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## A faster algorithm for computing motorcycle graphs (2013)

Venue: | PROC. 29TH SYMP. ON COMPUTATIONAL GEOMETRY, SOCG ’13, ACM |

Citations: | 4 - 1 self |

### Citations

10617 | Introduction to Algorithms
- Cormen, Leiserson, et al.
- 2001
(Show Context)
Citation Context ...number of points |Ak ∩pq| in the corresponding subarray. This gives a halving point h(p, q) with ρ = 3/4. The median of each subarray can be found in O(1) time, and their weighted median in O(C) time =-=[17]-=-, so the query time is dominated by the C binary searches. Thus, we can answer halving queries in O(C log n) time. 3.3 Bounded precision input The data structure for answering halving queries in Secti... |

2052 |
Computational geometry: An introduction
- Preparata, Shamos
- 1985
(Show Context)
Citation Context ...orting lines on both sides. (See Section 1.2.) The implementation of these data structures in different settings lead to different time bounds. For all our results, we use the standard real-RAM model =-=[31]-=-, that allows to perform arithmetic operations exactly on arbitrary real numbers. But for some of our results, we make the assumption that all input coordinates are O(log n)-bit rational numbers. It h... |

266 | Geometric range searching and its relatives
- Agarwal, Erickson
- 1999
(Show Context)
Citation Context ...rm a collection of h + 1 disjoint simple polygons. We could directly use known ray-shooting data structures [4, 24], which can be made dynamic at the expense of an extra nε factor in the running time =-=[1]-=-. In the following, we give a different approach, which leads to a better time bound when used as a subroutine of our algorithm. This approach takes advantage of the fact that the holes of P are fixed... |

126 | Ray shooting and parametric search.
- Agarwal, Matousek
- 1993
(Show Context)
Citation Context ...ral case, as presented in Section 1.2. So motorcycles have arbitrary starting position, destination point and velocity. For ray shooting, we can directly use a data structure by Agarwal and Matoušek =-=[2]-=-, which requires preprocessing time O(n4/3+ε), with update and query time O(n1/3+ε), for any ε > 0. For halving queries, we use known range searching data structures and parametric search, as in the w... |

99 | Efficient partition trees,
- Matousek
- 1992
(Show Context)
Citation Context ... motorcycle graph can be computed in O(n4/3+ε) time, for any ε > 0. It should be possible to replace the nε factor in the bounds of Lemma 4 with a polylogarithm using known range searching techniques =-=[11, 28]-=-, because we only need a static data structure for halving queries, but in any case we need a dynamic data structure for ray shooting queries, so it would not improve our overall time bounds. 3.2 C-Or... |

98 |
Ray shooting in polygons using geodesic triangulations,
- Chazelle, Edelsbrunner, et al.
- 1991
(Show Context)
Citation Context ...s advantage of the fact that the holes of P are fixed (only the tentative tracks are dynamic). We use a spanning tree with low crossing number, which is not a new idea in ray-shooting data structures =-=[13, 24]-=-. We pick one point on the boundary of each hole of P , and on the boundary of P . We connect these h + 1 points using a spanning tree T with low stabbing number [3], that is, a spanning tree such tha... |

96 |
A pedestrian approach to ray shooting: Shoot a ray, take a walk
- Hershberger, Suri
- 1995
(Show Context)
Citation Context ...As our input polygon has h holes, the boundary ∂P of P together with the tentative tracks form a collection of h + 1 disjoint simple polygons. We could directly use known ray-shooting data structures =-=[4, 24]-=-, which can be made dynamic at the expense of an extra nε factor in the running time [1]. In the following, we give a different approach, which leads to a better time bound when used as a subroutine o... |

70 | A novel type of skeleton for polygons
- Aichholzer, Aurenhammer, et al.
- 1995
(Show Context)
Citation Context ...l position. (See Figure 1.) It has been known since at least the 19th century; for instance, figures representing the straight skeleton can be found in the book by von Peschka [32]. Aichholzer et al. =-=[5]-=- gave the first efficient algorithms for computing the straight skeleton, and presented it as an alternative to the medial axis having only straight-line edges. The straight skeleton has found numerou... |

58 | Raising Roofs, Crashing Cycles & Playing Pool: Applications of a Data Structure for Finding Pairwise Interactions. In:
- Eppstein, Erickson
- 1999
(Show Context)
Citation Context ...on [27], architectural modeling [26], polyhedral surface reconstruction [7, 21, 29], biomedical image processing [16]. It also has a direct application to CAD, as it allows to compute offset polygons =-=[18]-=-. The straight skeleton has become a standard tool in geometric computing, and thus fast and robust software has been developed to compute it [10, 25, 30]. ∗King Abdullah University of Science and Tec... |

46 |
Coquillart S. 3D reconstruction of complex polyhedral shapes from contours using a simplified generalized Voronoi diagram
- JM, Perrin
(Show Context)
Citation Context ...t-line edges. The straight skeleton has found numerous applications in computer science, for instance to city model reconstruction [27], architectural modeling [26], polyhedral surface reconstruction =-=[7, 21, 29]-=-, biomedical image processing [16]. It also has a direct application to CAD, as it allows to compute offset polygons [18]. The straight skeleton has become a standard tool in geometric computing, and ... |

45 |
A strong and easily computable separation bound for arithmetic expressions involving square roots
- Burnikel, Fleischer, et al.
- 1997
(Show Context)
Citation Context ... vertices in this arrangement of bisectors cannot be too small. This distance can be shown to be at least 2−W , where W = 64(80w + 105) + 1 = O(w), by applying the separation bound by Burnikel et al. =-=[9]-=-. So we obtain a result for induced motorcycle graphs that is analogous to Theorem 7. Lemma 11. Given a polygon P whose input coordinates are w-bit rational numbers, we can compute the motorcycle grap... |

36 | Dynamic ray shooting and shortest paths in planar subdivisions via balanced geodesic triangulations
- Goodrich, Tamassia
- 1997
(Show Context)
Citation Context ...ng data structures are known. One such case is ray-shooting in a connected planar subdivision, which can be done in O(log2 n)-time per update and query using a data structure by Goodrich and Tamassia =-=[23]-=-. We can use this data structure if, for instance, all motorcycles move inside a simple polygon P , starting from its boundary. (So for all i, sidi ⊂ P , and si is on the boundary of P .) Then we perf... |

31 | I/O-efficient dynamic planar point location.
- Arge, Vahrenhold
- 2004
(Show Context)
Citation Context ...es. A first approach to answer our ray shooting queries is to use C instances of a data structure for vertical ray shooting in a planar subdivision. Several data structures are known for this problem =-=[6]-=-, we use a data structure by Cheng and Janardan [14] that takes O(log2 n) time per update and O(log n) time per query. So overall, we get Q(n) = O(C log2 n) with the terminology of Theorem 2. Alternat... |

28 |
Applications of a new space-partitioning technique
- Agarwal, Sharir
- 1993
(Show Context)
Citation Context ... ray-shooting data structures [13, 24]. We pick one point on the boundary of each hole of P , and on the boundary of P . We connect these h + 1 points using a spanning tree T with low stabbing number =-=[3]-=-, that is, a spanning tree such that any line crosses at most O( √ h) edges of T . This tree can be computed in O(n1+ε) time [3, Section 8]. We maintain a polygonal subdivision which is the overlay of... |

28 |
New results on dynamic planar point location
- Cheng, Janardan
- 1992
(Show Context)
Citation Context ...ries is to use C instances of a data structure for vertical ray shooting in a planar subdivision. Several data structures are known for this problem [6], we use a data structure by Cheng and Janardan =-=[14]-=- that takes O(log2 n) time per update and O(log n) time per query. So overall, we get Q(n) = O(C log2 n) with the terminology of Theorem 2. Alternatively, we can use C(C − 1) instances of a data struc... |

26 | Optimal partition trees.
- Chan
- 2012
(Show Context)
Citation Context ... motorcycle graph can be computed in O(n4/3+ε) time, for any ε > 0. It should be possible to replace the nε factor in the bounds of Lemma 4 with a polylogarithm using known range searching techniques =-=[11, 28]-=-, because we only need a static data structure for halving queries, but in any case we need a dynamic data structure for ray shooting queries, so it would not improve our overall time bounds. 3.2 C-Or... |

22 | Transdichotomous results in computational geometry, I: Point location in sublogarithmic time.
- Chan, Patrascu
- 2009
(Show Context)
Citation Context ...utational geometry problems under a bounded precision model (the word RAM), for instance the computation of Delaunay triangulations, convex hulls, polygon triangulation and line segment intersections =-=[8, 12]-=-. We first show in Section 3.3 that a motorcycle graph can be computed in O(n log3 n) time if the motorcycles move within a simple polygon, starting from its boundary. The only other nontrivial cases ... |

21 | Obdrzalek Straight skeleton implementation
- Felkel, S
- 1998
(Show Context)
Citation Context ...t-line edges. The straight skeleton has found numerous applications in computer science, for instance to city model reconstruction [27], architectural modeling [26], polyhedral surface reconstruction =-=[7, 21, 29]-=-, biomedical image processing [16]. It also has a direct application to CAD, as it allows to compute offset polygons [18]. The straight skeleton has become a standard tool in geometric computing, and ... |

20 | Straight-skeleton based contour interpolation
- BAREQUET, GOODRICH, et al.
- 2003
(Show Context)
Citation Context ...t-line edges. The straight skeleton has found numerous applications in computer science, for instance to city model reconstruction [27], architectural modeling [26], polyhedral surface reconstruction =-=[7, 21, 29]-=-, biomedical image processing [16]. It also has a direct application to CAD, as it allows to compute offset polygons [18]. The straight skeleton has become a standard tool in geometric computing, and ... |

19 | Interactive architectural modeling with procedural extrusions. - Kelly, Wonka - 2011 |

16 | Vigneron Motorcycle graphs and straight skeletons. Algorithmica
- Cheng, A
- 2007
(Show Context)
Citation Context ...is still very much open. The previously best known algorithms were the O(n17/11+ε)-time algorithm by Eppstein and Erickson [18], and the O(n3/2 log2 n)-time randomized algorithm by Cheng and Vigneron =-=[15]-=-. The only known lower bound is Ω(n log n), by a reduction from sorting [20]. In this paper, we give new subquadratic algorithms for computing straight skeletons. In particular, if all input coordinat... |

14 | Optimal dynamic vertical ray shooting in rectilinear planar subdivisions
- Giora, Kaplan
(Show Context)
Citation Context ...r ray shooting queries with a given direction, into segments with another direction: We just need to change the two coordinate axis to these two directions. Using a recent result by Giyora and Kaplan =-=[22]-=-, we obtain Q(n) = O(C2 log n). Halving queries. Our data structure for halving queries simply consists of a sorted list of motorcycles for each direction. So for each k ∈ 1, . . . , C, we have an arr... |

14 |
M.: Automatically generating large urban environments based on the footprint data of buildings
- LAYCOCK, DAY
(Show Context)
Citation Context ...presented it as an alternative to the medial axis having only straight-line edges. The straight skeleton has found numerous applications in computer science, for instance to city model reconstruction =-=[27]-=-, architectural modeling [26], polyhedral surface reconstruction [7, 21, 29], biomedical image processing [16]. It also has a direct application to CAD, as it allows to compute offset polygons [18]. T... |

12 | Delaunay triangulations in O(sort(n)) time and more.
- Buchin, Mulzer
- 2011
(Show Context)
Citation Context ...utational geometry problems under a bounded precision model (the word RAM), for instance the computation of Delaunay triangulations, convex hulls, polygon triangulation and line segment intersections =-=[8, 12]-=-. We first show in Section 3.3 that a motorcycle graph can be computed in O(n log3 n) time if the motorcycles move within a simple polygon, starting from its boundary. The only other nontrivial cases ... |

10 | Motorcycle graphs: Canonical quad mesh partitioning - Eppstein, Goodrich, et al. - 2008 |

8 |
Ray shooting amidst convex polygons in 2D
- Agarwal, Sharir
- 1996
(Show Context)
Citation Context ...As our input polygon has h holes, the boundary ∂P of P together with the tentative tracks form a collection of h + 1 disjoint simple polygons. We could directly use known ray-shooting data structures =-=[4, 24]-=-, which can be made dynamic at the expense of an extra nε factor in the running time [1]. In the following, we give a different approach, which leads to a better time bound when used as a subroutine o... |

7 | Angular bisector network, a simplified generalized voronoi diagram: Application to processing complex intersections in biomedical images - Cloppet, Oliva, et al. |

7 |
M.: Theoretical and Practical Results on Straight Skeletons of Planar Straight-Line Graphs
- Huber, Held
- 2011
(Show Context)
Citation Context ...ation to CAD, as it allows to compute offset polygons [18]. The straight skeleton has become a standard tool in geometric computing, and thus fast and robust software has been developed to compute it =-=[10, 25, 30]-=-. ∗King Abdullah University of Science and Technology (KAUST), Geometric Modeling and Scientific Visualization Center, Thuwal 23955-6900, Saudi Arabia. {antoine.vigneron, lie.yan}@kaust.edu.sa (a) Inp... |

4 | On Computing Straight Skeletons by Means of Kinetic Triangulations
- Palfrader, Held, et al.
(Show Context)
Citation Context ...ation to CAD, as it allows to compute offset polygons [18]. The straight skeleton has become a standard tool in geometric computing, and thus fast and robust software has been developed to compute it =-=[10, 25, 30]-=-. ∗King Abdullah University of Science and Technology (KAUST), Geometric Modeling and Scientific Visualization Center, Thuwal 23955-6900, Saudi Arabia. {antoine.vigneron, lie.yan}@kaust.edu.sa (a) Inp... |

2 | A CGAL implementation of the straight skeleton of a simple 2d polygon with holes
- CACCIOLA
- 2004
(Show Context)
Citation Context ...ation to CAD, as it allows to compute offset polygons [18]. The straight skeleton has become a standard tool in geometric computing, and thus fast and robust software has been developed to compute it =-=[10, 25, 30]-=-. ∗King Abdullah University of Science and Technology (KAUST), Geometric Modeling and Scientific Visualization Center, Thuwal 23955-6900, Saudi Arabia. {antoine.vigneron, lie.yan}@kaust.edu.sa (a) Inp... |

1 |
Crashing motorcycles efficiently. http://www.cs.uiuc.edu/~jeffe/open/ cycles.html
- Erickson
- 1998
(Show Context)
Citation Context .../11+ε)-time algorithm by Eppstein and Erickson [18], and the O(n3/2 log2 n)-time randomized algorithm by Cheng and Vigneron [15]. The only known lower bound is Ω(n log n), by a reduction from sorting =-=[20]-=-. In this paper, we give new subquadratic algorithms for computing straight skeletons. In particular, if all input coordinates are O(log n)-bit rational numbers, we give an O(n √ h+ 1 log3 n)-time ran... |

1 |
Kotirte Ebenen
- Peschka
(Show Context)
Citation Context ... parallel to its original position. (See Figure 1.) It has been known since at least the 19th century; for instance, figures representing the straight skeleton can be found in the book by von Peschka =-=[32]-=-. Aichholzer et al. [5] gave the first efficient algorithms for computing the straight skeleton, and presented it as an alternative to the medial axis having only straight-line edges. The straight ske... |