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Realistic Input Models for Geometric Algorithms
 IN PROC. 13TH ANNU. ACM SYMPOS. COMPUT. GEOM
, 1997
"... Many algorithms developed in computational geometry are needlessly complicated and slow because they have to be prepared for very complicated, hypothetical inputs. To avoid this, realistic models are needed that describe the properties that realistic inputs have, so that algorithms can de designed t ..."
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Cited by 93 (21 self)
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Many algorithms developed in computational geometry are needlessly complicated and slow because they have to be prepared for very complicated, hypothetical inputs. To avoid this, realistic models are needed that describe the properties that realistic inputs have, so that algorithms can de designed that take advantage of these properties. This can lead to algorithms that are provably efficient in realistic situations. We obtain some fundamental results in this research direction. In particular, we have the following results. ffl We show the relations between various models that have been proposed in the literature. ffl For several of these models, we give algorithms to compute the model parameter(s) for a given scene; these algorithms can be used to verify whether a model is appropriate for typical scenes in some application area. ffl As a case study, we give some experimental results on the appropriateness of some of the models for one particular type of scenes often encountered in ...
Approximating the Fréchet distance for realistic curves in near linear time
 In Proc. 26th Annu. ACM Sympos. Comput. Geom
, 2010
"... We present a simple and practical (1 + ε)approximation algorithm for the Fréchet distance between two polygonal curves in IRd. To analyze this algorithm we introduce a new realistic family of curves, cpacked curves, that is closed under simplification. We believe the notion of cpacked curves to ..."
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Cited by 21 (7 self)
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We present a simple and practical (1 + ε)approximation algorithm for the Fréchet distance between two polygonal curves in IRd. To analyze this algorithm we introduce a new realistic family of curves, cpacked curves, that is closed under simplification. We believe the notion of cpacked curves to be of independent interest. We show that our algorithm has near linear running time for cpacked polygonal curves, and similar results for other input models, such as low density polygonal curves. 1
Approximate map matching with respect to the Fréchet distance
 Proc. 7th Workshop on Algorithm Engeneering and Experiments
, 2011
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Persistent Predecessor Search and Orthogonal Point Location on the Word RAM
"... We answer a basic data structuring question (for example, raised by Dietz and Raman back in SODA 1991): can van Emde Boas trees be made persistent, without changing their asymptotic query/update time? We present a (partially) persistent data structure that supports predecessor search in a set of int ..."
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Cited by 14 (5 self)
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We answer a basic data structuring question (for example, raised by Dietz and Raman back in SODA 1991): can van Emde Boas trees be made persistent, without changing their asymptotic query/update time? We present a (partially) persistent data structure that supports predecessor search in a set of integers in {1,..., U} under an arbitrary sequence of n insertions and deletions, with O(log log U) expected query time and expected amortized update time, and O(n) space. The query bound is optimal in U for linearspace structures and improves previous nearO((log log U) 2) methods. The same method solves a fundamental problem from computational geometry: point location in orthogonal planar subdivisions (where edges are vertical or horizontal). We obtain the first static data structure achieving O(log log U) worstcase query time and linear space. This result is again optimal in U for linearspace structures and improves the previous O((log log U) 2) method by de Berg, Snoeyink, and van Kreveld (1992). The same result also holds for higherdimensional subdivisions that are orthogonal binary space partitions, and for certain nonorthogonal planar subdivisions such as triangulations without small angles. Many geometric applications follow, including improved query times for orthogonal range reporting for dimensions ≥ 3 on the RAM. Our key technique is an interesting new vanEmdeBoas–style recursion that alternates between two strategies, both quite simple.
Approximate Range Searching Using Binary Space Partitions
"... We show how any BSP tree TP for the endpoints of a set of n disjoint segments in the plane can be used to obtain a BSP tree of size O(n · depth(TP)) for the segments themselves, such that the rangesearching efficiency remains almost the same. We apply this technique to obtain a BSP tree of size O(n ..."
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Cited by 10 (5 self)
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We show how any BSP tree TP for the endpoints of a set of n disjoint segments in the plane can be used to obtain a BSP tree of size O(n · depth(TP)) for the segments themselves, such that the rangesearching efficiency remains almost the same. We apply this technique to obtain a BSP tree of size O(n log n) such that εapproximate range searching queries with any constantcomplexity convex query range can be answered in O(minε>0{(1/ε) + kε} log n) time, where kε is the number of segments intersecting the εextended range. The same result can be obtained for disjoint constantcomplexity curves, if we allow the BSP to use splitting curves along the given curves. We also describe how to construct a linearsize BSP tree for lowdensity scenes consisting of n objects in R d such that εapproximate range searching with any constantcomplexity convex query range can be done in O(log n + minε>0{(1/ε d−1) + kε}) time. Finally we show how to adapt our structures so that they become I/Oefficient.
Guarding Scenes against Invasive Hypercubes
, 1998
"... A set of points G is a guarding set for a set of objects O, if any hypercube not containing a point from G in its interior intersects at most objects of O. This definition underlies a new input model, that is both more general than de Berg's unclutteredness, and retains its main property: ..."
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Cited by 9 (3 self)
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A set of points G is a guarding set for a set of objects O, if any hypercube not containing a point from G in its interior intersects at most objects of O. This definition underlies a new input model, that is both more general than de Berg's unclutteredness, and retains its main property: a ddimensional scene satisfying the new model's requirements is known to have a linearsize binary space partition. We propose several algorithms for computing guarding sets, and evaluate them experimentally. One of them appears to be quite practical. 1. Introduction Recently de Berg et al. [4] brought together several of the realistic input models that have been proposed in the literature, namely fatness, low density, unclutteredness, and small simplecover 1 Supported by the ESPRIT IV LTR Project No. 21957 (CGAL). 2 Supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities. 3 Supported by the Netherlands' Organization for Scientific Research...
Models and Motion Planning
, 1998
"... We study the complexity of the motion planning problem for a boundedreach robot in the situation where the n obstacles in its workspace satisfy two of the realistic models proposed in the literature, namely unclutteredness and small simplecover complexity. We show that the maximum complexity of ..."
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Cited by 9 (2 self)
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We study the complexity of the motion planning problem for a boundedreach robot in the situation where the n obstacles in its workspace satisfy two of the realistic models proposed in the literature, namely unclutteredness and small simplecover complexity. We show that the maximum complexity of the free space of a robot with f degrees of freedom in the plane is #(n f/2 + n) for uncluttered environments as well as environments with small simplecover complexity. The maximum complexity of the free space of a robot moving in a threedimensional uncluttered environment is #(n 2f/3 +n). All these bounds fit nicely between the #(n) bound for the maximum freespace complexity for lowdensity environments and the #(n f ) bound for unrestricted environments. Surprisinglybecause contrary to the situation in the planethe maximum freespace complexity is #(n f ) for a threedimensional environment with small simplecover complexity.
Local polyhedra and geometric graphs
 In Proc. 14th ACMSIAM Sympos. on Discrete Algorithms
, 2003
"... 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 ed ..."
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Cited by 9 (0 self)
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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 axisaligned 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 d1 n). Finally, we describe efficient algorithms for computing Minkowski sums of local polyhedra in two and three dimensions.
On the union of κround objects in three and four dimensions
 Geom
, 2004
"... A compact set c in R d is κround if for every point p ∈ ∂c there exists a closed ball that contains p, is contained in c, and has radius κ diam c. We show that, for any fixed κ> 0, the combinatorial complexity of the union of n κround, not necessarily convex objects in R 3 (resp., in R 4) of co ..."
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Cited by 8 (5 self)
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A compact set c in R d is κround if for every point p ∈ ∂c there exists a closed ball that contains p, is contained in c, and has radius κ diam c. We show that, for any fixed κ> 0, the combinatorial complexity of the union of n κround, not necessarily convex objects in R 3 (resp., in R 4) of constant description complexity is O(n 2+ε) (resp., O(n 3+ε)) for any ε> 0, where the constant of proportionality depends on ε, κ, and the algebraic complexity of the objects. The bound is almost tight in the worst case. 1