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Marc van Kreveld

Bio: Marc van Kreveld is an academic researcher from Utrecht University. The author has contributed to research in topics: Bowyer–Watson algorithm & Pitteway triangulation. The author has an hindex of 39, co-authored 258 publications receiving 10560 citations. Previous affiliations of Marc van Kreveld include Carleton University & Free University of Berlin.


Papers
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Book
01 Jan 1997
TL;DR: In this article, an introduction to computational geometry focusing on algorithms is presented, which is related to particular applications in robotics, graphics, CAD/CAM, and geographic information systems.
Abstract: This introduction to computational geometry focuses on algorithms. Motivation is provided from the application areas as all techniques are related to particular applications in robotics, graphics, CAD/CAM, and geographic information systems. Modern insights in computational geometry are used to provide solutions that are both efficient and easy to understand and implement.

4,805 citations

Proceedings ArticleDOI
01 Aug 1997
TL;DR: This paper gives the first methods to obtain seed sets that are provably small in size based on a variant of the contour tree (or topographic change tree), and develops a simple approximation algorithm giving a seed set of size at most twice the size of the minimum once the contours tree is known.
Abstract: For 2D or 3D meshes that represent the domain of continuous function to the reals, the contours|or isosurfaces|of a speci ed value are an important way to visualize the function To nd such contours, a seed set can be used for the starting points from which the traversal of the contours can begin This paper gives the rst methods to obtain seed sets that are provably small in size They are based on a variant of the contour tree (or topographic change tree) We give a new, simple algorithm to compute such a tree in regular and irregular meshes that requires O(n logn) time in 2D for meshes with n elements, and in O(n) time in higher dimensions The additional storage overhead is proportial to the maximum size of any contour (linear in the worst case, but typically less) Given the contour tree, a minimum size seed set can be computed in roughly quadratic time Since in practice this can be excessive, we develop a simple approximation algorithm giving a seed set of size at most twice the size of the minimum It requires O(n log n) time and linear storage once the contour tree is known We also give experimental results, showing the size of the seed sets for several data sets

363 citations

Proceedings ArticleDOI
10 Nov 2006
TL;DR: This work considers the computational efficiency of computing two of the most basic spatio-temporal patterns in trajectories, namely flocks and meetings, and gives several exact and approximation algorithms.
Abstract: Moving point object data can be analyzed through the discovery of patterns. We consider the computational efficiency of computing two of the most basic spatio-temporal patterns in trajectories, namely flocks and meetings. The patterns are large enough subgroups of the moving point objects that exhibit similar movement and proximity for a certain amount of time. We consider the problem of computing a longest duration flock or meeting. We give several exact and approximation algorithms, and also show that some variants are as hard as MaxClique to compute and approximate.

296 citations

Journal ArticleDOI
TL;DR: This work investigates the problem of computing a large non-intersecting subset in a set of n rectangles in the plane and obtains a (1 + l/k)-approximati on in time O(n logn + n 2k-1) time, for any integer k/> 1.
Abstract: Motivated by the problem of labeling maps, we investigate the problem of computing a large non-intersecting subset in a set of n rectangles in the plane. Our results are as follows. In O(n log n) time, we can find an O(log n)factor approximation of the maximum subset in a set of n arbitrary axis-parallel rectangles in the plane. If all rectangles have unit height, we can find a 2-approximation in O(n logn) time. Extending this result, we obtain a (1 + l/k)-approximati on in time O(n logn + n 2k-1) time, for any integer k/> 1. © 1998 Elsevier Science B.V. All rights reserved.

241 citations

Proceedings Article
01 Jan 1997
TL;DR: In this paper, the problem of computing a large nonintersecting subset in a set of n rectangles in the plane was investigated and a (1 + l/k)-approximation was obtained in O(n logn + n 2k-1) time.
Abstract: Motivated by the problem of labeling maps, we investigate the problem of computing a large non-intersecting subset in a set of n rectangles in the plane. Our results are as follows. In O(n log n) time, we can find an O(log n)factor approximation of the maximum subset in a set of n arbitrary axis-parallel rectangles in the plane. If all rectangles have unit height, we can find a 2-approximation in O(n logn) time. Extending this result, we obtain a (1 + l/k)-approximati on in time O(n logn + n 2k-1) time, for any integer k/> 1. © 1998 Elsevier Science B.V. All rights reserved.

215 citations


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01 Jan 2002

9,314 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that given an integer k ≥ 1, (1 + ϵ)-approximation to the k nearest neighbors of q can be computed in additional O(kd log n) time.
Abstract: Consider a set of S of n data points in real d-dimensional space, Rd, where distances are measured using any Minkowski metric. In nearest neighbor searching, we preprocess S into a data structure, so that given any query point q∈ Rd, is the closest point of S to q can be reported quickly. Given any positive real ϵ, data point p is a (1 +ϵ)-approximate nearest neighbor of q if its distance from q is within a factor of (1 + ϵ) of the distance to the true nearest neighbor. We show that it is possible to preprocess a set of n points in Rd in O(dn log n) time and O(dn) space, so that given a query point q ∈ Rd, and ϵ > 0, a (1 + ϵ)-approximate nearest neighbor of q can be computed in O(cd, ϵ log n) time, where cd,ϵ≤d ⌈1 + 6d/ϵ⌉d is a factor depending only on dimension and ϵ. In general, we show that given an integer k ≥ 1, (1 + ϵ)-approximations to the k nearest neighbors of q can be computed in additional O(kd log n) time.

2,813 citations

Proceedings ArticleDOI
01 Aug 2001
TL;DR: A novel technique is proposed, called Topology Matching, in which similarity between polyhedral models is quickly, accurately, and automatically calculated by comparing Multiresolutional Reeb Graphs (MRGs), which operates well as a search key for 3D shape data sets.
Abstract: There is a growing need to be able to accurately and efficiently search visual data sets, and in particular, 3D shape data sets. This paper proposes a novel technique, called Topology Matching, in which similarity between polyhedral models is quickly, accurately, and automatically calculated by comparing Multiresolutional Reeb Graphs (MRGs). The MRG thus operates well as a search key for 3D shape data sets. In particular, the MRG represents the skeletal and topological structure of a 3D shape at various levels of resolution. The MRG is constructed using a continuous function on the 3D shape, which may preferably be a function of geodesic distance because this function is invariant to translation and rotation and is also robust against changes in connectivities caused by a mesh simplification or subdivision. The similarity calculation between 3D shapes is processed using a coarse-to-fine strategy while preserving the consistency of the graph structures, which results in establishing a correspondence between the parts of objects. The similarity calculation is fast and efficient because it is not necessary to determine the particular pose of a 3D shape, such as a rotation, in advance. Topology Matching is particularly useful for interactively searching for a 3D object because the results of the search fit human intuition well.

2,406 citations

Journal ArticleDOI
TL;DR: The class of point access methods, which are used to search sets of points in two or more dimensions, are presented and a discussion of theoretical and experimental results concerning the relative performance of various approaches are discussed.
Abstract: Search operations in databases require special support at the physical level. This is true for conventional databases as well as spatial databases, where typical search operations include the point query (find all objects that contain a given search point) and the region query (find all objects that overlap a given search region). More than ten years of spatial database research have resulted in a great variety of multidimensional access methods to support such operations. We give an overview of that work. After a brief survey of spatial data management in general, we first present the class of point access methods, which are used to search sets of points in two or more dimensions. The second part of the paper is devoted to spatial access methods to handle extended objects, such as rectangles or polyhedra. We conclude with a discussion of theoretical and experimental results concerning the relative performance of various approaches.

1,758 citations

Proceedings ArticleDOI
01 Aug 1999
TL;DR: The first distributed algorithms for routing that do not require duplication of packets or memory at the nodes and yet guarantee that a packet is delivered to its destination are described.
Abstract: We consider routing problems in ad hoc wireless networks modeled as unit graphs in which nodes are points in the plane and two nodes can communicate if the distance between them is less than some fixed unit. We describe the first distributed algorithms for routing that do not require duplication of packets or memory at the nodes and yet guarantee that a packet is delivered to its destination. These algorithms can be extended to yield algorithms for broadcasting and geocasting that do not require packet duplication. A byproduct of our results is a simple distributed protocol for extracting a planar subgraph of a unit graph. We also present simulation results on the performance of our algorithms.

1,537 citations