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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 (1 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
Proportional Contact Representations of Planar Graphs
"... Abstract. We study contact representations for planar graphs, with vertices represented by simple polygons and adjacencies represented by a pointcontact or a sidecontact between the corresponding polygons. Specifically, we consider proportional contact representations, where prespecified vertex w ..."
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Cited by 6 (5 self)
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Abstract. We study contact representations for planar graphs, with vertices represented by simple polygons and adjacencies represented by a pointcontact or a sidecontact between the corresponding polygons. Specifically, we consider proportional contact representations, where prespecified vertex weights must be represented by the areas of the corresponding polygons. Several natural optimization goals for such representations include minimizing the complexity of the polygons, the cartographic error, and the unused area. We describe constructive algorithms for proportional contact representations with optimal complexity for general planar graphs and planar 2segment graphs, which include maximal outerplanar graphs and partial 2trees. 1
Complexity of octagonal and rectangular cartograms
 In Proceedings of the 17th Canadian Conference on Computational Geometry
, 2005
"... In this paper, we study the complexity of rectangular cartograms, i.e., maps where every region is a rectangle, and which should be deformed such that given area requirements are satisfied. We study the closely related problem of cartograms with orthogonal octagons, and show that this problem is NP ..."
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Cited by 5 (0 self)
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In this paper, we study the complexity of rectangular cartograms, i.e., maps where every region is a rectangle, and which should be deformed such that given area requirements are satisfied. We study the closely related problem of cartograms with orthogonal octagons, and show that this problem is NPhard. From our proof, it also follows that rectangular cartograms are NPhard if we allow for the existence of a “sea”, i.e., a region of arbitrarily high complexity on the outside of the drawing. 1
Lineartime algorithms for proportional contact graph representations
, 2011
"... Abstract. In a proportional contact representation of a planar graph, each vertex is represented by a simple polygon with area proportional to a given weight, and edges are represented by adjacencies between the corresponding pairs of polygons. In this paper we study proportional contact representat ..."
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Cited by 3 (2 self)
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Abstract. In a proportional contact representation of a planar graph, each vertex is represented by a simple polygon with area proportional to a given weight, and edges are represented by adjacencies between the corresponding pairs of polygons. In this paper we study proportional contact representations that use rectilinear polygons without wasted areas (white space). In this setting, the best known algorithm for proportional contact representation of a maximal planar graph uses 12sided rectilinear polygons and takes�time. We describe a new algorithm that guarantees 10sided rectilinear polygons and runs in time. We also describe a lineartime algorithm for proportional contact representation of planar 3trees with 8sided rectilinear polygons and show that this optimal, as there exist planar 3trees that requires 8sided polygons. Finally, we show that a maximal outerplanar graph admits a proportional contact representation with 6sided rectilinear polygons when the outerboundary is a rectangle and with 4 sides otherwise. 1
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:
LinearTime Algorithms for Rectilinear Holefree Proportional Contact Representations
"... Abstract. A proportional contact representation of a planar graph is one where each vertex is represented by a simple polygon with area proportional to a given weight and adjacencies between polygons represent edges between the corresponding pairs of vertices. In this paper we study proportional con ..."
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Abstract. A proportional contact representation of a planar graph is one where each vertex is represented by a simple polygon with area proportional to a given weight and adjacencies between polygons represent edges between the corresponding pairs of vertices. In this paper we study proportional contact representations that use only rectilinear polygons and contain no unused area or hole. There is an algorithm that gives a holefree proportional contact representation of a maximal planar graph with 12sided rectilinear polygons in O(n log n) time. We improve this result by giving a lineartime algorithm that produces a holefree proportional contact representation of a maximal planar graph with a 10sided rectilinear polygons. For a planar 3tree we give a lineartime algorithm for a holefree proportional contact representation with 8sided rectilinear polygons. Furthermore, there exist a planar 3tree that requires 8sided polygons in any holefree contact representation with rectilinear polygons. A maximal outerplanar graph admits a holefree proportional contact representation with rectangles. 1
1 Introduction Complexity of Octagonal and Rectangular Cartograms
"... A cartogram is a type of map used to visualize data. In a map regions are displayed in their true shapes and with their exact relations with the adjacent regions. However, ..."
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A cartogram is a type of map used to visualize data. In a map regions are displayed in their true shapes and with their exact relations with the adjacent regions. However,
On rectangular cartograms
, 2005
"... www.elsevier.com/locate/comgeo A rectangular cartogram is a type of map where every region is a rectangle. The size of the rectangles is chosen such that their areas represent a geographic variable (e.g., population). Good rectangular cartograms are hard to generate: The area specifications for each ..."
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www.elsevier.com/locate/comgeo A rectangular cartogram is a type of map where every region is a rectangle. The size of the rectangles is chosen such that their areas represent a geographic variable (e.g., population). Good rectangular cartograms are hard to generate: The area specifications for each rectangle may make it impossible to realize correct adjacencies between the regions and so hamper the intuitive understanding of the map. We present the first algorithms for rectangular cartogram construction. Our algorithms depend on a precise formalization of region adjacencies and build upon existing VLSI layout algorithms. Furthermore, we characterize a nontrivial class of rectangular subdivisions for which exact cartograms can be computed efficiently. An implementation of our algorithms and various tests show that in practice, visually pleasing rectangular cartograms with small cartographic error can be generated effectively.