## On the efficiency of a local iterative algorithm to compute delaunay realizations (2008)

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Venue: | In Workshop on Experimental Algorithms (WEA |

Citations: | 8 - 1 self |

### BibTeX

@INPROCEEDINGS{Lillis08onthe,

author = {Kevin M. Lillis and Sriram V. Pemmaraju},

title = {On the efficiency of a local iterative algorithm to compute delaunay realizations},

booktitle = {In Workshop on Experimental Algorithms (WEA},

year = {2008}

}

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### Abstract

Abstract. Greedy routing protocols for wireless sensor networks (WSNs) are fast and efficient but in general cannot guarantee message delivery. Hence researchers are interested in the problem of embedding WSNs in low dimensional space (e.g., R 2) in a way that guarantees message delivery with greedy routing. It is well known that Delaunay triangulations are such embeddings. We present the algorithm FindAngles, which is a fast, simple, local distributed algorithm that computes a Delaunay triangulation from any given combinatorial graph that is Delaunay realizable. Our algorithm is based on a characterization of Delaunay realizability due to Hiroshima et al. (IEICE 2000). When compared to the PowerDiagram algorithm of Chen et al. (SoCG 2007) that embeds triangulations in the plane so as to permit successful greedy routing, our algorithm requires on average 1/6 th the number of iterations. FindAngles also scales linearly to larger networks and has a much faster distributed implementation than PowerDiagram, requiring just a single round of communication in each iteration. The PowerDiagram algorithm was proposed as an improvement on another algorithm due to Thurston (unpublished, 1988). Our experiments show that on average the PowerDiagram algorithm uses about 19 % fewer iterations than the Thurston algorithm, whereas our algorithm uses about 89 % fewer iterations. Experimentally, FindAngles exhibits well behaved convergence. Theoretically, we prove that with certain initial conditions the error term decreases monotonically. Taken together, these suggest our algorithm may have polynomial time convergence for certain classes of graphs. We note that our algorithm runs only on Delaunay realizable triangulations. This is not a significant concern because Hiroshima et al. (IEICE 2000) indicate that most combinatorial triangulations are indeed Delaunay realizable, which we have also observed experimentally: out of 5000 randomly generated combinatorial triangulations on 100 vertices, only one was not Delaunay realizable.

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Citation Context ...e are based on the Koebe Representation Theorem and we describe that and sketch the Thurston algorithm and the PowerDiagram algorithm in Section 2.2. 2.1 The HMS test for Delaunay Realizability Rivin =-=[18, 19]-=- presents a polynomial time test for Delaunay realizability that is based on ideas in hyperbolic geometry. In contrast, the HMS test is based on elementary Euclidean geometric ideas. We describe the H... |

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Citation Context ... . . , rn), the angle sum σr ′(vi) = 2π. Of course this update affects the angle sums at neighbors of vi. However, it is possible to show that this iterative process converges. Collins and Stephenson =-=[7]-=- suggest several improvements to this basic algorithm that reduce the number of iterations needed for convergence. This algorithm has an obvious distributed implementation in which, in each round, eac... |

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Citation Context ... the proof of existence does not give a clear indication of how to efficiently compute these embeddings. It should be noted that there are polynomial time algorithms to compute a Koebe representation =-=[14, 21]-=-, but these use the ellipsoid method and are therefore 3 Actually, the Papadimitriou-Ratajczak conjecture is more general and states that every 3-connected planar graph has a greedy embedding.not pra... |

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Citation Context ... Delaunay triangulations. In fact, Papadimitriou and Ratajczak [16] conjecture that every combinatorial triangulation has a greedy embedding 3 . This conjecture was very recently proved by Dhandapani =-=[8]-=-. The proof essentially depends on the Knaster-Kuratowski-Mazurkiewicz Theorem [12] that is known to be equivalent to the Brouwer Fixed Point Thorem. Due to this dependency, Dhandapani’s proof does no... |

12 |
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Citation Context ...t any theoretical bounds or experimental results. A second approach to the problem of finding embeddings that permit successful greedy geographic routing uses power diagrams and is due to Chen at al. =-=[5]-=-. Let P ⊆ R 2 be a planar set of points, with each point p ∈ P having an associated disk D(p) with center p and radius r(p) ≥ 0. The power distance from any point q ∈ R 2 to p, denoted power(q, p) is ... |

6 |
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Citation Context ...representation C = {C1, C2, . . . , Cn} of a combinatorial triangulation G = (V, E) can also be viewed as a power diagram whose planar dual is G. This is the starting point of the work of Chen et al. =-=[4, 5]-=-, who go on to point out that Koebe representations are special power diagrams in which disks corresponding to adjacent cells are mutually tangent. With this motivation, Chen et al. [4, 5] develop a l... |

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Citation Context ... the proof of existence does not give a clear indication of how to efficiently compute these embeddings. It should be noted that there are polynomial time algorithms to compute a Koebe representation =-=[14, 21]-=-, but these use the ellipsoid method and are therefore 3 Actually, the Papadimitriou-Ratajczak conjecture is more general and states that every 3-connected planar graph has a greedy embedding.not pra... |

4 |
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Citation Context ...ng whether or not a combinatorial triangulation is Delaunay realizable can be solved in polynomial time; for example by checking if a certain linear system of inequalities defined by Hiroshima et al. =-=[10]-=- has a feasible solution. However, as far as we know, the problem of actually finding a Delaunay realization does not have a polynomial time solution and seems rather difficult. In this paper we prese... |