## Local polyhedra and geometric graphs (2003)

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Venue: | In Proc. 14th ACM-SIAM Sympos. on Discrete Algorithms |

Citations: | 11 - 0 self |

### BibTeX

@INPROCEEDINGS{Erickson03localpolyhedra,

author = {Jeff Erickson},

title = {Local polyhedra and geometric graphs},

booktitle = {In Proc. 14th ACM-SIAM Sympos. on Discrete Algorithms},

year = {2003},

pages = {171--180},

publisher = {ACM Press}

}

### OpenURL

### Abstract

We introduce a new realistic input model for geometric graphs and nonconvex polyhedra. A geometric graph G is local if (1) the longest edge at every vertex v is only a constant factor longer than the distance from v to its Euclidean nearest neighbor and (2) the lengths of the longest and shortest edges differ by at most a polynomial factor. A polyhedron is local if all its faces are simplices and its edges form a local geometric graph. We show that any boolean combination of any two local polyhedra in IR d each with n vertices, can be computed in O(n log n) time, using a standard hierarchy of axis-aligned bounding boxes. Using results of de Berg, we also show that any local polyhedron in IR d has a binary space partition tree of size O(n log d-1 n). Finally, we describe efficient algorithms for computing Minkowski sums of local polyhedra in two and three dimensions.

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Citation Context ...hedra, the algorithms refine their hierarchies only to the coarsest level at which the resulting bounding volumes are disjoint. Beginning with Guttmann’s introduction of the R-tree in the early 1980=-=s [24], -=-several types of bounding volume hierarchies have been proposed and implemented [2, 22, 23, 26, 31, 33, 35]. Unfortunately, all of these methods—in fact, any related method that uses a hierarchy of ... |

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Citation Context ... resulting bounding volumes are disjoint. Beginning with Guttmann’s introduction of the R-tree in the early 1980s [24], several types of bounding volume hierarchies have been proposed and implemente=-=d [2, 22, 23, 26, 31, 33, 35]. Unfo-=-rtunately, all of these methods—in fact, any related method that uses a hierarchy of convex bounding volumes— can be forced to spend Ω(n 2 ) time to determine whether two n-vertex polyhedra inte... |

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Citation Context ...(n 4/3 ) lower bound for the polyhedron intersection problem. The strongest lower bound known for this problem is only Ω(n log n), in the algebraic decision tree and algebraic computation tree model=-=s [52, 4]-=-.s2 Jeff Erickson and folds; however, it forbids many long edges to be packed closely together. See Section 2 for more formal definitions and basic properties. We consider the complexity of three prob... |

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Citation Context ...erlay non-matching meshes of similar (or identical) surfaces [27, 28, 30]. In particular, this technique is efficient for the well-shaped tetrahedral meshes produced by Delaunay refinement algorithms =-=[44, 49]-=-, even if they contain slivers. 4 The Harpsicordion We now show that the results from the previous section are asymptotically optimal for polyhedra in IR 3 by constructing a pair of local polyhedra th... |

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Citation Context ... resulting bounding volumes are disjoint. Beginning with Guttmann’s introduction of the R-tree in the early 1980s [24], several types of bounding volume hierarchies have been proposed and implemente=-=d [2, 22, 23, 26, 31, 33, 35]. Unfo-=-rtunately, all of these methods—in fact, any related method that uses a hierarchy of convex bounding volumes— can be forced to spend Ω(n 2 ) time to determine whether two n-vertex polyhedra inte... |

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Citation Context ...ool for computing depth orders for rendering. Since their introduction, BSP trees have been used for many other applications in computer graphics, including shadow generation [12, 13], solid modeling =-=[37, 57]-=-, geometric data repair [36], and visibility culling for interactive walkthroughs [55]. As in the case for intersection detection, the worst-case complexity bounds for BSPs of polyhedra in IR 3 are qu... |

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Citation Context ...en locality and clutter in more detail in Section 5. We can also compare our model to quality metrics used for simplicial finite-element mesh generation. For example, Miller, Talmor, Teng, and others =-=[32, 34, 49, 54, 56]-=- define a triangulation to be well-shaped if the circumradius of each simplex is only a constant factor longer than the shortest edge of that simplex. Well-shaped triangulations have bounded local str... |

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Citation Context ...pend Ω(n 2 ) time to determine whether two n-vertex polyhedra intersect. The worst-case example consists of two polyhedra whose edges approximate a twisted grid, similar to a construction of Chazell=-=e [10, 41]-=-. (See Section 4.) Since worst-case efficient algorithms for detecting intersections seem unlikely, many authors have analyzed heuristics under the assumption that the input objects satisfy certain re... |

91 | Realistic Input Models for Geometric Algorithms
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Citation Context ...xplicit dependence on these parameters from our upper bounds. 2.3 Relationship to Other Input Models Several different models of realistic or well-shaped geometric data have been proposed in the past =-=[6, 7]-=-. Perhaps the most well-known realistic input model is fatness [59]. An object X is fat if any ball centered inside X either contains X or has a constant fraction of its volume inside X. Thus, fat obj... |

90 |
Merging BSP trees yields polyhedral set operations
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Citation Context ...ool for computing depth orders for rendering. Since their introduction, BSP trees have been used for many other applications in computer graphics, including shadow generation [12, 13], solid modeling =-=[37, 57]-=-, geometric data repair [36], and visibility culling for interactive walkthroughs [55]. As in the case for intersection detection, the worst-case complexity bounds for BSPs of polyhedra in IR 3 are qu... |

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Citation Context ...acker et al. [47], as a tool for computing depth orders for rendering. Since their introduction, BSP trees have been used for many other applications in computer graphics, including shadow generation =-=[12, 13]-=-, solid modeling [37, 57], geometric data repair [36], and visibility culling for interactive walkthroughs [55]. As in the case for intersection detection, the worst-case complexity bounds for BSPs of... |

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Citation Context ...en locality and clutter in more detail in Section 5. We can also compare our model to quality metrics used for simplicial finite-element mesh generation. For example, Miller, Talmor, Teng, and others =-=[32, 34, 49, 54, 56]-=- define a triangulation to be well-shaped if the circumradius of each simplex is only a constant factor longer than the shortest edge of that simplex. Well-shaped triangulations have bounded local str... |

72 | Convex partitions of polyhedra: a lower bound and worst-case optimal algorithm
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Citation Context ...pend Ω(n 2 ) time to determine whether two n-vertex polyhedra intersect. The worst-case example consists of two polyhedra whose edges approximate a twisted grid, similar to a construction of Chazell=-=e [10, 41]-=-. (See Section 4.) Since worst-case efficient algorithms for detecting intersections seem unlikely, many authors have analyzed heuristics under the assumption that the input objects satisfy certain re... |

67 |
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Citation Context ...e Figure 2. Figure 2. A local nonfat polygon, and a fat nonlocal polygon. Another realistic input model, introduced by van der Stappen in the context of motion planning [51], is low density; see also =-=[39, 48]. A-=- set of objects have density λ if any ball of radius r intersects at most λ objects with diameter r or greater. Most bounds for low density scenes depend linearly on λ. Lemma 2.4 implies that a loc... |

64 |
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Citation Context ...acker et al. [47], as a tool for computing depth orders for rendering. Since their introduction, BSP trees have been used for many other applications in computer graphics, including shadow generation =-=[12, 13]-=-, solid modeling [37, 57], geometric data repair [36], and visibility culling for interactive walkthroughs [55]. As in the case for intersection detection, the worst-case complexity bounds for BSPs of... |

60 |
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Citation Context ...f any leaf cell of B. The size of a BSP is the number of cuts, or equivalently, one less than the number of leaves. Fuchs et al. [21] introduced BSP trees, following earlier work by Schumacker et al. =-=[47]-=-, as a tool for computing depth orders for rendering. Since their introduction, BSP trees have been used for many other applications in computer graphics, including shadow generation [12, 13], solid m... |

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Citation Context ...(n 4/3 ) lower bound for the polyhedron intersection problem. The strongest lower bound known for this problem is only Ω(n log n), in the algebraic decision tree and algebraic computation tree model=-=s [52, 4]-=-.s2 Jeff Erickson and folds; however, it forbids many long edges to be packed closely together. See Section 2 for more formal definitions and basic properties. We consider the complexity of three prob... |

54 |
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Citation Context ...aligned bounding boxes of facets of P and Q, respectively. Since each polyhedron has O(n) facets, we can clearly calculate B1 and B2 in O(n) time. Using multidimensional range trees and segment trees =-=[16, 50], we can -=-find all pairs of intersecting boxes (�1, �2) ∈ B1 × B2 in time O(n log d−1 n + k), where k is the number of intersecting pairs. Finally, for each pair of intersecting boxes, we can test in O... |

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Citation Context ...r example, can local polyhedra be triangulated using only a near-linear number of simplices? How hard is constructing the triangulation of a local polyhedron with the minimum number of Steiner points =-=[45]-=- or tetrahedra [3]? How complex is the medial axis of a local polyhedron in the worst case? Acknowledgments. Thanks to Pankaj Agarwal, Mark de Berg, Leo Guibas, Sariel Har-Peled, and Mark van Kreveld ... |

52 | A.: Consistent solid and boundary representations from arbitrary polygonal data
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Citation Context ...or rendering. Since their introduction, BSP trees have been used for many other applications in computer graphics, including shadow generation [12, 13], solid modeling [37, 57], geometric data repair =-=[36], -=-and visibility culling for interactive walkthroughs [55]. As in the case for intersection detection, the worst-case complexity bounds for BSPs of polyhedra in IR 3 are quite pessimistic. Chazelle’s ... |

50 |
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Citation Context ...stest algorithm for deciding whether two static nonconvex polyhedra intersect, due to Pellegrini, runs in time O(n 8/5+ε ) [43]. For polyhedral terrains, the time bound can be improved to O(n 4/3+ε =-=) [11]. Pe-=-llegrini’s algorithm was generalized by Schömer and Thiel [46] to find the first collision between two translating polyhedra in time O(n 8/5+ε ), or between two rotating polyhedra in time O(n 5/3+... |

50 | Optimal binary space partitions for orthogonal objects - Paterson, Yao - 1992 |

48 | Nice point sets can have nasty Delaunay triangulations
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Citation Context ...whose edge graph is connected. For disconnected sets of simplices, however, our running-time analysis requires the spread of the vertices— the ratio between the largest and smallest pairwise distanc=-=e [20]��-=-�to be bounded by a polynomial in n. For example, given two local connected planar straight-line graphs, we can overlay them in O(n log n) time in two different ways. One method is to use the standard... |

46 | Efficient collision detection for moving polyhedra
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Citation Context ...a intersect, due to Pellegrini, runs in time O(n 8/5+ε ) [43]. For polyhedral terrains, the time bound can be improved to O(n 4/3+ε ) [11]. Pellegrini’s algorithm was generalized by Schömer and T=-=hiel [46] to -=-find the first collision between two translating polyhedra in time O(n 8/5+ε ), or between two rotating polyhedra in time O(n 5/3+ε ). To avoid directly checking all Ω(n 2 ) edge pairs, these algo... |

44 | Well-Spaced Points for Numerical Methods
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(Show Context)
Citation Context ...en locality and clutter in more detail in Section 5. We can also compare our model to quality metrics used for simplicial finite-element mesh generation. For example, Miller, Talmor, Teng, and others =-=[32, 34, 49, 54, 56]-=- define a triangulation to be well-shaped if the circumradius of each simplex is only a constant factor longer than the shortest edge of that simplex. Well-shaped triangulations have bounded local str... |

37 | Collision detection for deforming necklaces
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Citation Context ...s of objects, a standard bounding box heuristic culls out most non-intersecting pairs, provided the objects are fat (at least on average) and all about the same size [53, 60]. Agarwal, Guibas, et al. =-=[23, 1]-=- and independently Lotan et al. [33] recently showed that in a certain hierarchy of bounding spheres for well-behaved necklaces of balls, only O(n 4/3 ) pairs of balls can intersect. Haverkort et al. ... |

37 |
Motion planning amidst fat obstacles
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Citation Context ... and thus need not be local. See Figure 2. Figure 2. A local nonfat polygon, and a fat nonlocal polygon. Another realistic input model, introduced by van der Stappen in the context of motion planning =-=[51], -=-is low density; see also [39, 48]. A set of objects have density λ if any ball of radius r intersects at most λ objects with diameter r or greater. Most bounds for low density scenes depend linearly... |

35 |
separators: a unified geometric approach to graph partitioning
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Citation Context ... σ is less than some fixed constant and its global stretch Σ is less than some fixed polynomial in the number of vertices. Local graphs are a generalization of the civilized graphs considered by Ten=-=g [56], for -=-which Σ = O(1). The choice of the word “local” is meant to emphasize the much more important role of the local stretch; all of our complexity bounds are polynomial in σ but at most polylogarithm... |

33 | Linear size binary space partitions for uncluttered scenes
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(Show Context)
Citation Context ... optimal; our lower bound construction also implies that two simple relaxations of our model allow the worst-case quadratic behavior of arbitrary polyhedra. In Section 5, we apply a result of de Berg =-=[2]-=- to show that any local polyhedron in IR d has a binary space partition tree of size O(n log d-1 n). We develop upper and lower bounds on the complexity of Minkowski sums of local polyhedra in low dim... |

33 | New lower bounds for Hopcroft’s problem
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(Show Context)
Citation Context ... on a line? The main idea of the reduction is to replace each point and line with an infinitesimally thin spike. In light of this reduction and Erickson’s Ω(n 4/3 ) lower bound for Hopcroft’s pr=-=oblem [18]-=-, an algorithm that detects intersections in o(n 4/3 ) worst-case time appears unlikely. 1 In practice, one of the most popular techniques for intersection detection uses a hierarchy of bounding volum... |

33 | Analysis of a Bounding Box Heuristic for Object Intersection
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(Show Context)
Citation Context ...ve shown that for large collections of objects, a standard bounding box heuristic culls out most non-intersecting pairs, provided the objects are fat (at least on average) and all about the same size =-=[53, 60]-=-. Agarwal, Guibas, et al. [23, 1] and independently Lotan et al. [33] recently showed that in a certain hierarchy of bounding spheres for well-behaved necklaces of balls, only O(n 4/3 ) pairs of balls... |

28 |
Fast software for box intersection
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(Show Context)
Citation Context ...that k = O(n log n), so the overall running time of the algorithm is O(n log d−1 n). This algorithm can be made extremely practical, at least in low dimensions, by combining it with simple heuristic=-=s [61]-=-. In fact, we can actually compute the intersection, or any other boolean combination, of two local polyhedra within the same time bound, by performing an additional constant amount of work for each p... |

22 | Efficient maintenance and self-collision testing for kinematic chains. Symposium on Computational Geometry
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Citation Context |

22 |
Counting and reporting intersections of d-ranges
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Citation Context ...aligned bounding boxes of facets of P and Q, respectively. Since each polyhedron has O(n) facets, we can clearly calculate B1 and B2 in O(n) time. Using multidimensional range trees and segment trees =-=[16, 50], we can -=-find all pairs of intersecting boxes (�1, �2) ∈ B1 × B2 in time O(n log d−1 n + k), where k is the number of intersecting pairs. Finally, for each pair of intersecting boxes, we can test in O... |

18 | On the relative complexities of some geometric problems
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(Show Context)
Citation Context ...oid directly checking all Ω(n 2 ) edge pairs, these algorithms employ complex multilevel range-searching data structures that would be difficult (if not impossible) to implement efficiently. Erickso=-=n [17] p-=-roved that the polyhedron intersection problem is at least as hard (in the the algebraic decision tree model of computation) as Hopcroft’s problem: Given a set of points and lines in the plane, does... |

18 | On the boundary complexity of the union of fat triangles
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Citation Context ... more realistic measure than the maximum. (Although most realistic input models restrict the worst-case behavior of some parameter, a few results are known for sets of objects that are fat on average =-=[40, 60]-=-.) Higher-order local stretch, on the other hand, was more tightly concentrated around the mean. First- and tenth-order local stretch distributions for the dragon model are displayed in Figure 13; the... |

17 | Range searching in low-density environments
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(Show Context)
Citation Context ...e Figure 2. Figure 2. A local nonfat polygon, and a fat nonlocal polygon. Another realistic input model, introduced by van der Stappen in the context of motion planning [51], is low density; see also =-=[39, 48]. A-=- set of objects have density λ if any ball of radius r intersects at most λ objects with diameter r or greater. Most bounds for low density scenes depend linearly on λ. Lemma 2.4 implies that a loc... |

16 |
S.-H.: Generating well-shaped delaunay meshes
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Citation Context |

13 |
On Fatness and Fitness - Realistic Input Models for Geometric Algorithms
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(Show Context)
Citation Context ...ationship to Other Input Models Several different models of realistic or well-shaped geometric data have been proposed in the past [6, 7]. Perhaps the most well-known realistic input model is fatness =-=[59]-=-. An object X is fat if any ball centered inside X either contains X or has a constant fraction of its volume inside X. Thus, fat objects have no sharp spikes or folds. Local polyhedra, however, can h... |

11 |
BOXTREE: A hierarchical representation of surfaces
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(Show Context)
Citation Context ... resulting bounding volumes are disjoint. Beginning with Guttmann's introduction of the R-tree in the early 1980s [17], several types of bounding volume hierarchies have been proposed and implemented =-=[1, 15, 16, 19, 20, 22, 24]-=-. Unfortunately, all of these methods---in fact, any related method that uses a hierarchy of convex bounding volumes---can be forced to spend# n 2 ) time to determine whether two n-vertex polyhedra in... |

9 |
Common Refinement Based Data Transfer Between Non-matching Meshes
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(Show Context)
Citation Context ...ion of different physical quantities over a single domain, or to track the evolution of a domain over time. In these applications, solutions must be transferred efficiently between overlapping meshes =-=[29]-=-. A key step in the solution transfer process is identifying pairs of overlapping elements. If the meshes are local, we can find all such pairs in near-linear time, using recursive bisection to define... |

9 |
Impulse-based dynamic simulation. In The algorithmic foundations of robotics (Eds
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8 | Models and motion planning
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(Show Context)
Citation Context ...explicit dependence on these parameters from our upper bounds. 2.3 Relationship to Other Input Models Several di#erent models of realistic or well-shaped geometric data have been proposed in the past =-=[3, 4]-=-. Perhaps the most well-known realistic input model is fatness [42]. An object X is fat if any ball centered inside X either contains X or has a constant fraction of its volume inside X. Thus, fat obj... |

8 | Mesh association: Formulation and algorithms
- Jiao, Edelsbrunner, et al.
- 1999
(Show Context)
Citation Context ...ll such pairs in near-linear time, using recursive bisection to define a graded bounding volume hierarchy. Similar methods can be use to overlay non-matching meshes of similar (or identical) surfaces =-=[27, 28, 30]-=-. In particular, this technique is efficient for the well-shaped tetrahedral meshes produced by Delaunay refinement algorithms [44, 49], even if they contain slivers. 4 The Harpsicordion We now show t... |