## Computing the maximum detour and spanning ratio of planar chains, trees and cycles (2002)

Venue: | In Proc. 19th Internat. Symp. Theor. Aspects of C.Sc., LNCS 2285:250–261 |

Citations: | 21 - 1 self |

### BibTeX

@INPROCEEDINGS{Langerman02computingthe,

author = {Stefan Langerman and Pat Morin and Michael Soss},

title = {Computing the maximum detour and spanning ratio of planar chains, trees and cycles},

booktitle = {In Proc. 19th Internat. Symp. Theor. Aspects of C.Sc., LNCS 2285:250–261},

year = {2002},

pages = {250--261},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

Let G = (V, E) be an embedded connected graph with n vertices and m edges. Specifically, the vertex set V consists of points in R 2, and E consists

### Citations

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Citation Context ...he upper envelope of identical and identically oriented conessC ′ C vi ui Fig. 1. If ci contains uj then D(C, vi, vj) ≥ d. has been given at least two other names: additively-weighted Voronoi diagram =-=[7]-=- and Johnson-Mehl crystal growth diagram [9]. It is known that f consists of O(n) pieces and can be computed in O(n log n) time using a sweep line algorithm [7]. Thus, the decision problem of determin... |

143 | Spanning trees and spanners
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Citation Context ...sity. 1 Here and thoughout, ∪S is shorthand for � e∈S e.sThe maximum detour and spanning ratio play important roles in the analysis of online routing algorithms [3,8] and the construction of spanners =-=[6]-=-. In the former case, the goal is to find paths that minimize maximum detour. In the latter, the goal is to construct graphs with few edges that minimize the spanning ratio. 1.1 Related Work Recently,... |

102 |
Decomposable searching problems I: Static-to-dynamic transformation
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Citation Context ...a structure is redundant. In general, this can be treated as a decomposable search problem and a data structure with O(n 3 2 log n) update time can be obtained using the technique of Bentley and Saxe =-=[2]-=-. However, in our application insertions and deletions are done in a FIFO manner, so that the ith point inserted will be the ith point deleted. We will use this property to obtain a data structure tha... |

51 | Geometric applications of a randomized optimization technique
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Citation Context ...gram 2 Subquadratic time algorithms were independently found by Agarwal et al [1], see below.sthat seems not to have been considered previously. We then apply a general optimization technique of Chan =-=[4]-=- to convert the decision algorithm into an optimization algorithm. We also show that more complicated structures can sometimes be treated by using multiple invocations of the above technique. As examp... |

49 | Approximating the stretch factor of Euclidean graphs
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Citation Context ... paper we describe O(n log n) time algorithms for computing the maximum detour and spanning ratio of a planar polygonal path. These algorithms solve open problems posed in at least two previous works =-=[5,10]-=-. We also generalize these algorithms to obtain O(n log 2 n) time algorithms for computing the maximum detour and spanning ratio of planar trees and cycles. 1 Introduction Let G = (V, E) be an embedde... |

37 | Online routing in Geometric Graphs
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Citation Context ...was affiliated with SOCS, McGill University. 1 Here and thoughout, ∪S is shorthand for � e∈S e.sThe maximum detour and spanning ratio play important roles in the analysis of online routing algorithms =-=[3,8]-=- and the construction of spanners [6]. In the former case, the goal is to find paths that minimize maximum detour. In the latter, the goal is to construct graphs with few edges that minimize the spann... |

36 |
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Citation Context ...ly oriented conessC ′ C vi ui Fig. 1. If ci contains uj then D(C, vi, vj) ≥ d. has been given at least two other names: additively-weighted Voronoi diagram [7] and Johnson-Mehl crystal growth diagram =-=[9]-=-. It is known that f consists of O(n) pieces and can be computed in O(n log n) time using a sweep line algorithm [7]. Thus, the decision problem of determining whether the spanning ratio of C is at le... |

28 | earching for the kernel of a polygon: A competitive strategy using self-approaching curves. 11th Annu
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Citation Context ...was affiliated with SOCS, McGill University. 1 Here and thoughout, ∪S is shorthand for � e∈S e.sThe maximum detour and spanning ratio play important roles in the analysis of online routing algorithms =-=[3,8]-=- and the construction of spanners [6]. In the former case, the goal is to find paths that minimize maximum detour. In the latter, the goal is to construct graphs with few edges that minimize the spann... |

22 |
A fast algorithm for approximating the detour of a polygonal chain
- Ebbers-Baumann, Klein, et al.
(Show Context)
Citation Context ... paper we describe O(n log n) time algorithms for computing the maximum detour and spanning ratio of a planar polygonal path. These algorithms solve open problems posed in at least two previous works =-=[5,10]-=-. We also generalize these algorithms to obtain O(n log 2 n) time algorithms for computing the maximum detour and spanning ratio of planar trees and cycles. 1 Introduction Let G = (V, E) be an embedde... |

12 | Computing the detour of polygonal curves
- Agarwal, Klein, et al.
- 2002
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Citation Context ...nes is infinite, and corresponds to computing the additively-weighted Voronoi diagram of line segments in the plane, a diagram 2 Subquadratic time algorithms were independently found by Agarwal et al =-=[1]-=-, see below.sthat seems not to have been considered previously. We then apply a general optimization technique of Chan [4] to convert the decision algorithm into an optimization algorithm. We also sho... |

2 |
A decomposition of multi-dimension point-sets with applications to k-nearest-neighbors and N-body potential fields
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Citation Context ...the problem of approximating the spanning ratio can be reduced to O(n) approximate shortest path queries on G. Their results make use of the well-separated pair decomposition of Callahan and Kosaraju =-=[2]-=- and hold even for graphs embedded in R d . The authors also show that approximating the spanning ratio requires Ω(n log n) time in the algebraic decision tree model of computation, even for chains in... |

1 |
Classes of graphs which approximate hte complete euclidean graph
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Citation Context ...ing the maximum detour of a polygonal chain, and give an O(n log n) time algorithm that finds an ɛ-approximation to the maximum detour. Their result is based on the θ-graph spanner of Keilsand Gutwin =-=[7]-=-. 3 New Results In this paper we give randomized algorithms with O(n log n) expected running time that compute the exact spanning ratio or maximum detour of a polygonal chain with n vertices. These ar... |