## Localized construction of bounded degree and planar spanner for wireless ad hoc networks (2003)

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Venue: | In DIALM-POMC |

Citations: | 71 - 8 self |

### BibTeX

@INPROCEEDINGS{Wang03localizedconstruction,

author = {Yu Wang and Xiang-yang Li},

title = {Localized construction of bounded degree and planar spanner for wireless ad hoc networks},

booktitle = {In DIALM-POMC},

year = {2003},

pages = {59--68},

publisher = {ACM}

}

### Years of Citing Articles

### OpenURL

### Abstract

We propose a novel localized algorithm that constructs a bounded degree and planar spanner for wireless ad hoc networks modeled by unit disk graph (UDG). Every node only has to know its 2-hop neighbors to find the edges in this new structure. Our method applies the Yao structure on the local Delaunay graph [21] in an ordering that are computed locally. This new structure has the following attractive properties: (1) it is a planar graph; (2) its node degree is bounded from above by a positive constant 19 + ⌈ 2π α ⌉; (3) it is a t-spanner (given any two nodes u and v, there is a path connecting them in the structure such that its length is no more than t ≤ max { π α,πsin 2 2 +1}·Cdel times of the shortest path in UDG); (4) it can be constructed locally and is easy to maintain when the nodes move around; (5) moreover, we show that the total communication cost is O(n), where n is the number of wireless nodes, and the computation cost of each node is at most O(d log d), where d is its 2-hop neighbors in the original unit disk graph. Here Cdel is the spanning ratio of the Delaunay triangulation, which is at most 4 √ 3 9 π. And the adjustable parameter α satisfies 0 <α<π/3. In addition, experiments are conducted to show this topology is efficient in practice, compared with other well-known topologies used in wireless ad hoc networks. Previously, only centralized method [5] of constructing bounded degree planar spanner is known, with degree bound 27 and spanning ratio t ≃ 10.02. The distributed implementation of their centralized method takes O(n 2) communications in the worst case. No localized methods were known previously for constructing bounded degree planar spanner.

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Citation Context ...hops distance of u in the unit-disk graph UDG(V ). An edge uv is called constrained Gabriel edge if �uv� ≤ 1 and the open disk using uv as diameter does not contain any node from V . It is well known =-=[25]-=- that the constrained Gabriel graph is a subgraph of the Delaunay triangulation, more precisely, GG(V ) ⊆ UDel(V ). Recall that a triangle △uvw belongs to the Delaunay triangulation Del(V ) if its cir... |

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Citation Context ...or wireless ad hoc networks has draw considerable attentions recently [11, 13, 20, 22, 23, 26, 27, 29, 30]. Topology control methods try to maintain a structure that can be used for efficient routing =-=[7, 14, 15]-=- or improve the overall networking performance [11, 20, 27]. Different structures with different properties have been proposed recently in the literature. In this paper, we will focus on the construct... |

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Citation Context ...or wireless ad hoc networks has draw considerable attentions recently [11, 13, 20, 22, 23, 26, 27, 29, 30]. Topology control methods try to maintain a structure that can be used for efficient routing =-=[7, 14, 15]-=- or improve the overall networking performance [11, 20, 27]. Different structures with different properties have been proposed recently in the literature. In this paper, we will focus on the construct... |

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Citation Context ...on cost in wireless networks is to send out the signal while the receiving cost of a message is neglected here. Topology control for wireless ad hoc networks has draw considerable attentions recently =-=[11, 13, 20, 22, 23, 26, 27, 29, 30]-=-. Topology control methods try to maintain a structure that can be used for efficient routing [7, 14, 15] or improve the overall networking performance [11, 20, 27]. Different structures with differen... |

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Citation Context ...raphs are Θ(n) and Θ( √ n) respectively. They are precisely n − 1 and √ n − 1 actually [28]. Recently, some researchers [22, 30] proposed to construct the wireless network topology based on Yao graph =-=[31]-=- (also called θ-graph [4]). It is known that the length stretch factor and the node out-degree of Yao graph are bounded by some positive constants. But as Li et al. mentioned in [22], all these three ... |

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Citation Context ...nt bounded length stretch factor. However, all these graphs [22, 23, 30] are not guaranteed to be planar. In [21] Li, et al. proposed a planar spanner localized Delaunay triangulations (LDel), and in =-=[10]-=- Gao et al. proposed a planar spanner Restricted Delaunay Graph for wireless ad hoc networks. However both of them could have unbounded node degree. The structure constructed by Hu [13] may not be a s... |

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Citation Context ...nner for a given nodes set V , for t = (1 + π) · Cdel � 10.02, such that the node degree is bounded from above by 27. Hereafter, we use Cdel to denote the spanning ratio of the Delaunay triangulation =-=[9, 16, 17]-=-. As we knew, this algorithm is the first method to compute a planar spanner of bounded degree. However the distributed implementation of this centralized method takes O(n 2 ) communications in the wo... |

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Citation Context ... denoted by △p1p2p3. An edge uv is called constrained Gabriel edge (or simply Gabriel edge here) if �uv� ≤ 1 and the open disk using uv as diameter does not contain any node from V . It is well known =-=[32]-=- that the constrained Gabriel graph is a subgraph of the Delaunay triangulation, more precisely, GG(V ) ⊆ UDel(V ). Recall that a triangle △uvw belongs to the Delaunay triangulation Del(V ) if its cir... |

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Citation Context ...t of points V , the unit Delaunay triangulation, denoted by UDel(V ), is the graph obtained by removing all edges of the Delaunay triangulation Del(V ) that are longer than one unit. It was proved in =-=[36]-=-, [21] that UDel(V ) is a t-spanner of UDG(V ). Li et al. [1] proved that graph UDel(V ) is a subgraph of the k-localized Delaunay graph LDel (k) (V ). Graph LDel (1) (V ) is not a planar graph, and L... |

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Citation Context ...nner for a given nodes set V , for t = (1 + π) · Cdel � 10.02, such that the node degree is bounded from above by 27. Hereafter, we use Cdel to denote the spanning ratio of the Delaunay triangulation =-=[9, 16, 17]-=-. As we knew, this algorithm is the first method to compute a planar spanner of bounded degree. However the distributed implementation of this centralized method takes O(n 2 ) communications in the wo... |

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Citation Context ...wi and wj in the ordered neighborhood of u, which are added by processing u. For example, in Figure 2, D(w1, w7) = w1w2w3w4w5w6w7. We can bound the length of D(wi, wj) by π/2�wiwj� by the argument in =-=[5, 6]-=-. In [6], it is shown that the length of D(wi, wj) is at most π/2 times �wiwj�, provided that (1) the straight-line segment between wi and wj lies outside the Voronoi region induced by u, and (2) that... |

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Citation Context ...f topology control is to increase the longevity of such networks which can be obtained by designing power efficient algorithms [3], [4], [5], [6], [7], [8]. One effective approach [4], [5], [6], [8], =-=[9]-=-, [10], [11], [12], [13], [14] is to maintain only a linear number of links using a localized construction method. In other words, we construct a sparse distributed structure as network topology for t... |

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Citation Context ...ecause of limited capacity and resources on each node. Thus a main goal of topology control is to increase the longevity of such networks which can be obtained by designing power efficient algorithms =-=[3]-=-, [4], [5], [6], [7], [8]. One effective approach [4], [5], [6], [8], [9], [10], [11], [12], [13], [14] is to maintain only a linear number of links using a localized construction method. In other wor... |

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Citation Context ...rds, these graphs are sparser, which is also verified by Figure 10. Recall that theoretically, only Y G ∗ (V ), BP S1(V ) and BP S2(V ) have bounded node degree (both for indegree and out-degree). In =-=[22, 23]-=-, Li et al. gave an example to show that RNG(V ), GG(V ), Y G(V ) and LDel(V ) could have large node degree (in-degree for Y G(V )). Notice that in our experiments since the wireless nodes are randoml... |

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Citation Context ...oc networks modeled by unit disk graph (UDG). Every node only has to know its 2-hop neighbors to find the edges in this new structure. Our method applies the Yao structure on the local Delaunay graph =-=[21]-=- in an ordering that are computed locally. This new structure has the following attractive properties: (1) it is a planar graph; (2) its node degree is bounded from above by a positive constant 19 + ⌈... |

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Citation Context ...d is the number of 1-hop or 2-hop neighbors. In [7, 15], two planar subgraphs relative neighborhood graph (RNG) and Gabriel graph (GG) are used as underlying network topologies. However, Bose, et al. =-=[3]-=- proved that the length stretch factors of these two graphs are Θ(n) and Θ( √ n) respectively. They are precisely n − 1 and √ n − 1 actually [28]. Recently, some researchers [22, 30] proposed to const... |

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Citation Context ...n 9 addition, experiments are conducted to show this topology is efficient in practice, compared with other well-known topologies used in wireless ad hoc networks. Previously, only centralized method =-=[5]-=- of constructing bounded degree planar spanner is known, with degree bound 27 and spanning ratio t � 10.02. The distributed implementation of their centralized method takes O(n 2 ) communications in t... |

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Citation Context ...er, we gave the first method to construct LDel (2) using O(n) messages. ALGORITHM 2. Construct LDel (2) Locally 1. Every node u collects the location information of N2(u) based on an efficient method =-=[8]-=- described later. It computes the Delaunay triangulation Del(N2(u)) of its 2-neighbors N2(u), including u itself. 2. For each edge uv of Del(N2(u)), let △uvw and △uvz be two triangles incident on uv. ... |

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Citation Context ...nner for a given nodes set V , for t = (1 + π) · Cdel � 10.02, such that the node degree is bounded from above by 27. Hereafter, we use Cdel to denote the spanning ratio of the Delaunay triangulation =-=[9, 16, 17]-=-. As we knew, this algorithm is the first method to compute a planar spanner of bounded degree. However the distributed implementation of this centralized method takes O(n 2 ) communications in the wo... |

36 | Distributed spanners with bounded degree for wireless ad hoc networks
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Citation Context ...or wireless ad hoc networks has draw considerable attentions recently [11, 13, 20, 22, 23, 26, 27, 29, 30]. Topology control methods try to maintain a structure that can be used for efficient routing =-=[7, 14, 15]-=- or improve the overall networking performance [11, 20, 27]. Different structures with different properties have been proposed recently in the literature. In this paper, we will focus on the construct... |

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20 |
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Citation Context ...rks is however very useful when computing local information. The approach by Calinescu [8] is based on the specific connected dominating set (virtual backbone) introduced by Alzoubi, Wan, and Frieder =-=[1]-=-. This connected dominating set is based on a maximal independent set (MIS). In Calinescu’s algorithm, each node uses its adjacent node(s) in the MIS to broadcast its relevant information (its ID and ... |

20 | Ordered theta graphs
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Citation Context ... respectively. They are precisely n − 1 and √ n − 1 actually [28]. Recently, some researchers [22, 30] proposed to construct the wireless network topology based on Yao graph [31] (also called θ-graph =-=[4]-=-). It is known that the length stretch factor and the node out-degree of Yao graph are bounded by some positive constants. But as Li et al. mentioned in [22], all these three graphs can not guarantee ... |

17 | Efficient construction of low weighted bounded degree planar spanner
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Citation Context ...pute a planar spanner of bounded degree. However the distributed implementation of this centralized method takes O(n 2 ) communications in the worst case for a set V of n nodes. Recently, Li and Wang =-=[24]-=- improved this by giving a centralized method that constructs a planar structure with degree bounded by at most 19 + ⌈ 2π ⌉ and the spanning ratio at most t ≤ max{ π 2 α , π sin α 2 + 1} · Cdel. Here ... |

15 | Ophir Frieder. Sparse power efficient topology for wireless networks - Li, Wan, et al. - 2002 |

12 | Ophir Frieder, Message-optimal connected dominating sets in mobile ad hoc networks - Alzoubi, Wan - 2002 |

11 | Improved greedy algorithms for constructing sparse geometric spanners - Gudmundsson, Levcopoulos, et al. - 2000 |

7 |
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Citation Context ...as underlying network topologies. However, Bose, et al. [3] proved that the length stretch factors of these two graphs are Θ(n) and Θ( √ n) respectively. They are precisely n − 1 and √ n − 1 actually =-=[28]-=-. Recently, some researchers [22, 30] proposed to construct the wireless network topology based on Yao graph [31] (also called θ-graph [4]). It is known that the length stretch factor and the node out... |

7 |
X-Y, Frieder O. Distributed spanner with bounded degree for wireless networks
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(Show Context)
Citation Context ...ease the longevity of such networks which can be obtained by designing power efficient algorithms [3], [4], [5], [6], [7], [8]. One effective approach [4], [5], [6], [8], [9], [10], [11], [12], [13], =-=[14]-=- is to maintain only a linear number of links using a localized construction method. In other words, we construct a sparse distributed structure as network topology for the wireless network. However, ... |