## Computing the detour and spanning ratio of paths, trees and cycles in 2D and 3D

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@MISC{Agarwal_computingthe,

author = {Pankaj K. Agarwal and Rolf Klein and Christian Knauer and Stefan Langerman and Pat Morin and Micha Sharir and Michael Soss},

title = {Computing the detour and spanning ratio of paths, trees and cycles in 2D and 3D},

year = {}

}

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### Abstract

The detour and spanning ratio of a graph � embedded in �� � measure how well � approximates Euclidean space and the complete Euclidean graph, respectively. In this paper we describe �������������� � time algorithms for computing the detour and spanning ratio of a planar polygonal path. By generalizing these algorithms, we obtain ���������������� �-time algorithms for computing the detour or spanning ratio of planar trees and cycles. Finally, we develop subquadratic algorithms for computing the detour and spanning ratio for paths, cycles, and trees embedded in �� � , and show that computing the detour in �� � is at least as hard as Hopcroft’s problem.

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Citation Context ... follows: We compute the Voronoi diagram ¤ ©¦¥ § £¦¥ � in ¢s� � � ¥ © � � � ¥ ¡ � �£¢ � © � ¢ ��¡ £¥¤�� ¡�© � � ¥ ¨ ��� time [7]. By using the red-blue-merge algorithm of Guibas et al. [15] (see also =-=[11, 25]-=-), we compute the sets of faces for all , which in turn gives us the sets for all . By the Combination Lemma of Guibas et al. [15], , and the set can be computed in ¢ � � © £¥¤¨§�©���¤�� � time. Final... |

333 |
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Citation Context ...e minimization diagram of , the projection of the lower envelope of onto the ¦ § ¤ ¥ ©¦¥¨§�£¦¥�� -plane, is the additive-weight Voronoi diagram of , under the weight function ¢ ��� . For a point time =-=[13]-=-. ¥ , let ¤ ©¦¥©§�£���� denote the Voronoi cell of � in ¤ ©¦¥�§ £¦¥�� . ¤ ©¦¥©§�£¦¥�� can be computed in ¢ £¥¤¨§�©���¤�� We first test whether ¤ ©¦¥�§�£���� is nonempty for every vertex � � ¥ . If not... |

260 | Epsilon-nets and simplex range queries
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Citation Context ... � � ¦ ¦ ¦ § � § ¦ � � § § � � � ¦ , thereby implying that � � §���� £�� . Similar � £ arguments imply that (iii) or (iv) implies (i). £ Using Lemma 4.2(iv) and the standard random-sampling technique =-=[16]-=-, we construct a four-level data structure to decide whether � §�� ��� � . The first level constructs a complete bipartite decom£�� position for the ¡�£���§���� � �s� � £�� � §�� � � � � ¨ set . The s... |

251 | Geometric range searching and its relatives
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Citation Context ... ¤ points £�s� � in a -dimensional convex polyhedron defined by the ¤ intersection of halfspaces. This problem can be solved in ¢ ������� � time using a data structure for halfspace-emptiness queries =-=[1]-=-. Using £¥¤ � Chan’s technique, as in the planar case, we can compute � itself within the same £� �� asymptotic time bound. Finally, as for the planar case, the algorithm can be extended to compute th... |

232 | Applying parallel computation algorithms in the design of serial algorithms
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Citation Context ...io of a polygonal chain with vertices embedded in can be computed in ¢ £¥¤¨§�©�� ¤�� randomized expected time. Remark. One can obtain an alternative deterministic solution that uses parametric search =-=[22]-=-, and runs in time ¢ � ¤�� , for some constant � . However, the resulting algorithm is considerably £¥¤¨§�©�� more involved on top of being slightly less efficient. We therefore omit its description. ... |

144 | Voronoi diagrams
- Aurenhammer, Klein
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Citation Context ... � ��� � © � ¨ lies below all the cones of � The algorithm thus proceeds as follows: We compute the Voronoi diagram ¤ ©¦¥ § £¦¥ � in ¢s� � � ¥ © � � � ¥ ¡ � �£¢ � © � ¢ ��¡ £¥¤�� ¡�© � � ¥ ¨ ��� time =-=[7]-=-. By using the red-blue-merge algorithm of Guibas et al. [15] (see also [11, 25]), we compute the sets of faces for all , which in turn gives us the sets for all . By the Combination Lemma of Guibas e... |

125 | Guibas L.: Discrete geometric shapes: Matching, interpolation, and approximation
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Citation Context ...curves is at most � , then their Fr échet distance is at most ����� times their Hausdorff distance. The Fr échet and Hausdorff distances are two commonly used similarity measures for geometric shapes =-=[5]-=-. Although the Hausdorff distance works well for planar regions, the Fr échet distance is more suitable to measure the similarity of two curves [5]. However, the Fr échet distance is much harder to co... |

89 | Applications of parametric searching in geometric optimization
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Citation Context ...ree ¤ with edges � � in can be computed in randomized expected time ¢ ����������� � , for any � ��� . £¥¤ Remark. We remark that it is also possible to use the parametric search technique [22], as in =-=[3]-=-, to obtain a deterministic alternative solution. This however (a) results in a considerably more involved algorithm, and (b) requires us to derandomize the decision algorithm, i.e., its vertical deco... |

57 |
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Citation Context ...dified to cover the case where we know a priori that the polygonal chains we are given as input do not self-intersect. The construction uses techniques presented in Erickson [12]. ¨¤£ �¦¥¨§�© � ¥ � � =-=[21]-=-. Without loss of generality, we may assume that none of the given lines is § -vertical. We begin by sorting the lines in � in increasing order of their slopes and the points insin increasing lexicogr... |

50 |
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Citation Context ...hus proceeds as follows: We compute the Voronoi diagram ¤ ©¦¥ § £¦¥ � in ¢s� � � ¥ © � � � ¥ ¡ � �£¢ � © � ¢ ��¡ £¥¤�� ¡�© � � ¥ ¨ ��� time [7]. By using the red-blue-merge algorithm of Guibas et al. =-=[15]-=- (see also [11, 25]), we compute the sets of faces for all , which in turn gives us the sets for all . By the Combination Lemma of Guibas et al. [15], , and the set can be computed in ¢ � � © £¥¤¨§�©�... |

49 | Geometric applications of a randomized optimization technique
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Citation Context ... � £� �§�¥�� . It thus suffices to describe an algorithm £�s��¡ for the decision problem: Given � ��� a parameter , determine whether � . We will £� �§�¥������ then use a randomized technique by Chan =-=[9]-=- to compute the actual value of � . £� �§�¥�� 2.2 Decision algorithmsfrom� to� � � � � ¥�¦ £ ¥ �s� � £���§���� We orient . For a given parameter , we describe an algorithm that determines whether for ... |

48 | Approximating the stretch factor of Euclidean graphs
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(Show Context)
Citation Context ...rossing-free graphs in � � . Even if the input graphsis a simple path in � � , no subquadratic-time algorithm has previously been known for computing its detour or spanning ratio. Narasimhan and Smid =-=[23]-=- study the problem of approximating the spanning ratio of an arbitrary geometric graph � � in . They give a ¢ -time algorithm that computes an £�������� -approximate £¥¤¨§�©���¤�� value of the spannin... |

40 |
The complexity and construction of many faces in arrangements of lines and of segments
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(Show Context)
Citation Context ... follows: We compute the Voronoi diagram ¤ ©¦¥ § £¦¥ � in ¢s� � � ¥ © � � � ¥ ¡ � �£¢ � © � ¢ ��¡ £¥¤�� ¡�© � � ¥ ¨ ��� time [7]. By using the red-blue-merge algorithm of Guibas et al. [15] (see also =-=[11, 25]-=-), we compute the sets of faces for all , which in turn gives us the sets for all . By the Combination Lemma of Guibas et al. [15], , and the set can be computed in ¢ � � © £¥¤¨§�©���¤�� � time. Final... |

34 | On line routing in geometric graphs
- Morin
- 2001
(Show Context)
Citation Context ...known condition (apart from convexity) under which a linear relationship between the two measures is known. Analyzing on-line navigation strategies also often involves estimating the detour of curves =-=[8, 17]-=-. Sometimes the geometric properties of curves allow us to infer upper bounds on their detour [4, 18, 24], but these results do not lead to efficient computation of the detour of the curve. Related wo... |

33 | New lower bounds for Hopcroft’s problem
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Citation Context ...¤ ������� � , for any ����� . Using the same £¥¤�� and cycles. We also show that it is unlikely that an ��£¥¤ � 2swhich a lower bound of £ £¥¤ � ��� � , in a special model of computation, is given in =-=[12]-=-. Preliminary versions of this work appeared in [2, 20]; the 2-dimensional algorithm described in [20] is significantly different from the one presented here. 2 Polygonal Chains in the Plane Let the g... |

31 | Almost tight upper bounds for vertical decompositions in four dimensions
- Koltun
- 2011
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Citation Context ...y large constant independent of , and compute the vertical decomposition of the arrangement of the surfaces semi-algebraic of constant description complexity. Hence, we can apply the result of Koltun =-=[19]-=-, to conclude that has ¢ � � � � � ¡ ¡ ��© cells, for � � ¨ any . For each � � � cell , let � ��� £��§¦ £ � � � ��¨ ¨ , let � © ¥ � be the set of edges � for which the surface £ £ � � � crosses ¨ , an... |

28 | Searching for the kernel of a polygon - a competitive strategy
- Icking, Klein
- 1995
(Show Context)
Citation Context ...known condition (apart from convexity) under which a linear relationship between the two measures is known. Analyzing on-line navigation strategies also often involves estimating the detour of curves =-=[8, 17]-=-. Sometimes the geometric properties of curves allow us to infer upper bounds on their detour [4, 18, 24], but these results do not lead to efficient computation of the detour of the curve. Related wo... |

22 | Comparison of distance measures for planar curves
- Alt, Knauer, et al.
(Show Context)
Citation Context ...tions influence the nature of the problem considerably. In this paper we are studying both, detour and spanning ratio. The case ofsbeing a planar polygonal chain is of particular interest. Alt et al. =-=[6]-=- proved that if the detour of two planar curves is at most � , then their Fr échet distance is at most ����� times their Hausdorff distance. The Fr échet and Hausdorff distances are two commonly used ... |

21 | A fast algorithm for approximating the detour of a polygonal chain - Ebbers-Baumann, Klein, et al. |

21 | Computing the maximum detour and spanning ratio of planar paths, trees, and cycles
- Langerman, Morin, et al.
- 2002
(Show Context)
Citation Context ...cellence in Geometric Computing at Tel Aviv University, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. � Some of these results have appeared in a preliminary form in =-=[2, 20]-=-. � Department of Computer Science, Duke University, Durham, NC 27708-0129, U.S.A.,pankaj@cs.duke.edu. � Institut für Informatik I, Universität Bonn, Römerstraße 164, D-53117 Bonn, Germany, rolf.klein... |

16 | Curves with increasing chords
- Rote
- 1994
(Show Context)
Citation Context ...known. Analyzing on-line navigation strategies also often involves estimating the detour of curves [8, 17]. Sometimes the geometric properties of curves allow us to infer upper bounds on their detour =-=[4, 18, 24]-=-, but these results do not lead to efficient computation of the detour of the curve. Related work. Recently, researchers have become interested in computing the detour and spanning ratio of embedded g... |

12 | Computing the detour of polygonal curves
- Agarwal, Klein, et al.
- 2002
(Show Context)
Citation Context ...cellence in Geometric Computing at Tel Aviv University, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. � Some of these results have appeared in a preliminary form in =-=[2, 20]-=-. � Department of Computer Science, Duke University, Durham, NC 27708-0129, U.S.A.,pankaj@cs.duke.edu. � Institut für Informatik I, Universität Bonn, Römerstraße 164, D-53117 Bonn, Germany, rolf.klein... |

8 | Self-approaching curves
- Icking, Klein, et al.
- 1995
(Show Context)
Citation Context ...known. Analyzing on-line navigation strategies also often involves estimating the detour of curves [8, 17]. Sometimes the geometric properties of curves allow us to infer upper bounds on their detour =-=[4, 18, 24]-=-, but these results do not lead to efficient computation of the detour of the curve. Related work. Recently, researchers have become interested in computing the detour and spanning ratio of embedded g... |

4 | Generalized selfapproaching curves
- Aichholzer, Aurenhammer, et al.
(Show Context)
Citation Context ...known. Analyzing on-line navigation strategies also often involves estimating the detour of curves [8, 17]. Sometimes the geometric properties of curves allow us to infer upper bounds on their detour =-=[4, 18, 24]-=-, but these results do not lead to efficient computation of the detour of the curve. Related work. Recently, researchers have become interested in computing the detour and spanning ratio of embedded g... |

1 |
üne. Umwege in Polygonen
- Gr
- 2002
(Show Context)
Citation Context ...er bounds, it was shown by Narasimhan and Smid [23] that computing the spanning ratio of a planar polygonal chain requires £ £¥¤¨§�©���¤�� time if self-overlapping chains are allowed as input. Gr üne =-=[14]-=- has shown that the same lower bound holds if the input is restricted to polygonal chains that are monotonic, hence simple. It is unknown whether the £ £¥¤¨§�©�� ¤�� lower bound also holds for computi... |