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Monotone Paths in Line Arrangements with a Small Number of Directions

Adrian Dumitrescu
2004 Discrete & Computational Geometry  
We give subquadratic bounds on the maximum length of an Ü-monotone path in an arrangement of Ò lines with at most ÐÓ ÐÓ Ò directions, where is a suitable constant.  ...  For instance, the maximum length of an Ü-monotone path in an arrangement of Ò lines having at most ten slopes is Ç´Ò ¿ µ.  ...  The author is grateful to William Steiger for suggesting this restricted version of the problem.  ... 
doi:10.1007/s00454-004-1106-6 fatcat:zxyeysrjgrhenek4um3nbq2ld4

Lower bounds on the length of monotone paths in arrangements

Jiří Matoušek
1991 Discrete & Computational Geometry  
We show that the maximal number of turns of an x-monotone path in an arrangement of n lines is f~(n 5/3) and the maximal number of turns of an x-monotone path in arrangement of n pseudolines is f~(n2/log  ...  In this paper we consider one such property, the maximal possible length of an x-monotone polygonal line (path) composed of edges of the arrangement; the length is measured as the number of turns of the  ...  Acknowledgment I thank a referee for bringing the manuscript [CMS] to my attention and for comments which I found helpful in the presentation of the results.  ... 
doi:10.1007/bf02574679 fatcat:ccxi3ojcpvgxzdqkbdxnajsuty

Long monotone paths in line arrangements

Jazsef Balogh, Oded Regev, Clifford Smyth, William Steiger, Mario Szegedy
2003 Proceedings of the nineteenth conference on Computational geometry - SCG '03  
We show how to construct an arrangement of n lines having a monotone path of length (n 2−(d/ √ log n) ), where d > 0 is some constant, and thus nearly settle the long standing question on monotone path  ...  length in line arrangements.  ...  A path is monotone in direction (a, b) if its sequence of vertices is monotone when projected orthogonally along the line with equation ay − bx = 0.  ... 
doi:10.1145/777811.777812 fatcat:4lwspzhqwbgs3dfjlqkgbqnee4

Long monotone paths in line arrangements

Jazsef Balogh, Oded Regev, Clifford Smyth, William Steiger, Mario Szegedy
2003 Proceedings of the nineteenth conference on Computational geometry - SCG '03  
We show how to construct an arrangement of n lines having a monotone path of length (n 2−(d/ √ log n) ), where d > 0 is some constant, and thus nearly settle the long standing question on monotone path  ...  length in line arrangements.  ...  A path is monotone in direction (a, b) if its sequence of vertices is monotone when projected orthogonally along the line with equation ay − bx = 0.  ... 
doi:10.1145/777792.777812 dblp:conf/compgeom/BaloghRSSS03 fatcat:6cr4a33xknekfltoyejvbmgjwi

Long Monotone Paths in Line Arrangements

J�zsef Balogh, Oded Regev, Clifford Smyth, William Steiger, Mario Szegedy
2004 Discrete & Computational Geometry  
We show how to construct an arrangement of n lines having a monotone path of length (n 2−(d/ √ log n) ), where d > 0 is some constant, and thus nearly settle the long standing question on monotone path  ...  length in line arrangements.  ...  A path is monotone in direction (a, b) if its sequence of vertices is monotone when projected orthogonally along the line with equation ay − bx = 0.  ... 
doi:10.1007/s00454-004-1119-1 fatcat:7lavbutrkzgvbpuxkn2vifa7jq

On some monotone path problems in line arrangements

Adrian Dumitrescu
2005 Computational geometry  
We estimate the minimum length of a longest monotone path in an arrangement of n lines, where length counts the number of turns on the path.  ...  When length is defined as the size of a convex/concave chain in the arrangement an exact bound is obtained.  ...  From the opposite direction, subquadratic upper bounds have been recently obtained only for line arrangements with a small number of slopes [3] . So the general problem is still open.  ... 
doi:10.1016/j.comgeo.2005.01.001 fatcat:qpo2kvjjbbhwjljju5fmj3z244

Monotone Simultaneous Embeddings of Upward Planar Digraphs

Oswin Aichholzer, Thomas Hackl, Sarah Lutteropp, Tamara Mchedlidze, Alexander Pilz, Birgit Vogtenhuber
2015 Journal of Graph Algorithms and Applications  
For more than three paths, we present a polynomial-time algorithm that, given any number of paths and predefined directions of monotonicity, decides whether the paths admit a monotone simultaneous embedding  ...  On the other hand, we show that already for three paths, any monotone simultaneous embedding might need a grid whose size is exponential in the number of vertices.  ...  Acknowledgements We thank anonymous referees for helpful comments and for making us aware of the related work [3] .  ... 
doi:10.7155/jgaa.00350 fatcat:46tdmqjuovdj7i2gzg3qucpdce

Verifiable implementations of geometric algorithms using finite precision arithmetic

Victor J. Milenkovic
1988 Artificial Intelligence  
Data normalization is applied to the problem of modeling polygonal regions in the plane, and the hidden variable method is used to calculate arrangements of lines. ,I  ...  The first method, data normalization, transforms the geometric structure into a configuration for which all finite precision calculations yield correct answers.  ...  Such a modeling system would only introduce a small error dependent on the number of lines, not the number of operations.  ... 
doi:10.1016/0004-3702(88)90061-6 fatcat:e7sdqwc7jjc4rfers2ny3yb4za

Monotone Paths in Planar Convex Subdivisions [chapter]

Adrian Dumitrescu, Günter Rote, Csaba D. Tóth
2012 Lecture Notes in Computer Science  
Then, there is a path with at least Ω(log(n/k)/ log log(n/k)) edges that is monotone in some direction. This bound is also the best possible.  ...  Then, starting from every vertex there is a path with at least Ω(log ∆ n) edges that is monotone in some direction. This bound is the best possible.  ...  Dumitrescu [5] proved that every simple arrangement of n lines admits a monotone path of length at least n in the convex subdivision generated by n lines.  ... 
doi:10.1007/978-3-642-32241-9_21 fatcat:zowky65uxrh4njdiyvt4mik2te

Counting Carambolas

Adrian Dumitrescu, Maarten Löffler, André Schulz, Csaba D. Tóth
2015 Graphs and Combinatorics  
Configurations of interest include convex polygons, star-shaped polygons and monotone paths. We also consider related problems for directed planar straight-line graphs.  ...  We give upper and lower bounds on the maximum and minimum number of geometric configurations of various kinds present (as subgraphs) in a triangulation of n points in the plane.  ...  Given a plane straight-line graph G on n vertices, the lines passing through the O(n) edges of G induce a line arrangement with O(n 2 ) faces. Choose a face f of the arrangement, and a vertex v of G.  ... 
doi:10.1007/s00373-015-1621-7 fatcat:cf76frghsvgilmfrcrkxl3rb2q

Monotone Paths in Planar Convex Subdivisions and Polytopes [chapter]

Adrian Dumitrescu, Günter Rote, Csaba D. Tóth
2013 Fields Institute Communications  
Then, there is a path with at least Ω(log(n/k)/ log log(n/k)) edges that is monotone in some direction. This bound is also the best possible.  ...  Then, starting from every vertex there is a path with at least Ω(log ∆ n) edges that is monotone in some direction. This bound is the best possible.  ...  Therefore we could not control the entrance into sibling subtrees for the nodes in which the monotone path starts or ends. In this section, we construct another polytope Q with a recursive structure.  ... 
doi:10.1007/978-3-319-00200-2_6 fatcat:mvoj5ertlfcm7p2lb7hewlwdom

Dealing with Difficult Instances of Object Rearrangement

Athanasios Krontiris, Kostas Bekris
2015 Robotics: Science and Systems XI  
The second contribution is the use of either the monotone or of the new non-monotone method as a local planner in the context of a higher-level task planner that searches the space of object placements  ...  Rearranging multiple objects is a critical skill for robots so that they can effectively deal with clutter in human spaces.  ...  Any opinions or findings expressed in this paper do not necessarily reflect the views of the sponsors. The authors would like to thank the anonymous RSS reviewers for their comments. BIBLIOGRAPHY  ... 
doi:10.15607/rss.2015.xi.045 dblp:conf/rss/KrontirisB15 fatcat:twbe35llgbbxbc4k37c2olgz3y

Automated torch path planning using polygon subdivision for solid freeform fabrication based on welding

Rajeev Dwivedi, Radovan Kovacevic
2004 Journal of manufacturing systems  
The suggested approach describes a method based on the subdivision of a two-dimensional (2-D) polygonal section into a set of monotone polygons to generate a continuous path for material deposition.  ...  This paper proposes a method for torch path planning applicable to SFF based on welding with an emphasis on minimum human intervention.  ...  Figure lOc shows a zig-zag path with the number of segments equal to 7, and the number of vertices equal to 8; the same path mapped to a closed zig-zag path, as shown in Figure lOb, has 8 segments and  ... 
doi:10.1016/s0278-6125(04)80040-2 fatcat:yn7q6gp4lrcw7aiieolht57hai

Dynamic motion planning in low obstacle density environments [chapter]

Robert-Paul Berretty, Mark Overmars, A. Frank van der Stappen
1997 Lecture Notes in Computer Science  
The weak and realistic low obstacle density (L.O.D.) assumption results in linear complexity in the number of obstacles of the free space [11] .  ...  We will show that in this situation a cell decomposition of the free space of size On 2 n log 2 n can be computed in On 2 n log 2 n time.  ...  If a hyper-plane P, that is orthogonal to the time direction, intersects a subcell in a number of disconnected regions, then there might be configurations in that cannot be connected with a time-monotone  ... 
doi:10.1007/3-540-63307-3_44 fatcat:ajkskp7nenes5aogrvxbw24tvq

GENERALIZING MONOTONICITY: ON RECOGNIZING SPECIAL CLASSES OF POLYGONS AND POLYHEDRA

PROSENJIT BOSE, MARC VAN KREVELD
2005 International journal of computational geometry and applications  
Abstract A simple polyhedron is weakly-monotonic in direction d provided that the intersection of the polyhedron and any plane with normal d is simply-connected (i.e. empty, a point, a line-segment or  ...  Given a simple n-vertex polygon P , we can determine whether or not a line can be swept over P in a continuous manner but with varying direction, such that any position of intersects P in at most two edges  ...  Recall that a simple polygon P is monotone in direction d if the intersection of P with any line having normal d is a convex set (i.e. empty, a point, or a line-segment).  ... 
doi:10.1142/s0218195905001877 fatcat:nhibqr2y5fbvtlqb5q2cznswky
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