Online Packing to Minimize Area or Perimeter

We consider online packing problems where we get a stream of axis-parallel rectangles. The rectangles have to be placed in the plane without overlapping, and each rectangle must be placed without knowing the subsequent rectangles. The goal is to minimize the perimeter or the area of the axis-parallel bounding box of the rectangles. We either allow rotations by 90^? or translations only. For the perimeter version we give algorithms with an absolute competitive ratio slightly less than 4 when only translations are allowed and when rotations are also allowed. We then turn our attention to minimizing the area and show that the competitive ratio of any algorithm is at least ?(?n), where n is the number of rectangles in the stream, and this holds with and without rotations. We then present algorithms that match this bound in both cases and the competitive ratio is thus optimal to within a constant factor. We also show that the competitive ratio cannot be bounded as a function of Opt. We then consider two special cases. The first is when all the given rectangles have aspect ratios bounded by some constant. The particular variant where all the rectangles are squares and we want to minimize the area of the bounding square has been studied before and an algorithm with a competitive ratio of 8 has been given [Fekete and Hoffmann, Algorithmica, 2017]. We improve the analysis of the algorithm and show that the ratio is at most 6, which is tight. The second special case is when all edges have length at least 1. Here, the ?(?n) lower bound still holds, and we turn our attention to lower bounds depending on Opt. We show that any algorithm for the translational case has a competitive ratio of at least ?(?{Opt}). If rotations are allowed, we show a lower bound of ?(?{Opt}). For both versions, we give algorithms that match the respective lower bounds: With translations only, this is just the algorithm from the general case with competitive ratio O(?n) = O(?{Opt}). If rotations are allowed, we give an algorithm with competitive ratio O(min{?n,?{Opt}}), thus matching both lower bounds simultaneously

The State-of-the-Art of Set Visualization

Sets comprise a generic data model that has been used in a variety of data analysis problems. Such problems involve analysing and visualizing set relations between multiple sets defined over the same collection of elements. However, visualizing sets is a non-trivial problem due to the large number of possible relations between them. We provide a systematic overview of state-of-the-art techniques for visualizing different kinds of set relations. We classify these techniques into six main categories according to the visual representations they use and the tasks they support. We compare the categories to provide guidance for choosing an appropriate technique for a given problem. Finally, we identify challenges in this area that need further research and propose possible directions to address these challenges. Further resources on set visualization are available at http://www.setviz.net

Geometric-based Optimization Algorithms for Cable Routing and Branching in Cluttered Environments

The need for designing lighter and more compact systems often leaves limited space for planning routes for the connectors that enable interactions among the system’s components. Finding optimal routes for these connectors in a densely populated environment left behind at the detail design stage has been a challenging problem for decades. A variety of deterministic as well as heuristic methods has been developed to address different instances of this problem. While the focus of the deterministic methods is primarily on the optimality of the final solution, the heuristics offer acceptable solutions, especially for such problems, in a reasonable amount of time without guaranteeing to find optimal solutions. This study is an attempt to furthering the efforts in deterministic optimization methods to tackle the routing problem in two and three dimensions by focusing on the optimality of final solutions. The objective of this research is twofold. First, a mathematical framework is proposed for the optimization of the layout of wiring connectors in planar cluttered environments. The problem looks at finding the optimal tree network that spans multiple components to be connected with the aim of minimizing the overall length of the connectors while maximizing their common length (for maintainability and traceability of connectors). The optimization problem is formulated as a bi-objective problem and two solution methods are proposed: (1) to solve for the optimal locations of a known number of breakouts (where the connectors branch out) using mixed-binary optimization and visibility notion and (2) to find the minimum length tree that spans multiple components of the system and generates the optimal layout using the previously-developed convex hull based routing. The computational performance of these methods in solving a variety of problems is further evaluated. Second, the problem of finding the shortest route connecting two given nodes in a 3D cluttered environment is considered and addressed through deterministically generating a graphical representation of the collision-free space and searching for the shortest path on the found graph. The method is tested on sample workspaces with scattered convex polyhedra and its computational performance is evaluated. The work demonstrates the NP-hardness aspect of the problem which becomes quickly intractable as added components or increase in facets are considered

The Impact of Visual Aesthetics on the Utility, Affordance, and Readability of Network Graphs

The readability of networks – how different visual design elements affect the understanding of network data – has been central in network visualization research. However, existing studies have mainly focused on readability induced by topological mapping (based on different layouts) and overlooked the effect of visual aesthetics. Proposed is a novel experimental framework to study how different network aesthetic choices affect users' abilities of understanding the network structures. The visual aesthetics are grouped in two forms: 1) visual encoding (where the aesthetic mapping depends on the underlying network data) and 2) visual styling (where the aesthetics are applied independent of underlying data). Users are given a simple task – identifying most connected nodes in a network – in a hybrid experimental setting where the visual aesthetic choices are tested in a within-subject manner while the network topologies are tested in a between-subject manner based on a randomized blocking design. This novel experimental design ensures an efficient decoupling of the influence of network topology on readability tests. The utility of different visual aesthetics is measured comprehensively based on task performance (accuracy and time), eye-tracking data, and user feedback (perceived affordance). The results show differential readability effects among choices of visual aesthetics. Particularly, node based visual encoding significantly enhances network readability; specifically, glyphs allow participants to create more robust strategies in their utilization. The study contributes to both the understanding of the role of visual aesthetics in network visualization design and the experimental design for testing the network readability

Knowledge-based energy functions for computational studies of proteins

This chapter discusses theoretical framework and methods for developing knowledge-based potential functions essential for protein structure prediction, protein-protein interaction, and protein sequence design. We discuss in some details about the Miyazawa-Jernigan contact statistical potential, distance-dependent statistical potentials, as well as geometric statistical potentials. We also describe a geometric model for developing both linear and non-linear potential functions by optimization. Applications of knowledge-based potential functions in protein-decoy discrimination, in protein-protein interactions, and in protein design are then described. Several issues of knowledge-based potential functions are finally discussed.Comment: 57 pages, 6 figures. To be published in a book by Springe

Measuring and improving the readability of network visualizations

Network data structures have been used extensively for modeling entities and their ties across such diverse disciplines as Computer Science, Sociology, Bioinformatics, Urban Planning, and Archeology. Analyzing networks involves understanding the complex relationships between entities as well as any attributes, statistics, or groupings associated with them. The widely used node-link visualization excels at showing the topology, attributes, and groupings simultaneously. However, many existing node-link visualizations are difficult to extract meaning from because of (1) the inherent complexity of the relationships, (2) the number of items designers try to render in limited screen space, and (3) for every network there are many potential unintelligible or even misleading visualizations. Automated layout algorithms have helped, but frequently generate ineffective visualizations even when used by expert analysts. Past work, including my own described herein, have shown there can be vast improvements in network visualizations, but no one can yet produce readable and meaningful visualizations for all networks. Since there is no single way to visualize all networks effectively, in this dissertation I investigate three complimentary strategies. First, I introduce a technique called motif simplification that leverages the repeating patterns or motifs in a network to reduce visual complexity. I replace common, high-payoff motifs with easily understandable glyphs that require less screen space, can reveal otherwise hidden relationships, and improve user performance on many network analysis tasks. Next, I present new Group-in-a-Box layouts that subdivide large, dense networks using attribute- or topology-based groupings. These layouts take group membership into account to more clearly show the ties within groups as well as the aggregate relationships between groups. Finally, I develop a set of readability metrics to measure visualization effectiveness and localize areas needing improvement. I detail optimization recommendations for specific user tasks, in addition to leveraging the readability metrics in a user-assisted layout optimization technique. This dissertation contributes an understanding of why some node-link visualizations are difficult to read, what measures of readability could help guide designers and users, and several promising strategies for improving readability which demonstrate that progress is possible. This work also opens several avenues of research, both technical and in user education

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

Large bichromatic point sets admit empty monochromatic 4-gons

We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).Postprint (published version