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50
A Framework for Dynamic Graph Drawing
 CONGRESSUS NUMERANTIUM
, 1992
"... Drawing graphs is an important problem that combines flavors of computational geometry and graph theory. Applications can be found in a variety of areas including circuit layout, network management, software engineering, and graphics. The main contributions of this paper can be summarized as follows ..."
Abstract

Cited by 544 (40 self)
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Drawing graphs is an important problem that combines flavors of computational geometry and graph theory. Applications can be found in a variety of areas including circuit layout, network management, software engineering, and graphics. The main contributions of this paper can be summarized as follows: ffl We devise a model for dynamic graph algorithms, based on performing queries and updates on an implicit representation of the drawing, and we show its applications. ffl We present several efficient dynamic drawing algorithms for trees, seriesparallel digraphs, planar stdigraphs, and planar graphs. These algorithms adopt a variety of representations (e.g., straightline, polyline, visibility), and update the drawing in a smooth way.
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 86 (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.
Clustering with Constraints: Feasibility Issues and the kMeans Algorithm
, 2005
"... Recent work has looked at extending the kMeans algorithm to incorporate background information in the form of instance level mustlink and cannotlink constraints. We introduce two ways of specifying additional background information in the form of # and # constraints that operate on all instances ..."
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Cited by 66 (8 self)
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Recent work has looked at extending the kMeans algorithm to incorporate background information in the form of instance level mustlink and cannotlink constraints. We introduce two ways of specifying additional background information in the form of # and # constraints that operate on all instances but which can be interpreted as conjunctions or disjunctions of instance level constraints and hence are easy to implement. We present complexity results for the feasibility of clustering under each type of constraint individually and several types together. A key finding is that determining whether there is a feasible solution satisfying all constraints is, in general, NPcomplete. Thus, an iterative algorithm such as kMeans should not try to find a feasible partitioning at each iteration. This motivates our derivation of a new version of the kMeans algorithm that minimizes the constrained vector quantization error but at each iteration does not attempt to satisfy all constraints. Using standard UCI datasets, we find that using constraints improves accuracy as others have reported, but we also show that our algorithm reduces the number of iterations until convergence. Finally, we illustrate these benefits and our new constraint types on a complex real world object identification problem using the infrared detector on an Aibo robot.
the Complexity and Distributed Construction of EnergyEfficient Broadcast Trees
 in Static and Ad Hoc Wireless Networks,” Proc. CISS
, 2002
"... Abstract—This paper addresses the energyefficient broadcasting problem in ad hoc wireless networks. First, we show that finding the minimumenergy broadcast tree is NPcomplete. We then develop a distributed clustering algorithm that computes energyefficient broadcast trees in polynomial time. Our ..."
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Cited by 32 (3 self)
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Abstract—This paper addresses the energyefficient broadcasting problem in ad hoc wireless networks. First, we show that finding the minimumenergy broadcast tree is NPcomplete. We then develop a distributed clustering algorithm that computes energyefficient broadcast trees in polynomial time. Our distributed algorithm computes all N possible broadcast trees simultaneously, while requiring O(N 2) messages to be exchanged between nodes. We compare our algorithm’s performance to the bestknown centralized algorithm, and show that it constructs trees consuming, on average, only 18 % more energy. We also consider the possibility of having multiple source nodes that can be used to broadcast the message and adapt our algorithm to compute energyefficient broadcast trees with multiple source nodes. We observe a reduction in the amount of energy needed to form the broadcast tree that is linear in the number of source nodes. Index Terms—Broadcast, complexity, energy efficiency, wireless networks. I.
Finding Compact Coordinate Representations for Polygons and Polyhedra
 IBM Journal of Research and Development
, 1989
"... Practical solid modeling systems are plagued by numerical problems that arise from using floatingpoint arithmetic. For example, polyhedral solids are often represented by a combination of geometric and combinatorial information. The geometric information might consist of explicit plane equations, wi ..."
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Cited by 23 (4 self)
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Practical solid modeling systems are plagued by numerical problems that arise from using floatingpoint arithmetic. For example, polyhedral solids are often represented by a combination of geometric and combinatorial information. The geometric information might consist of explicit plane equations, with floatingpoint coefficients; the combinatorial information might consist of face, edge, and vertex adjacencies and orientations, with edges defined by faceface adjacencies and vertices by edgeedge adjacencies. Problems arise when numerical error in geometric operations causes the geometric information to become inconsistent with the combinatorial information. These problems could be avoided by using exact arithmetic instead of floatingpoint arithmetic. However, some operations, like rotation, increase the number of bits required to represent the plane equation coefficients. Since the execution time of exact arithmetic operators increases with the number of bits in the operands, the inc...
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 17 (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...
Comparing and Evaluating Layout Algorithms within GraphEd
 J. Visual Languages and Computing
, 1995
"... This paper is organized as follows. In section 2, we present an overview on the GraphEd system and the implemented graph drawing algorithms. Section 3 explains our evaluation experiments, and Section 4 shows our results. In Section 5 we give a subjective ranking of layout criteria. 2 GraphEd ..."
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Cited by 16 (2 self)
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This paper is organized as follows. In section 2, we present an overview on the GraphEd system and the implemented graph drawing algorithms. Section 3 explains our evaluation experiments, and Section 4 shows our results. In Section 5 we give a subjective ranking of layout criteria. 2 GraphEd
An Experimental Comparison of Three Graph Drawing Algorithms (Extended Abstract)
, 1995
"... In this paper we present an extensive experimental study... ..."
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Cited by 16 (5 self)
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In this paper we present an extensive experimental study...
Complexity of Inference in Graphical Models
"... It is wellknown that inference in graphical models is hard in the worst case, but tractable for models with bounded treewidth. We ask whether treewidth is the only structural criterion of the underlying graph that enables tractable inference. In other words, is there some class of structures with u ..."
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Cited by 14 (0 self)
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It is wellknown that inference in graphical models is hard in the worst case, but tractable for models with bounded treewidth. We ask whether treewidth is the only structural criterion of the underlying graph that enables tractable inference. In other words, is there some class of structures with unbounded treewidth in which inference is tractable? Subject to a combinatorial hypothesis due to Robertson et al. (1994), we show that low treewidth is indeed the only structural restriction that can ensure tractability. Thus, even for the “best case” graph structure, there is no inference algorithm with complexity polynomial in the treewidth. 1
OutputSensitive Reporting of Disjoint Paths
, 1996
"... A kpath query on a graph consists of computing k vertexdisjoint paths between two given vertices of the graph, whenever they exist. In this paper, we study the problem of performing kpath queries, with k < 3, in a graph G with n vertices. We denote with the total length of the paths reported. ..."
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Cited by 12 (2 self)
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A kpath query on a graph consists of computing k vertexdisjoint paths between two given vertices of the graph, whenever they exist. In this paper, we study the problem of performing kpath queries, with k < 3, in a graph G with n vertices. We denote with the total length of the paths reported. For k < 3, we present an optimal data structure for G that uses O(n) space and executes kpath queries in outputsensitive O() time. For triconnected planar graphs, our results make use of a new combinatorial structure that plays the same role as bipolar (st) orientations for biconnected planar graphs. This combinatorial structure also yields an alternative construction of convex grid drawings of triconnected planar graphs.