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MaxMin DCluster Formation in Wireless Ad Hoc Networks
 IN PROCEEDINGS OF IEEE INFOCOM
, 2000
"... An ad hoc network may be logically represented as a set of clusters. The clusterheads form a dhop dominating set. Each node is at most d hops from a clusterhead. Clusterheads form a virtual backbone and may be used to route packets for nodes in their cluster. Previous heuristics restricted themselv ..."
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Cited by 195 (3 self)
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An ad hoc network may be logically represented as a set of clusters. The clusterheads form a dhop dominating set. Each node is at most d hops from a clusterhead. Clusterheads form a virtual backbone and may be used to route packets for nodes in their cluster. Previous heuristics restricted themselves to 1hop clusters. We show that the minimum dhop dominating set problem is NPcomplete. Then we present a heuristic to form dclusters in a wireless ad hoc network. Nodes are assumed to have nondeterministic mobility pattern. Clusters are formed by diffusing node identities along the wireless links. When the heuristic terminates, a node either becomes a clusterhead, or is at most d wireless hops away from its clusterhead. The value of d is a parameter of the heuristic. The heuristic can be run either at regular intervals, or whenever the network configuration changes. One of the features of the heuristic is that it tends to reelect existing clusterheads even when the network configurat...
On the Computational Complexity of Upward and Rectilinear Planarity Testing (Extended Abstract)
, 1994
"... A directed graph is upward planar if it can be drawn in the plane such that every edge is a monotonically increasing curve in the vertical direction, and no two edges cross. An undirected graph is rectilinear planar if it can be drawn in the plane such that every edge is a horizontal or vertical se ..."
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Cited by 84 (4 self)
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A directed graph is upward planar if it can be drawn in the plane such that every edge is a monotonically increasing curve in the vertical direction, and no two edges cross. An undirected graph is rectilinear planar if it can be drawn in the plane such that every edge is a horizontal or vertical segment, and no two edges cross. Testing upward planarity and rectilinear planarity are fundamental problems in the effective visualization of various graph and network structures. In this paper we show that upward planarity testing and rectilinear planarity testing are NPcomplete problems. We also show that it is NPhard to approximate the minimum number of bends in a planar orthogonal drawing of an nvertex graph with an O(n 1\Gammaffl ) error, for any ffl ? 0.
An Experimental Comparison of Four Graph Drawing Algorithms
, 1995
"... In this paper we present an extensive experimental study comparing four generalpurpose graph drawing algorithms. The four algorithms take as input general graphs (with no restrictions whatsoever on connectivity, planarity, etc.) and construct orthogonal grid drawings, which are widely used in so ..."
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Cited by 56 (8 self)
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In this paper we present an extensive experimental study comparing four generalpurpose graph drawing algorithms. The four algorithms take as input general graphs (with no restrictions whatsoever on connectivity, planarity, etc.) and construct orthogonal grid drawings, which are widely used in software and database visualization applications. The test data (available by anonymous ftp) are 11,582 graphs, ranging from 10 to 100 vertices, which have been generated from a core set of 112 graphs used in "reallife" software engineering and database applications. The experiments
Computing Orthogonal Drawings with the Minimum Number of Bends
 IEEE Transactions on Computers
, 1998
"... We describe a branchandbound algorithm for computing an orthogonal grid drawing with the minimum number of bends of a biconnected planar graph. Such algorithm is based on an efficient enumeration schema of the embeddings of a planar graph and on several new methods for computing lower bounds o ..."
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Cited by 28 (10 self)
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We describe a branchandbound algorithm for computing an orthogonal grid drawing with the minimum number of bends of a biconnected planar graph. Such algorithm is based on an efficient enumeration schema of the embeddings of a planar graph and on several new methods for computing lower bounds of the number of bends. We experiment such algorithm on a large test suite and compare the results with the stateoftheart. The experiments show how minimizing the number of bends strongly improves several quality measures of the effectiveness of the drawing. We also present a graphic tool with animation that embodies the algorithm and allows interacting with all the phases of the computation. 1 Introduction Various graphic standards have been proposed to draw graphs, each one devoted to a specific class of applications. An extensive literature on the subject can be found in [4, 23, 2]. In particular, an orthogonal drawing maps each edge into a chain of horizontal and vertical segmen...
ThreeDimensional Orthogonal Graph Drawing
, 2000
"... vi Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Tables . . . . . . . . . . . . ..."
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Cited by 27 (10 self)
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vi Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii List of Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv I Orthogonal Graph Drawing 1 1
Algorithms for Drawing Clustered Graphs
, 1997
"... In the mid 1980s, graphics workstations became the main platforms for software and information engineers. Since then, visualization of relational information has become an essential element of software systems. Graphs are commonly used to model relational information. They are depicted on a graphics ..."
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Cited by 25 (2 self)
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In the mid 1980s, graphics workstations became the main platforms for software and information engineers. Since then, visualization of relational information has become an essential element of software systems. Graphs are commonly used to model relational information. They are depicted on a graphics workstation as graph drawings. The usefulness of the relational model depends on whether the graph drawings effectively convey the relational information to the users. This thesis is concerned with finding good drawings of graphs. As the amount of information that we want to visualize becomes larger and the relations become more complex, the classical graph model tends to be inadequate. Many extended models use a node hierarchy to help cope with the complexity. This thesis introduces a new graph model called the clustered graph. The central theme of the thesis is an investigation of efficient algorithms to produce good drawings for clustered graphs. Although the criteria for judging the qua...
Planar Polyline Drawings with Good Angular Resolution
 Graph Drawing (Proc. GD '98), volume 1547 of LNCS
, 1998
"... . We present a linear time algorithm that constructs a planar polyline grid drawing of any plane graph with n vertices and maximum degree d on a (2n \Gamma 5) \Theta ( 3 2 n \Gamma 7 2 ) grid with at most 5n \Gamma 15 bends and minimum angle ? 2 d . In the constructed drawings, every edge h ..."
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Cited by 22 (1 self)
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. We present a linear time algorithm that constructs a planar polyline grid drawing of any plane graph with n vertices and maximum degree d on a (2n \Gamma 5) \Theta ( 3 2 n \Gamma 7 2 ) grid with at most 5n \Gamma 15 bends and minimum angle ? 2 d . In the constructed drawings, every edge has at most three bends and length O(n). To our best knowledge, this algorithm achieves the best simultaneous bounds concerning the grid size, angular resolution, and number of bends for planar grid drawings of highdegree planar graphs. Besides the nice theoretical features, the practical drawings are aesthetically very pleasing. An implementation of our algorithm is available with the AGDLibrary (Algorithms for Graph Drawing) [2, 1]. Our algorithm is based on ideas by Kant for polyline grid drawings for triconnected plane graphs [23]. In particular, our algorithm significantly improves upon his bounds on the angular resolution and the grid size for nontriconnected plane graphs....
A Numerical Optimization Approach to General Graph Drawing
, 1999
"... Graphs are ubiquitous, finding applications in domains ranging from software engineering to computational biology. While graph theory and graph algorithms are some of the oldest, most studied fields in computer science, the problem of visualizing graphs is comparatively young. This problem, known as ..."
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Cited by 19 (0 self)
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Graphs are ubiquitous, finding applications in domains ranging from software engineering to computational biology. While graph theory and graph algorithms are some of the oldest, most studied fields in computer science, the problem of visualizing graphs is comparatively young. This problem, known as graph drawing, is that of transforming combinatorial graphs into geometric drawings for the purpose of visualization. Most published algorithms for drawing general graphs model the drawing problem with a physical analogy, representing a graph as a system of springs and other physical elements and then simulating the relaxation of this physical system. Solving the graph drawing problem involves both choosing a physical model and then using numerical optimization to simulate the physical system. In this
Turnregularity and optimal area drawings of orthogonal representations
, 2000
"... Given an orthogonal representation H with n vertices and bends, we study the problem of computing a planar orthogonal drawing of H with small area. This problem has direct applications to the development of practical graph drawing techniques for information visualization and VLSI layout. In this pap ..."
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Cited by 17 (5 self)
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Given an orthogonal representation H with n vertices and bends, we study the problem of computing a planar orthogonal drawing of H with small area. This problem has direct applications to the development of practical graph drawing techniques for information visualization and VLSI layout. In this paper, we introduce the concept of turnregularity of an orthogonal representation H, provide combinatorial characterizations of it, and show that if H is turnregular (i.e., all its faces are turnregular), then a planar orthogonal drawing of H with minimum area can be computed in O(n) time, and a planar orthogonal drawing of H with minimum area and minimum total edge length within that area can be computed in O(n 7/4 log n) time. We also apply our theoretical results to the design and implementation of new practical heuristic methods for constructing planar orthogonal drawings. An experimental study conducted on a test suite of orthogonal representations of randomly generated biconnected 4planar graphs shows that the percentage of turnregular faces is quite high and that our heuristic drawing methods perform better than previous ones.
Algorithms for AreaEfficient Orthogonal Drawings
 Computational Geometry: Theory and Applications
, 1996
"... An orthogonal drawing of a graph is a drawing such that nodes are placed on grid points and edges are drawn as sequences of vertical and horizontal segments. In this paper we present linear time algorithms that produce orthogonal drawings of graphs with n nodes. If the maximum degree is four, then t ..."
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Cited by 16 (4 self)
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An orthogonal drawing of a graph is a drawing such that nodes are placed on grid points and edges are drawn as sequences of vertical and horizontal segments. In this paper we present linear time algorithms that produce orthogonal drawings of graphs with n nodes. If the maximum degree is four, then the drawing produced by our first algorithm needs area at most (roughly) 0:76n 2 , and introduces at most 2n + 2 bends. Also, each edge of such a drawing has at most two bends. Our algorithm is based on forming and placing pairs of vertices of the graph. If the maximum degree is three, then the drawing produced by our second algorithm needs at most (roughly) 1 4 n 2 area and, if the graph is biconnected, at most b n 2 c + 3 bends. These upper bounds match the upper bounds known for planar graphs of maximum degree 3. This algorithm produces optimal drawings (within a constant of 2) with respect to the number of bends, since there is a lower bound of n 2 + 1 in the number of bends fo...