Helmut Alt - Academia.edu (original) (raw)

Papers by Helmut Alt

Research paper thumbnail of Computational Discrete Mathematics

Lecture Notes in Computer Science, 2001

Research paper thumbnail of On some connection problems in straight-line segment arrangements

Research paper thumbnail of Approximate Motion Planning and the Complexity of the Boundary of the Union of Simple Geometric Figures

Given a satisfiable Boolean formula in 2-CNF, it is NP-hard to find a satisfying assignment that ... more Given a satisfiable Boolean formula in 2-CNF, it is NP-hard to find a satisfying assignment that contains a minimum number of true variables. A polynomial-time approximation algorithm is given that finds an assignment with at most twice as many true variables as necessary. The algorithm also works for a weighted generalization of the problem. An application to the optimal stable roommates problem is given in detail, and other applications are mentioned.

Research paper thumbnail of The Computational Geometry of Comparing Shapes

Springer eBooks, 2009

... At the upper and right boundaries of each cell we can mark those intervals that can be reache... more ... At the upper and right boundaries of each cell we can mark those intervals that can be reached by a monotone path from the origin, if this information is available for the lower and ... The first major result in this direction was obtained by Godau in his Ph.D. thesis [8] who showed ...

Research paper thumbnail of Thoughts on Barnette's Conjecture

arXiv (Cornell University), Dec 13, 2013

We prove a new sufficient condition for a cubic 3-connected planar graph to be Hamiltonian. This ... more We prove a new sufficient condition for a cubic 3-connected planar graph to be Hamiltonian. This condition is most easily described as a property of the dual graph. Let G be a planar triangulation. Then the dual G * is a cubic 3-connected planar graph, and G * is bipartite if and only if G is Eulerian. We prove that if the vertices of G are (improperly) coloured blue and red, such that the blue vertices cover the faces of G, there is no blue cycle, and every red cycle contains a vertex of degree at most 4, then G * is Hamiltonian. This result implies the following special case of Barnette's Conjecture: if G is an Eulerian planar triangulation, whose vertices are properly coloured blue, red and green, such that every red-green cycle contains a vertex of degree 4, then G * is Hamiltonian. Our final result highlights the limitations of using a proper colouring of G as a starting point for proving Barnette's Conjecture. We also explain related results on Barnette's Conjecture that were obtained by Kelmans and for which detailed self-contained proofs have not been published.

Research paper thumbnail of Middle curves based on discrete Fréchet distance

Computational Geometry, 2020

Given a set of polygonal curves, we present algorithms for computing a middle curve that serves a... more Given a set of polygonal curves, we present algorithms for computing a middle curve that serves as a representative for the entire set of curves. We require that the middle curve consists of vertices of the input curves and that it minimizes the maximum discrete Fréchet distance to all input curves. We consider three dierent variants of a middle curve depending on in which order vertices of the input curves may occur on the middle curve, and provide algorithms for computing each variant.

Research paper thumbnail of Piecewise linear approximation of Be´zier-curves

Proceedings of the thirteenth annual symposium on Computational geometry - SCG '97, 1997

While geometric objects are often designed and represented as smooth curves or surfaces, many eff... more While geometric objects are often designed and represented as smooth curves or surfaces, many efficient algorithms (e.g. rendering tools) require their input in piecewise linear form. Therefore it becomes necessary to efficiently approximate objects of the first type by objects of the second type. We will inveAigate this problem for B6zier-curves in two or higher dimensions, i.e. parameterized curves of the form n C(t) =~Bi,n(t)pi,

Research paper thumbnail of Nearest Neighbor Search in High Dimensions

Research paper thumbnail of 3 D-Disk-Packing ∗

In this article, we consider the problem of finding in three dimensions a minimum volume axispara... more In this article, we consider the problem of finding in three dimensions a minimum volume axisparallel cuboid container into which a given set of unit size disks can be packed under translations. The problem is neither known to be NP-hard nor to be in NP. We give a constant factor approximation algorithm based on reduction to finding a shortest Hamiltonian path in a weighted graph. As a byproduct, we can show that there is no finite size container into which all unit disks can be packed simultaneously.

Research paper thumbnail of Executive Summary Computational Geometry 1 Computational Geometry Evolution

From March 8 to March 13, 2009, the Dagstuhl Seminar 09111 Computational Geometry was held in Sch... more From March 8 to March 13, 2009, the Dagstuhl Seminar 09111 Computational Geometry was held in Schloss Dagstuhl Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The rst section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available. 09111 Executive Summary Computational Geometry 1 Computational Geometry Evolution The eld of computational geometry is concerned with the design, analysis, and implementation of algorithms for geometric problems, which arise in a wide range of areas, including computer graphics, CAD, robotics computer vision, image processing, spatial databases, GIS, molecular biology, and sensor networks. Since the mid 1980s, computational geometry has arisen as a...

Research paper thumbnail of Computational Aspects of Packing Problems

Packing problems have been investigated in mathematics since centuries. For polygons, circles, or... more Packing problems have been investigated in mathematics since centuries. For polygons, circles, or other objects bounded by algebraic curves or surfaces it can be argued that packing problems are computable. However, even some of the simplest versions in the plane turn out to be NP-hard unless the number of objects to be packed is bounded. This article is a survey on results achieved about computability and complexity of packing problems, about approximation algorithms, and about very natural packing problems whose computational complexity is unknown. Packing objects is a quite natural problem and has been investigated in mathematics and operations research for a long time. Applications concern the physical nonoverlapping packing of concrete objects during storage or transportation but also in two dimensions how to eciently cut prescribed pieces from cloth or sheet metal while minimizing waste. Even more abstract problems like, e.g., ecient scheduling with respect to time and space c...

Research paper thumbnail of Packing 2D Disks into a 3D Container

In this article, we consider the problem of finding in three dimensions a minimum volume axis-par... more In this article, we consider the problem of finding in three dimensions a minimum volume axis-parallel box into which a given set of unit size disks can be packed under translations. The problem is neither known to be NP-hard nor to be in NP. We give a constant factor approximation algorithm based on reduction to finding a shortest Hamiltonian path in a weighted graph. As a byproduct, we can show that there is no finite size container into which all unit disks can be packed simultaneously.

Research paper thumbnail of Minimum Cell Connection in Line Segment Arrangements

International Journal of Computational Geometry & Applications, 2017

We study the complexity of the following cell connection problems in segment arrangements. Given ... more We study the complexity of the following cell connection problems in segment arrangements. Given a set of straight-line segments in the plane and two points [Formula: see text] and [Formula: see text] in different cells of the induced arrangement: [(i)] compute the minimum number of segments one needs to remove so that there is a path connecting [Formula: see text] to [Formula: see text] that does not intersect any of the remaining segments; [(ii)] compute the minimum number of segments one needs to remove so that the arrangement induced by the remaining segments has a single cell. We show that problems (i) and (ii) are NP-hard and discuss some special, tractable cases. Most notably, we provide a near-linear-time algorithm for a variant of problem (i) where the path connecting [Formula: see text] to [Formula: see text] must stay inside a given polygon [Formula: see text] with a constant number of holes, the segments are contained in [Formula: see text], and the endpoints of the segm...

Research paper thumbnail of A Middle Curve Based on Discrete Fréchet Distance

LATIN 2016: Theoretical Informatics, 2016

Given a set of polygonal curves we seek to find a "middle curve" that represents the set of curve... more Given a set of polygonal curves we seek to find a "middle curve" that represents the set of curves. We ask that the middle curve consists of points of the input curves and that it minimizes the discrete Fréchet distance to the input curves. We develop algorithms for three different variants of this problem.

Research paper thumbnail of Approximating Minimum-Area Rectangular and Convex Containers for Packing Convex Polygons

Lecture Notes in Computer Science, 2015

DOI to the publisher's website. • The final author version and the galley proof are versions of t... more DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the "Taverne" license above, please follow below link for the End User Agreement:

Research paper thumbnail of Placing your Coins on a Shelf

J. Comput. Geom., 2018

We consider the problem of packing a family of disks "on a shelf", that is, such that e... more We consider the problem of packing a family of disks "on a shelf", that is, such that each disk touches the xxx-axis from above and such that no two disks overlap. We prove that the problem of minimizing the distance between the leftmost point and the rightmost point of any disk is NP-hard. On the positive side, we show how to approximate this problem within a factor of 4/3 in O(nlogn)O(n \log n)O(nlogn) time, and provide an O(nlogn)O(n \log n)O(nlogn)-time exact algorithm for a special case, in particular when the ratio between the largest and smallest radius is at most four.

Research paper thumbnail of Scandinavian Thins on Top of Cake: New and Improved Algorithms for Stacking and Packing

Theory of Computing Systems, 2014

We show how to compute the smallest rectangle that can enclose any polygon, from a given set of p... more We show how to compute the smallest rectangle that can enclose any polygon, from a given set of polygons, in nearly linear time; we also present a PTAS for the problem, as well as a linear-time algorithm for the case when the polygons are rectangles themselves. We prove that finding a smallest convex polygon that encloses any of the given polygons is NP-hard, and give a PTAS for minimizing the perimeter of the convex enclosure. We also give efficient algorithms to find the smallest rectangle simultaneously enclosing a given pair of convex polygons.

Research paper thumbnail of Approximating Smallest Containers for Packing Three-Dimensional Convex Objects

International Journal of Computational Geometry & Applications

We investigate the problem of computing a minimal-volume container for the non-overlapping packin... more We investigate the problem of computing a minimal-volume container for the non-overlapping packing of a given set of three-dimensional convex objects. Already the simplest versions of the problem are [Formula: see text]-hard so that we cannot expect to find polynomial time algorithms to determine the exact solution. We give constant ratio approximation algorithms for packing axis-parallel (rectangular) cuboids under translation into an axis-parallel (rectangular) cuboid as container, for packing cuboids under rigid motions into an axis-parallel cuboid or into an arbitrary convex container, and for packing convex polyhedra under rigid motions into an axis-parallel cuboid or arbitrary convex container. This work gives the first approximability results for the computation of minimum volume containers for the objects described.

Research paper thumbnail of Complexity arguments in algebraic language theory

RAIRO. Informatique théorique

Complexity arguments in algebraic language theory RAIRO-Informatique théorique, tome 13, n o 3 (1... more Complexity arguments in algebraic language theory RAIRO-Informatique théorique, tome 13, n o 3 (1979), p. 217-225. <http://www.numdam.org/item?id=ITA_1979__13_3_217_0> © AFCET, 1979, tous droits réservés. L'accès aux archives de la revue « RAIRO-Informatique théorique » implique l'accord avec les conditions générales d'utilisation (http://www.numdam. org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

Research paper thumbnail of Probabilistic Matching of polygons

We analyze a probabilistic algorithm for matching plane compact sets with suciently nice boundari... more We analyze a probabilistic algorithm for matching plane compact sets with suciently nice boundaries under translations and rigid motions (rotation and translation). Given shapes A and B, the algorithm computes a transformation t such that with high prob- ability the area of overlap of t(A) and B is close to maximal. We give a time bound that does not depend

Research paper thumbnail of Computational Discrete Mathematics

Lecture Notes in Computer Science, 2001

Research paper thumbnail of On some connection problems in straight-line segment arrangements

Research paper thumbnail of Approximate Motion Planning and the Complexity of the Boundary of the Union of Simple Geometric Figures

Given a satisfiable Boolean formula in 2-CNF, it is NP-hard to find a satisfying assignment that ... more Given a satisfiable Boolean formula in 2-CNF, it is NP-hard to find a satisfying assignment that contains a minimum number of true variables. A polynomial-time approximation algorithm is given that finds an assignment with at most twice as many true variables as necessary. The algorithm also works for a weighted generalization of the problem. An application to the optimal stable roommates problem is given in detail, and other applications are mentioned.

Research paper thumbnail of The Computational Geometry of Comparing Shapes

Springer eBooks, 2009

... At the upper and right boundaries of each cell we can mark those intervals that can be reache... more ... At the upper and right boundaries of each cell we can mark those intervals that can be reached by a monotone path from the origin, if this information is available for the lower and ... The first major result in this direction was obtained by Godau in his Ph.D. thesis [8] who showed ...

Research paper thumbnail of Thoughts on Barnette's Conjecture

arXiv (Cornell University), Dec 13, 2013

We prove a new sufficient condition for a cubic 3-connected planar graph to be Hamiltonian. This ... more We prove a new sufficient condition for a cubic 3-connected planar graph to be Hamiltonian. This condition is most easily described as a property of the dual graph. Let G be a planar triangulation. Then the dual G * is a cubic 3-connected planar graph, and G * is bipartite if and only if G is Eulerian. We prove that if the vertices of G are (improperly) coloured blue and red, such that the blue vertices cover the faces of G, there is no blue cycle, and every red cycle contains a vertex of degree at most 4, then G * is Hamiltonian. This result implies the following special case of Barnette's Conjecture: if G is an Eulerian planar triangulation, whose vertices are properly coloured blue, red and green, such that every red-green cycle contains a vertex of degree 4, then G * is Hamiltonian. Our final result highlights the limitations of using a proper colouring of G as a starting point for proving Barnette's Conjecture. We also explain related results on Barnette's Conjecture that were obtained by Kelmans and for which detailed self-contained proofs have not been published.

Research paper thumbnail of Middle curves based on discrete Fréchet distance

Computational Geometry, 2020

Given a set of polygonal curves, we present algorithms for computing a middle curve that serves a... more Given a set of polygonal curves, we present algorithms for computing a middle curve that serves as a representative for the entire set of curves. We require that the middle curve consists of vertices of the input curves and that it minimizes the maximum discrete Fréchet distance to all input curves. We consider three dierent variants of a middle curve depending on in which order vertices of the input curves may occur on the middle curve, and provide algorithms for computing each variant.

Research paper thumbnail of Piecewise linear approximation of Be´zier-curves

Proceedings of the thirteenth annual symposium on Computational geometry - SCG '97, 1997

While geometric objects are often designed and represented as smooth curves or surfaces, many eff... more While geometric objects are often designed and represented as smooth curves or surfaces, many efficient algorithms (e.g. rendering tools) require their input in piecewise linear form. Therefore it becomes necessary to efficiently approximate objects of the first type by objects of the second type. We will inveAigate this problem for B6zier-curves in two or higher dimensions, i.e. parameterized curves of the form n C(t) =~Bi,n(t)pi,

Research paper thumbnail of Nearest Neighbor Search in High Dimensions

Research paper thumbnail of 3 D-Disk-Packing ∗

In this article, we consider the problem of finding in three dimensions a minimum volume axispara... more In this article, we consider the problem of finding in three dimensions a minimum volume axisparallel cuboid container into which a given set of unit size disks can be packed under translations. The problem is neither known to be NP-hard nor to be in NP. We give a constant factor approximation algorithm based on reduction to finding a shortest Hamiltonian path in a weighted graph. As a byproduct, we can show that there is no finite size container into which all unit disks can be packed simultaneously.

Research paper thumbnail of Executive Summary Computational Geometry 1 Computational Geometry Evolution

From March 8 to March 13, 2009, the Dagstuhl Seminar 09111 Computational Geometry was held in Sch... more From March 8 to March 13, 2009, the Dagstuhl Seminar 09111 Computational Geometry was held in Schloss Dagstuhl Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The rst section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available. 09111 Executive Summary Computational Geometry 1 Computational Geometry Evolution The eld of computational geometry is concerned with the design, analysis, and implementation of algorithms for geometric problems, which arise in a wide range of areas, including computer graphics, CAD, robotics computer vision, image processing, spatial databases, GIS, molecular biology, and sensor networks. Since the mid 1980s, computational geometry has arisen as a...

Research paper thumbnail of Computational Aspects of Packing Problems

Packing problems have been investigated in mathematics since centuries. For polygons, circles, or... more Packing problems have been investigated in mathematics since centuries. For polygons, circles, or other objects bounded by algebraic curves or surfaces it can be argued that packing problems are computable. However, even some of the simplest versions in the plane turn out to be NP-hard unless the number of objects to be packed is bounded. This article is a survey on results achieved about computability and complexity of packing problems, about approximation algorithms, and about very natural packing problems whose computational complexity is unknown. Packing objects is a quite natural problem and has been investigated in mathematics and operations research for a long time. Applications concern the physical nonoverlapping packing of concrete objects during storage or transportation but also in two dimensions how to eciently cut prescribed pieces from cloth or sheet metal while minimizing waste. Even more abstract problems like, e.g., ecient scheduling with respect to time and space c...

Research paper thumbnail of Packing 2D Disks into a 3D Container

In this article, we consider the problem of finding in three dimensions a minimum volume axis-par... more In this article, we consider the problem of finding in three dimensions a minimum volume axis-parallel box into which a given set of unit size disks can be packed under translations. The problem is neither known to be NP-hard nor to be in NP. We give a constant factor approximation algorithm based on reduction to finding a shortest Hamiltonian path in a weighted graph. As a byproduct, we can show that there is no finite size container into which all unit disks can be packed simultaneously.

Research paper thumbnail of Minimum Cell Connection in Line Segment Arrangements

International Journal of Computational Geometry & Applications, 2017

We study the complexity of the following cell connection problems in segment arrangements. Given ... more We study the complexity of the following cell connection problems in segment arrangements. Given a set of straight-line segments in the plane and two points [Formula: see text] and [Formula: see text] in different cells of the induced arrangement: [(i)] compute the minimum number of segments one needs to remove so that there is a path connecting [Formula: see text] to [Formula: see text] that does not intersect any of the remaining segments; [(ii)] compute the minimum number of segments one needs to remove so that the arrangement induced by the remaining segments has a single cell. We show that problems (i) and (ii) are NP-hard and discuss some special, tractable cases. Most notably, we provide a near-linear-time algorithm for a variant of problem (i) where the path connecting [Formula: see text] to [Formula: see text] must stay inside a given polygon [Formula: see text] with a constant number of holes, the segments are contained in [Formula: see text], and the endpoints of the segm...

Research paper thumbnail of A Middle Curve Based on Discrete Fréchet Distance

LATIN 2016: Theoretical Informatics, 2016

Given a set of polygonal curves we seek to find a "middle curve" that represents the set of curve... more Given a set of polygonal curves we seek to find a "middle curve" that represents the set of curves. We ask that the middle curve consists of points of the input curves and that it minimizes the discrete Fréchet distance to the input curves. We develop algorithms for three different variants of this problem.

Research paper thumbnail of Approximating Minimum-Area Rectangular and Convex Containers for Packing Convex Polygons

Lecture Notes in Computer Science, 2015

DOI to the publisher's website. • The final author version and the galley proof are versions of t... more DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the "Taverne" license above, please follow below link for the End User Agreement:

Research paper thumbnail of Placing your Coins on a Shelf

J. Comput. Geom., 2018

We consider the problem of packing a family of disks "on a shelf", that is, such that e... more We consider the problem of packing a family of disks "on a shelf", that is, such that each disk touches the xxx-axis from above and such that no two disks overlap. We prove that the problem of minimizing the distance between the leftmost point and the rightmost point of any disk is NP-hard. On the positive side, we show how to approximate this problem within a factor of 4/3 in O(nlogn)O(n \log n)O(nlogn) time, and provide an O(nlogn)O(n \log n)O(nlogn)-time exact algorithm for a special case, in particular when the ratio between the largest and smallest radius is at most four.

Research paper thumbnail of Scandinavian Thins on Top of Cake: New and Improved Algorithms for Stacking and Packing

Theory of Computing Systems, 2014

We show how to compute the smallest rectangle that can enclose any polygon, from a given set of p... more We show how to compute the smallest rectangle that can enclose any polygon, from a given set of polygons, in nearly linear time; we also present a PTAS for the problem, as well as a linear-time algorithm for the case when the polygons are rectangles themselves. We prove that finding a smallest convex polygon that encloses any of the given polygons is NP-hard, and give a PTAS for minimizing the perimeter of the convex enclosure. We also give efficient algorithms to find the smallest rectangle simultaneously enclosing a given pair of convex polygons.

Research paper thumbnail of Approximating Smallest Containers for Packing Three-Dimensional Convex Objects

International Journal of Computational Geometry & Applications

We investigate the problem of computing a minimal-volume container for the non-overlapping packin... more We investigate the problem of computing a minimal-volume container for the non-overlapping packing of a given set of three-dimensional convex objects. Already the simplest versions of the problem are [Formula: see text]-hard so that we cannot expect to find polynomial time algorithms to determine the exact solution. We give constant ratio approximation algorithms for packing axis-parallel (rectangular) cuboids under translation into an axis-parallel (rectangular) cuboid as container, for packing cuboids under rigid motions into an axis-parallel cuboid or into an arbitrary convex container, and for packing convex polyhedra under rigid motions into an axis-parallel cuboid or arbitrary convex container. This work gives the first approximability results for the computation of minimum volume containers for the objects described.

Research paper thumbnail of Complexity arguments in algebraic language theory

RAIRO. Informatique théorique

Complexity arguments in algebraic language theory RAIRO-Informatique théorique, tome 13, n o 3 (1... more Complexity arguments in algebraic language theory RAIRO-Informatique théorique, tome 13, n o 3 (1979), p. 217-225. <http://www.numdam.org/item?id=ITA_1979__13_3_217_0> © AFCET, 1979, tous droits réservés. L'accès aux archives de la revue « RAIRO-Informatique théorique » implique l'accord avec les conditions générales d'utilisation (http://www.numdam. org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

Research paper thumbnail of Probabilistic Matching of polygons

We analyze a probabilistic algorithm for matching plane compact sets with suciently nice boundari... more We analyze a probabilistic algorithm for matching plane compact sets with suciently nice boundaries under translations and rigid motions (rotation and translation). Given shapes A and B, the algorithm computes a transformation t such that with high prob- ability the area of overlap of t(A) and B is close to maximal. We give a time bound that does not depend