex degree of a graph on uniform points
- Appel, Russo
- 1997
(Show Context)
Citation Context ...ility phenomenon. As in the case of the Erdös-Rényi random graph model [24], there is a phase transition in the random geometric graph model at which the graph becomes connected with high probabilit=-=y [25]. -=-Penrose [26] generalizes this to kconnectivity with the result that if Gn(r) has a minimum vertex degree of k then with high probability Gn(r) is kconnected. � log n Since it is was proved in [27] f... |

21 | Sensor and network topologies of formations with direction, bearing and angle information between agents
- Eren, Whiteley, et al.
- 2003
(Show Context)
Citation Context ...e Construction of Generically Globally Rigid Graphs It is possible to derive useful sufficient conditions and inductive constructions for generically globally rigid graphs in spaces of all dimensions =-=[12]-=-, [18]. d e One simple construction inserts new nodes of degree d +1 into existing generically globally rigid formations to create larger generically globally rigid formations. Since we will use this ... |

15 | Scalable Analysis and Design of Ad Hoc Networks via Random Graph Theory
- Farago
- 2002
(Show Context)
Citation Context ... We formally analyzed the performance of network localization in networks of randomly placed nodes. Even though some researchers have studied random graphs in sensor networks, e.g., [45], [46], [47], =-=[48],-=- the focus is mainly on routing but not on localization. In [47], Díaz, Petit and Serna analyzed the performance of localization for optical sensor networks. However, their analysis is for the case i... |

8 | Random Scaled Sector Graphs - Díaz, Petit, et al. - 2002 |

1 |
On generic global rigidity,” preprint
- Connelly
- 2003
(Show Context)
Citation Context ...implies the graph is generically globally rigid. In 3-space, whether having one generically globally rigid formation is enough to show that the graph is generically globally rigid is an open question =-=[18]-=-. As with generic rigidity, we do not have a generalization of Theorem 6 to higher dimensions. However, it extends as a necessary but not sufficient condition. Theorem 7 ([13], [18]): If a graph with ... |

1 |
Random scaled sector graphs,” Tech. Rep. LSI-02-47-R, Departament de Llenguatges i Sistemes Inform‘atics, Universitat Polit‘ecnica de Catalunya
- Díaz, Petit, et al.
- 2002
(Show Context)
Citation Context ... many areas. We formally analyzed the performance of network localization in networks of randomly placed nodes. Even though some researchers have studied random graphs in sensor networks, e.g., [45], =-=[46],-=- [47], [48], the focus is mainly on routing but not on localization. In [47], Díaz, Petit and Serna analyzed the performance of localization for optical sensor networks. However, their analysis is fo... |

1 | Global Positioning System; Theory md Pmctice, Fourrh Edition - Hofmann-Wellenhof, Lichtenegger, et al. - 1997 |