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StraightLine Drawings on Restricted Integer Grids in Two and Three Dimensions (Extended Abstract)
, 2002
"... This paper investigates the following question: Given an integer grid phi, where phi is a proper subset of the integer plane or a proper subset of the integer 3d space, which graphs admit straightline crossingfree drawings with vertices located at the grid points of phi? We characterize the trees t ..."
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Cited by 38 (4 self)
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This paper investigates the following question: Given an integer grid phi, where phi is a proper subset of the integer plane or a proper subset of the integer 3d space, which graphs admit straightline crossingfree drawings with vertices located at the grid points of phi? We characterize the trees that can be drawn on a two dimensional c * n × k grid, where k and c are given integer constants, and on a two dimensional grid consisting of k parallel horizontal lines of infinite length. Motivated by the results on the plane we investigate restrictions of the integer grid in 3 dimensions and show that every outerplanar graph with n vertices can be drawn crossingfree with straight lines in linear volume on a grid called a prism. This prism consists of 3n integer grid points and is universal  it supports all outerplanar graphs of n vertices. This is the first algorithm that computes crossingfree straight line 3d drawings in linear volume for a nontrivial family of planar graphs. We also show that there exist planar graphs that cannot be drawn on the prism and that extension to a n × 2 × 2 integer grid, called a box, does not admit the entire class of planar graphs.
On Linear Layouts of Graphs
, 2004
"... In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A kstack (resp... ..."
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Cited by 31 (19 self)
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In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A kstack (resp...
On the Parameterized Complexity of Layered Graph Drawing
 PROC. 5TH ANNUAL EUROPEAN SYMP. ON ALGORITHMS (ESA '01
, 2001
"... We consider graph drawings in which vertices are assigned to layers and edges are drawn as straight linesegments between vertices on adjacent layers. We prove that graphs admitting crossingfree hlayer drawings (for fixed h) have bounded pathwidth. We then use a path decomposition as the basis for ..."
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Cited by 21 (9 self)
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We consider graph drawings in which vertices are assigned to layers and edges are drawn as straight linesegments between vertices on adjacent layers. We prove that graphs admitting crossingfree hlayer drawings (for fixed h) have bounded pathwidth. We then use a path decomposition as the basis for a lineartime algorithm to decide if a graph has a crossingfree hlayer drawing (for fixed h). This algorithm is extended to solve a large number of related problems, including allowing at most k crossings, or removing at most r edges to leave a crossingfree drawing (for fixed k or r). If the number of crossings or deleted edges is a nonfixed parameter then these problems are NPcomplete. For each setting, we can also permit downward drawings of directed graphs and drawings in which edges may span multiple layers, in which case the total span or the maximum span of edges can be minimized. In contrast to the socalled Sugiyama method for layered graph drawing, our algorithms do not assume a preassignment of the vertices to layers.
An Efficient Fixed Parameter Tractable Algorithm for 1Sided Crossing Minimization
 ALGORITHMICA
, 2004
"... We give an O(ϕ k · n 2) fixed parameter tractable algorithm for the 1SIDED CROSSING MINIMIZATION problem. The constant ϕ in the running time is the golden ratio ϕ = (1 + √ 5)/2 ≈ 1.618. The constant k is the parameter of the problem: the number of allowed edge crossings. ..."
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Cited by 11 (4 self)
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We give an O(ϕ k · n 2) fixed parameter tractable algorithm for the 1SIDED CROSSING MINIMIZATION problem. The constant ϕ in the running time is the golden ratio ϕ = (1 + √ 5)/2 ≈ 1.618. The constant k is the parameter of the problem: the number of allowed edge crossings.
Parameterized algorithmics for linear arrangement problems
, 2005
"... We discuss different variants of linear arrangement problems from a parameterized perspective. More specifically, we concentrate on developing simple search tree algorithms for these problems. However, the analysis of these algorithms is sometimes not so easy. Key words: Parameterized algorithms, li ..."
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Cited by 7 (2 self)
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We discuss different variants of linear arrangement problems from a parameterized perspective. More specifically, we concentrate on developing simple search tree algorithms for these problems. However, the analysis of these algorithms is sometimes not so easy. Key words: Parameterized algorithms, linear arrangement problems 1
A parameterized algorithm for upward planarity testing
 In Annual European Symposium on Algorithms (Proc. ESA ’04
, 2004
"... author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii We can visualize a graph by producing a geometric representation of the graph in which eac ..."
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Cited by 6 (0 self)
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author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii We can visualize a graph by producing a geometric representation of the graph in which each node is represented by a single point on the plane, and each edge is represented by a curve that connects its two endpoints. Directed graphs are often used to model hierarchical structures; in order to visualize the hierarchy represented by such a graph, it is desirable that a drawing of the graph reflects this hierarchy. This can be achieved by drawing all the edges in the graph such that they all point in an upwards direction. A graph that has a drawing in which all edges point in an upwards direction and in which no edges cross is known as an upward planar graph. Unfortunately, testing if a graph is upward planar is NPcomplete. Parameterized complexity is a technique used to find efficient algorithms for hard
Twolayer planarization: Improving on parameterized algorithmics
 SOFSEM, volume 3381 of LNCS
, 2005
"... A bipartite graph is biplanar if the vertices can be placed on two parallel lines in the plane such that there are no edge crossings when edges are drawn as straightline segments connecting vertices on one line to vertices on the other line. We study two problems: • 2Layer Planarization: can k edg ..."
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Cited by 5 (2 self)
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A bipartite graph is biplanar if the vertices can be placed on two parallel lines in the plane such that there are no edge crossings when edges are drawn as straightline segments connecting vertices on one line to vertices on the other line. We study two problems: • 2Layer Planarization: can k edges be deleted from a given graph G so that the remaining graph is biplanar? • 1Layer Planarization: same question, but the order of the vertices on one layer is fixed. Improving on earlier works of Dujmović et al. (Proc. Graph Drawing GD 2001, pp. 1–15, 2002), we solve the 2Layer Planarization problem in O(k 2 · 5.1926 k + G) time and the 1Layer Planarization problem in O(k 3 · 2.5616 k + G  2) time. Moreover, we derive a small problem kernel for 1Layer Planarization.
Experiments with the fixedparameter approach for twolayer planarization
 In [15
, 2003
"... Abstract. We present computational results of an implementation based on the fixed parameter tractability (FPT) approach for biplanarizing graphs. These results show that the implementation can efficiently minimum biplanarizing sets containing up to about 18 edges, thus making it comparable to previ ..."
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Cited by 4 (2 self)
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Abstract. We present computational results of an implementation based on the fixed parameter tractability (FPT) approach for biplanarizing graphs. These results show that the implementation can efficiently minimum biplanarizing sets containing up to about 18 edges, thus making it comparable to previous integer linear programming approaches. We show how our implementation slightly improves the theoretical running time to O(6 bpr (G) + G). Finally, we explain how our experimental work predicts how performance on sparse graphs may be improved. 1
Sorting Based Data Centric Storage
"... Abstract — Datacentric storage is a very important concept for sensor networks, where data of the same type are aggregated and stored in the same set of nodes. It is essential for many sensornet applications because it supports efficient innetwork query and processing. Multiple approaches have bee ..."
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Cited by 1 (1 self)
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Abstract — Datacentric storage is a very important concept for sensor networks, where data of the same type are aggregated and stored in the same set of nodes. It is essential for many sensornet applications because it supports efficient innetwork query and processing. Multiple approaches have been proposed so far. Their main technique is the hashing technique, where a hashing function is used to map data with the same key value to the same geometric location, and sensors closest to the location are made to store the data. Such solutions are elegant and efficient for implementation. However, two difficulties still remain: load balancing and the support for range queries. When the data of some key values are more abundant than data of other key values, or when sensors are not uniformly placed in the geometric space, some sensors can store substantially more data than other sensors. Since hashing functions map data with similar key values to independent locations, to query a range of data, multiple query messages need to be sent, even if the data of some key value in the range do not exist. In addition to the above two difficulties, obtaining the locations of sensors is also a nontrivial task. In this paper, we propose a new datacentric storage method based on sorting. Our method is robust for different network models and works for unlocalized homogeneous sensor networks, i.e., it requires no location information and no super nodes that have significantly more resources than other nodes. The idea is to sort the data in the network based on their key values, so that queries – including range queries – can be easily answered. The sorting method balances the storage load very well, and we present a sorting algorithm that is both decentralized and very efficient. We present both rigorous theoretical analysis and extensive simulations for analyzing its performance. They show that the sortingbased method has excellent performance for both communication and storage. I.
An Algorithm for 3Dbiplanar Graph Drawing
"... We introduce the concept of 3Dbiplanar drawing in which we partition a graph into two planar induced subgraphs. Our goal is to find such a partition with the minimum number of edges between the two partitions. We prove that this problem is NPcomplete and present a randomized parameterized algorith ..."
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We introduce the concept of 3Dbiplanar drawing in which we partition a graph into two planar induced subgraphs. Our goal is to find such a partition with the minimum number of edges between the two partitions. We prove that this problem is NPcomplete and present a randomized parameterized algorithm with O(n k) time, where k is the ratio of the optimal solution to the mincut size of the graph. 1