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37
LEDA: A Platform for Combinatorial and Geometric Computing
, 1999
"... We give an overview of the LEDA platform for combinatorial and geometric computing and an account of its development. We discuss our motivation for building LEDA and to what extent we have reached our goals. We also discuss some recent theoretical developments. This paper contains no new technical ..."
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Cited by 726 (46 self)
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We give an overview of the LEDA platform for combinatorial and geometric computing and an account of its development. We discuss our motivation for building LEDA and to what extent we have reached our goals. We also discuss some recent theoretical developments. This paper contains no new technical material. It is intended as a guide to existing publications about the system. We refer the reader also to our webpages for more information.
Graph visualization and navigation in information visualization: A survey
 In IEEE Transactions on Visualization and Computer Graphics
, 2000
"... ..."
Efficient Algorithms for Petersen's Matching Theorem
, 1999
"... Petersen's theorem is a classic result in matching theory from 1891, stating that every 3regular bridgeless graph has a perfect matching. Our work explores efficient algorithms for finding perfect matchings in such graphs. Previously, the only relevant matching algorithms were for general gra ..."
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Cited by 29 (3 self)
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Petersen's theorem is a classic result in matching theory from 1891, stating that every 3regular bridgeless graph has a perfect matching. Our work explores efficient algorithms for finding perfect matchings in such graphs. Previously, the only relevant matching algorithms were for general graphs, and the fastest algorithm ran in O(n^3/2) time for 3regular graphs. We have developed an O(n log^4 n)time algorithm for perfect matching in a 3regular bridgeless graph. When the graph is also planar, we have as the main result of our paper an optimal O(n)time algorithm. We present three applications of this result: terrain guarding, adaptive mesh refinement, and quadrangulation.
Some Algorithmic Problems in Polytope Theory
 IN ALGEBRA, GEOMETRY, AND SOFTWARE SYSTEMS
, 2003
"... Convex polytopes, i.e.. the intersections of finitely many closed affine halfspaces in R^d, are important objects in various areas of mathematics and other disciplines. In particular, the compact... ..."
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Cited by 22 (1 self)
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Convex polytopes, i.e.. the intersections of finitely many closed affine halfspaces in R^d, are important objects in various areas of mathematics and other disciplines. In particular, the compact...
A New Planarity Test
, 1999
"... Given an undirected graph, the planarity testing problem is to determine whether the graph can be drawn in the plane without any crossing edges. Linear time planarity testing algorithms have previously been designed by Hopcroft and Tarjan, and by Booth and Lueker. However, their approaches are quite ..."
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Cited by 22 (2 self)
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Given an undirected graph, the planarity testing problem is to determine whether the graph can be drawn in the plane without any crossing edges. Linear time planarity testing algorithms have previously been designed by Hopcroft and Tarjan, and by Booth and Lueker. However, their approaches are quite involved. Several other approaches have also been developed for simplifying the planariy test. In this paper, we developed a very simple linear time testing algorithm based only on a depthfirst search tree. When the given graph is not planar, our algorithm immediately produces explicit Kuratowski's subgraphs. A new data structure, PCtrees, is introduced, which can be viewed as abstract subembeddings of actual planar embeddings. A graphreduction technique is adopted so that the embeddings for the planar biconnected components constructed at each iteration never have to be changed. The recognition and embedding are actually done simultaneously in our algorithm 1 . The implementation of o...
Inserting an Edge Into a Planar Graph
 Algorithmica
, 2000
"... Computing a crossing minimum drawing of a given planar graph G augmented by an additional edge e in which all crossings involve e, has been a long standing open problem in graph drawing. Alternatively, the problem can be stated as finding a planar combinatorial embedding of a planar graph G in which ..."
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Cited by 19 (9 self)
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Computing a crossing minimum drawing of a given planar graph G augmented by an additional edge e in which all crossings involve e, has been a long standing open problem in graph drawing. Alternatively, the problem can be stated as finding a planar combinatorial embedding of a planar graph G in which the given edge e can be inserted with the minimum number of crossings. Many problems concerned with the optimization over the set of all combinatorial embeddings of a planar graph turned out to be NPhard. Surprisingly, we found a conceptually simple linear time algorithm based on SPQRtrees, which is able to find a crossing minimum solution.
An Algorithm for StraightLine Drawing of Planar Graphs
, 1995
"... Abstract. We present a new algorithm for drawing planar graphs on the plane. It can be viewed as a generalization of the algorithm of Chrobak and Payne, which, in turn, is based on an algorithm by de Fraysseix, Pach, and Pollack. Our algorithm improves the previous ones in that it does not require a ..."
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Cited by 15 (0 self)
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Abstract. We present a new algorithm for drawing planar graphs on the plane. It can be viewed as a generalization of the algorithm of Chrobak and Payne, which, in turn, is based on an algorithm by de Fraysseix, Pach, and Pollack. Our algorithm improves the previous ones in that it does not require a preliminary triangulation step; triangulation proves problematic in drawing graphs “nicely, ” as it has the tendency to ruin the structure of the input graph. The new algorithm retains the positive features of the previous algorithms: it embeds a biconnected graph of n vertices on a grid of size (2n − 4) × (n − 2) in linear time. We have implemented the algorithm as part of a software system for drawing graphs nicely. Key Words.
From Algorithms to Working Programs On the Use of Program Checking in LEDA
 IN PROC. INT. CONF. ON MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE (MFCS 98
, 1998
"... We report on the use of program checking in the LEDA library of efficient data types and algorithms. ..."
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Cited by 9 (3 self)
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We report on the use of program checking in the LEDA library of efficient data types and algorithms.
Terrain Guarding is NPHard
, 2009
"... A set G of points on a 1.5dimensional terrain, also known as an xmonotone polygonal chain, is said to guard the terrain if every point on the terrain is seen by a point in G. Two points on the terrain see each other if and only if the line segment between them is never strictly below the terrain. ..."
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Cited by 8 (0 self)
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A set G of points on a 1.5dimensional terrain, also known as an xmonotone polygonal chain, is said to guard the terrain if every point on the terrain is seen by a point in G. Two points on the terrain see each other if and only if the line segment between them is never strictly below the terrain. The minimum terrain guarding problem asks for a minimum guarding set for the given input terrain. Using a reduction from PLANAR 3SAT we prove that the decision version of this problem is NPhard. This solves a significant open problem and complements recent positive approximability results for the optimization problem. 1