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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.
Semantic word cloud representations: Hardness and approximation algorithms
 In Proc. 11th Latin American Theoret. Inform. Symp. (LATIN’14), volume 8392 of LNCS
, 2014
"... Abstract. We study a geometric representation problem, where we are given a set B of axisaligned rectangles (boxes) with fixed dimensions and a graph with vertex set B. The task is to place the rectangles without overlap such that two rectangles touch if the graph contains an edge between them. We ..."
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Abstract. We study a geometric representation problem, where we are given a set B of axisaligned rectangles (boxes) with fixed dimensions and a graph with vertex set B. The task is to place the rectangles without overlap such that two rectangles touch if the graph contains an edge between them. We call this problem CONTACT REPRESENTATION OF WORD NETWORKS (CROWN). It formalizes the geometric problem behind drawing word clouds in which semantically related words are close to each other. Here, we represent words by rectangles and semantic relationships by edges. We show that CROWN is strongly NPhard even if restricted to trees and weakly NPhard if restricted to stars. We also consider the optimization problem MAXCROWN where each adjacency induces a certain profit and the task is to maximize the sum of the profits. For this problem, we present constantfactor approximations for several graph classes, namely stars, trees, planar graphs, and graphs of bounded degree. Finally, we evaluate the algorithms experimentally and show that our best method improves upon the best existing heuristic by 45%.
On Semantic Word Cloud Representation
"... Abstract. We study the problem of computing semanticpreserving word clouds in which semantically related words are close to each other. While several heuristic approaches have been described in the literature, we formalize the underlying geometric algorithm problem: Word Rectangle Adjacency Contac ..."
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Abstract. We study the problem of computing semanticpreserving word clouds in which semantically related words are close to each other. While several heuristic approaches have been described in the literature, we formalize the underlying geometric algorithm problem: Word Rectangle Adjacency Contact (WRAC). In this model each word is a rectangle with fixed dimensions, and the goal is to represent semantically related word pairs by contacts between their corresponding rectangles. We design and analyze efficient polynomialtime algorithms for variants of the WRAC problem, show that some general variants are NPhard, and describe several approximation algorithms. Finally, we experimentally demonstrate that our theoreticallysound algorithms outperform the early heuristics. 1
3D Proportional Contact Representations of Graphs
"... In 3D contact representations, the vertices of a graph are represented by 3D polyhedra and the edges are realized by nonzeroarea common boundaries between corresponding polyhedra. While contact representations with cuboids have been studied in the literature, we consider a novel generalization of ..."
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In 3D contact representations, the vertices of a graph are represented by 3D polyhedra and the edges are realized by nonzeroarea common boundaries between corresponding polyhedra. While contact representations with cuboids have been studied in the literature, we consider a novel generalization of the problem in which vertices are represented by axisaligned polyhedra that are union of two cuboids. In particular, we study the weighted (proportional) version of the problem, where the volumes of the polyhedra and the areas of the common boundaries realize prespecified vertex and edge weights. For some classes of graphs (e.g., outerplanar, planar bipartite, planar, complete), we provide algorithms to construct such representations for arbitrary given weights. We also show that not all graphs can be represented in 3D with axisaligned polyhedra of constant complexity.
Improved Approximation Algorithms for Box Contact Representations ⋆
"... Abstract. We study the following geometric representation problem: Given a graph whose vertices correspond to axisaligned rectangles with fixed dimensions, arrange the rectangles without overlaps in the plane such that two rectangles touch if the graph contains an edge between them. This problem is ..."
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Abstract. We study the following geometric representation problem: Given a graph whose vertices correspond to axisaligned rectangles with fixed dimensions, arrange the rectangles without overlaps in the plane such that two rectangles touch if the graph contains an edge between them. This problem is called CONTACT REPRESENTATION OF WORD NETWORKS (CROWN) since it formalizes the geometric problem behind drawing word clouds in which semantically related words are close to each other. CROWN is known to be NPhard, and there are approximation algorithms for certain graph classes for the optimization version, MAXCROWN, in which realizing each desired adjacency yields a certain profit. We present the first O(1)approximation algorithm for the general case, when the input is a complete weighted graph, and for the bipartite case. Since the subgraph of realized adjacencies is necessarily planar, we also consider several planar graph classes (namely stars, trees, outerplanar, and planar graphs), improving upon the known results. For some graph classes, we also describe improvements in the unweighted case, where each adjacency yields the same profit. Finally, we show that the problem is APXhard on bipartite graphs of bounded maximum degree. 1