Improved Approximation Algorithms for Box Contact Representations ⋆

Abstract. We study the following geometric representation problem: Given a graph whose vertices correspond to axis-aligned 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 NP-hard, and there are approximation algorithms for certain graph classes for the optimization version, MAX-CROWN, 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 APX-hard on bipartite graphs of bounded maximum degree.

Piecewise Laplacian-based Projection for Interactive Data Exploration and Organization

Multidimensional projection has emerged as an important visualization tool in applications involving the visual analysis of high-dimensional data. However, high precision projection methods are either computationally expensive or not flexible enough to enable feedback from user interaction into the projection process. A built-in mechanism that dynamically adapts the projection based on direct user intervention would make the technique more useful for a larger range of applications and data sets. In this paper we propose the Piecewise Laplacian-based Projection (PLP), a novel multidimensional projection technique, that, due to the local nature of its formulation, enables a versatile mechanism to interact with projected data and to allow interactive changes to alter the projection map dynamically, a capability unique of this technique. We exploit the flexibility provided by PLP in two interactive projection-based applications, one designed to organize pictures visually and another to build music playlists. These applications illustrate the usefulness of PLP in handling high-dimensional data in a flexible and highly visual way. We also compare PLP with the currently most promising projections in terms of precision and speed, showing that it performs very well also according to these quality criteria.Radiolog

InVis: A Tool for Interactive Visual Data Analysis

Abstract. We present InVis 1, a tool to visually analyse data by interactively shaping a two dimensional embedding of it. Traditionally, embedding techniques focus on finding one fixed embedding, which emphasizes a single aspects of the data. In contrast, our application enables the user to explore the structures of a dataset by observing and controlling a projection of it. Ultimately it provides a way to search and find an embedding, emphasizing aspects that the user desires to highlight

Structural Decomposition Trees: Semantic and Practical Implications

Abstract. The visualization of high-dimensional data is a challenging research topic. Existing approaches can usually be assigned to either relation or value visualizations. Merging approaches from both classes into a single integrated strategy, Structural Decomposition Trees (SDTs) represent a completely novel visualization approach for high-dimensional data. Although this method is new and promising, statements on how to use and apply the technique in the context of real-world applications are still missing. This paper discusses how SDTs can be interpreted and interacted with to gain insights about the data more effectively. First, it is discussed what properties about the data can be obtained by an interpretation of the initial projection. These statements are also valid for other projections based on principal components analysis, addressing a frequent problem when applying this technique. Further, a detailed and task-oriented interaction guideline shows how provided interaction methods can be utilized effectively for data exploration. The results obtained by an application of these guidelines in air quality research indicate that much insight can be gained even for large and complex data sets. This justifies and further motivates the usefulness and wide applicability of SDTs as a novel visualization approach for high-dimensional data

Temporal Multivariate Networks

Networks that evolve over time, or dynamic graphs, have been of interest to the areas of information visualization and graph drawing for many years. Typically, its the structure of the dynamic graph that evolves as vertices and edges are added or removed from the graph. In a multivariate scenario, however, attributes play an important role and can also evolve over time. In this chapter, we characterize and survey methods for visualizing temporal multivariate networks. We also explore future applications and directions for this emerging area in the fields of information visualization and graph drawing