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On rectilinear duals for vertexweighted plane graphs
 In GD ’05: Proceedings of the Symposium on Graph Drawing
, 2005
"... Abstract. Let G =(V,E) be a plane triangulated graph where each vertex is assigned a positive weight. A rectilinear dual of G is a partition of a rectangle into V  simple rectilinear regions, one for each vertex, such that two regions are adjacent if and only if the corresponding vertices are conn ..."
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Cited by 17 (2 self)
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Abstract. Let G =(V,E) be a plane triangulated graph where each vertex is assigned a positive weight. A rectilinear dual of G is a partition of a rectangle into V  simple rectilinear regions, one for each vertex, such that two regions are adjacent if and only if the corresponding vertices are connected by an edge in E. A rectilinear dual is called a cartogram if the area of each region is equal to the weight of the corresponding vertex. We show that every vertexweighted plane triangulated graph G admits a cartogram of constant complexity, that is, a cartogram where the number of vertices of each region is constant. 1
Octagonal drawings of plane graphs with prescribed face areas
 COMPUTATIONAL GEOMETRY: THEORY AND APPLICATIONS
, 2009
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Computing Cartograms with Optimal Complexity
"... In a rectilinear dual of a planar graph vertices are represented by simple rectilinear polygons and edges are represented by sidecontact between the corresponding polygons. A rectilinear dual is called a cartogram if the area of each region is equal to a prespecified weight of the corresponding ve ..."
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Cited by 8 (7 self)
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In a rectilinear dual of a planar graph vertices are represented by simple rectilinear polygons and edges are represented by sidecontact between the corresponding polygons. A rectilinear dual is called a cartogram if the area of each region is equal to a prespecified weight of the corresponding vertex. The complexity of a cartogram is determined by the maximum number of corners (or sides) required for any polygon. In a series of papers the polygonal complexity of such representations for maximal planar graphs has been reduced from the initial 40 to 34, then to 12 and very recently to the currently best known 10. Here we describe a construction with 8sided polygons, which is optimal in terms of polygonal complexity as 8sided polygons are sometimes necessary. Specifically, we show how to compute the combinatorial structure and how to refine the representation into an areauniversal rectangular layout in linear time. The exact cartogram can be computed from the areauniversal rectangular layout with numerical iteration, or can be approximated with a hillclimbing heuristic. We also describe an alternative construction for Hamiltonian maximal planar graphs, which allows us to directly compute the cartograms in linear time. Moreover, we prove that even for Hamiltonian graphs 8sided rectilinear polygons are necessary, by constructing a nontrivial lower bound example. The complexity of the cartograms can be reduced to 6 if the Hamiltonian path has the extra property that it is onelegged, as in outerplanar graphs. Thus, we have optimal representations (in terms of both polygonal complexity and running time) for Hamiltonian maximal planar and maximal outerplanar graphs.
Orthogonal cartograms with few corners per face
, 2010
"... We give an algorithm to create orthogonal drawings of 3connected 3regular planar graphs such that each interior face of the graph is drawn with a prescribed area. This algorithm produces a drawing with at most 12 corners per face and 4 bends per edge, which improves the previous known result of 34 ..."
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Cited by 4 (1 self)
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We give an algorithm to create orthogonal drawings of 3connected 3regular planar graphs such that each interior face of the graph is drawn with a prescribed area. This algorithm produces a drawing with at most 12 corners per face and 4 bends per edge, which improves the previous known result of 34 corners per face.
Quantitative Measures for Cartogram Generation Techniques Submission No. # 327
"... Cartograms are used to visualize geographically distributed data by scaling the regions of a map (e.g., US states) such that their areas are proportional to some data associated with them (e.g., population). Thus the cartogram computation problem can be considered as a map deformation problem where ..."
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Cartograms are used to visualize geographically distributed data by scaling the regions of a map (e.g., US states) such that their areas are proportional to some data associated with them (e.g., population). Thus the cartogram computation problem can be considered as a map deformation problem where the input is a planar polygonal map M and an assignment of some positive weight for each region. The goal is to create a deformed map M′, where the area of each region realizes the weight assigned to it (no cartographic error) while the overall map remains readable and recognizable (e.g., the topology, relative positions and shapes of the regions remain as close to those before the deformation as possible). Although several such measures of cartogram quality are wellknown, different cartogram generation methods optimize different features and there is no standard set of quantitative metrics. In this paper we define such a set of seven quantitative measures, designed to evaluate how faithfully a cartogram represents the desired weights and to estimate the readability of the final representation. We then study several cartogramgeneration algorithms and compare them in terms of these quantitative measures. 1.
Drawing Social Networks Using AreaLabeling Rectangular
"... People are linked together by social networks, whose complexity depends upon a lot of factors. To our understanding, most visualization interfaces for social networks have not been designed for reflecting their factors so far. As a result, this paper tries to solve such a problem by rectangular cart ..."
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People are linked together by social networks, whose complexity depends upon a lot of factors. To our understanding, most visualization interfaces for social networks have not been designed for reflecting their factors so far. As a result, this paper tries to solve such a problem by rectangular cartogram, which is a kind of geographical visualization interface using rectangles to represent regions in a map. Besides the relative position of each rectangle can reflect actual geographical related positions, one of the main features of rectangular cartograms is to use the area size or the shape of each rectangle to reflect the information of its corresponding region, e.g., the population in that region. This paper proposes a layout approach for rectangular cartograms with area labeling for social networks, in which each region has a minimumwidth constraint for accommodating a text label. To satisfy the practical use for social networks, we apply a genetic algorithm to finding the minimumwidth arealabeling rectangular cartogram under some constraints. By doing so, we can visualize the labeling text on each rectangle and observe the information represented by its area size or shape at the same time. Furthermore, the proposed approach is applied to visualizing the distribution of the Facebook popularity of an enterprise in Taiwan. From the cartogram, the text label on each region can be read directly and the relation among regions as well as their popularity can be visualized at the same time, so that the enterprise can improve the regions with poor popularity by the help from the regions with high popularity.
Optimal Binary Space Partitions by
, 2006
"... A Binary Space Partition (BSP) is a scheme for recursively dividing a configuration of objects by hyperplanes until all objects are separated. Objects can be cut into fragments by this process. We present some results on optimal size Binary Space Partitions in two dimensions. In this thesis we show ..."
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A Binary Space Partition (BSP) is a scheme for recursively dividing a configuration of objects by hyperplanes until all objects are separated. Objects can be cut into fragments by this process. We present some results on optimal size Binary Space Partitions in two dimensions. In this thesis we show that we cannot always get an optimal BSP for a set of disjoint line segments if we only use fixed partition lines, which are partition lines that go through at least two endpoints of fragments. We also proof that the best BSP that only uses fixed partition lines cuts at most 3 times as much line segments as the optimal BSP. We provide an algorithm that computes an optimal BSP for a rectangular subdivision in O(n 5) time – using dynamic programming – and discuss some heuristics to improve this. We generalize the algorithm so it works for a larger class of optimality criteria, including size and depth. We give experimental results based on randomly generated rectangular
How to Visualize the Kroot Name Server
, 2012
"... We present a novel paradigm to visualize the evolution of the service provided by one of the most popular root name servers, called Kroot, operated by the RIPE Network Coordination Centre (RIPE NCC) and distributed in several locations (instances) worldwide. Our approach can be usedtoeither monitor ..."
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We present a novel paradigm to visualize the evolution of the service provided by one of the most popular root name servers, called Kroot, operated by the RIPE Network Coordination Centre (RIPE NCC) and distributed in several locations (instances) worldwide. Our approach can be usedtoeither monitor what happenedduringaprescribed time interval or observe the status of the service in near realtime. We visualize how and when the clients of Kroot migrate from one instance to another, how the workload associated with each instance changes over time, and what are the instances that contribute to offer the service to a selected Internet Service Provider. In addition, the visualization aims at distinguishing usual from unusual operational patterns. This helps not only to improve the quality of the service but also to spot securityrelated issues and to investigate unexpected routing changes. Submitted:
DOI: 10.7155/jgaa.00267 How to Visualize the Kroot Name Server
, 2012
"... We present a novel paradigm to visualize the evolution of the service provided by one of the most popular root name servers, called Kroot, operated by the RIPE Network Coordination Centre (RIPE NCC) and distributed in several locations (instances) worldwide. Our approach can be usedtoeither monitor ..."
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We present a novel paradigm to visualize the evolution of the service provided by one of the most popular root name servers, called Kroot, operated by the RIPE Network Coordination Centre (RIPE NCC) and distributed in several locations (instances) worldwide. Our approach can be usedtoeither monitor what happenedduringaprescribed time interval or observe the status of the service in near realtime. We visualize how and when the clients of Kroot migrate from one instance to another, how the workload associated with each instance changes over time, and what are the instances that contribute to offer the service to a selected Internet Service Provider. In addition, the visualization aims at distinguishing usual from unusual operational patterns. This helps not only to improve the quality of the service but also to spot securityrelated issues and to investigate unexpected routing changes. Submitted: