Given a plane graph G, we wish to draw it in the plane in such a way that the vertices of G are represented as grid points, and the edges are represented as straight-line segments between their endpoints. An additional objective is to minimize the size of the resulting grid. It is known that each plane graph can be drawn in such a way in a (n \Gamma 2) \Theta (n \Gamma 2) grid (for n 3), and that no grid smaller than (2n=3 \Gamma 1) \Theta (2n=3 \Gamma 1) can be used for this purpose, if n is a multiple of 3. In fact, for all n 3, each dimension of the resulting grid needs to be at least b2(n \Gamma 1)=3c, even if the other one is allowed to be unbounded. In this paper we show that this bound is tight by presenting a grid drawing algorithm that produces drawings of width b2(n \Gamma 1)=3c. The height of the produced drawings is bounded by 4b2(n \Gamma 1)=3c \Gamma 1. Our algorithm runs in linear time and is easy to implement. 1 Introduction The problem of automatic graph drawing ha...

Rooted trees are usually drawn planar and upward, i.e., without crossings and without any parent placed below its child. In this paper we investigate the area requirement of planar upward drawings of rooted trees. We give tight upper and lower bounds on the area of various types of drawings, and provide linear-time algorithms for constructing optimal area drawings. Let T be a bounded-degree rooted tree with N nodes. Our results are summarized as follows: ffl We show that T admits a planar polyline upward grid drawing with area O(N ), and with width O(N ff ) for any prespecified constant ff such that 0 ! ff ! 1. ffl If T is a binary tree, we show that T admits a planar orthogonal upward grid drawing with area O(N log log N ). ffl We show that if T is ordered, it admits an O(N log N)-area planar upward grid drawing that preserves the left-to-right ordering of the children of each node. ffl We show that all of the above area bounds are asymptotically optimal in the worst case. ffl ...

This paper examines an infinite family of proximity drawings of graphs called open and closed fi-drawings, first defined by Kirkpatrick and Radke [15, 21] in the context of computational morphology. Such proximity drawings include as special cases the well-known Gabriel, relative neighborhood and strip drawings. Complete characterizations of those trees that admit open fi-drawings for 0 fi ! fi ! 1 or closed fi-drawings for 0 fi ! fi 1 are given, as well as partial characterizations for other values of fi. For the intervals of fi in which complete characterizations are given, it can be determined in linear time whether a tree admits an open or closed fi-drawing, and, if so, such a drawing can be computed in linear time in the real RAM model. Finally, a complete characterization of all graphs which admit closed strip drawings is given.

Dynamic Fisheye Views: Combining Dynamic Queries and Mapping with Database View Definition Doctor of Philosophy, 1996 Department of Computer Science University of Toronto Information (or data) visualization refers to the graphical presentation of information with the aim of providing the viewer with a qualitative understanding of the information contents. Two common graphical methods for depicting multidimensional data are scatterplots (that show relationships among data dimensions) and node-andlink diagrams or graphs (that show relationships among individual data points). One strategy to aid understanding of multidimensional data is to map data dimensions or attributes (e.g., temperature, population, location) to graphic properties (e.g., colour, size, position). Dynamic mapping (DM) is an interactive technique that permits mappings to be adjusted dynamically. The dynamic query (DQ) is a related interactive technique that applies the principles of direct manipulation to the sea...

. In this paper we present a method to obtain optimal h-v and inclusion drawings in parallel. Based on parallel tree contraction, our method computes optimal (with respect to a class of cost functions of the enclosing rectangle) drawings in O(log 2 n) parallel time by using a polynomial number of EREW processors. The method can be extended to compute optimal inclusion layouts in the case where each leaf l of the tree is represented by rectangle l x \Theta l y . Our method also yields an NC algorithm for the slicing floorplanning problem. Whether this problem was in NC was an open question [2]. 1 Introduction In this paper we examine drawings of rooted binary trees. We study the h-v drawing convention studied by Crescenzi, Di Battista and Piperno [3] and Eades, Lin and Lin [7]. Our results extend to the inclusion convention [6], and to slicing floorplanning [10, 2]. The drawing of a rooted binary tree using the h-v drawing convention is a planar grid drawing in which tree nodes are ...

Abstract: The proper visualization and monitoring of their (ongoing) business processes is crucial for any enterprise. Thus a broad spectrum of processes has to be visualized ranging from simple, short–running processes to complex long–running ones (consisting of up to hundreds of activities). In any case, users shall be able to quickly understand the logic behind a process and to get a quick overview of related tasks. One practical problem arises when different fragments of a business process are scattered over several systems where they are often modeled using different process meta models (e.g., High–Level Petri Nets). The challenge is to find an integrated and user–friendly visualization for these business processes. In this paper we discover use cases relevant in this context. Since existing graph layout approaches have focused on general graph drawing so far we further develop a specific approach for layouting business process graphs. The work presented in this paper is embedded within a larger project (Proviado) on the visualization of automotive processes. 1

Graph Drawing (also known as Graph Visualization) tackles the problem of representing graphs on a visual medium such as computer screen, printer etc. Many applications such as software engineering, data base design, project planning, VLSI design, multimedia etc., have data structures that can be represented as graphs. With the ever increasing complexity of these and new applications, and availability of hardware supporting visualization, the area of graph drawing is increasingly getting more attention from both practitioners and researchers. In a typical drawing of a graph, the vertices are represented as symbols such as circles, dots or boxes, etc., and the edges are drawn as continuous curves joining their end points. Often, the edges are simply drawn as (straight- or poly-) lines joining their end points (and hence the title of this thesis), followed by an optional transformation into smooth curves. The goal of research in graph drawing is to develop techniques for constructing good...

Tree layout has received considerable attention because of its practical importance. Arguably the most common drawing convention is the (ordered) layered tree convention for rooted trees in which the layout is required to preserve the relative order of a node’s children. However, in some applications preserving the ordering of children is not important, and considerably more compact layout can be achieved if this requirement is dropped. Here we introduce the unordered layered tree drawing convention for binary rooted trees and show that determining a minimal width drawing for this convention is NP-complete.

A family of proximity drawings of graphs called open and closed &beta;-drawings, first defined in [16], and including the Gabriel, relative neighborhood and strip drawings, are investigated. Complete characterizations of which trees admit open &beta;-drawings for 0 &le &beta; &le; ... or closed &beta;-drawings for 0 &le; &beta; ... are given, as well as partial characterizations for other values of &beta;. For &beta; < &infin; in the intervals in which complete characterizations are given, it can be determined in linear time whether a tree admits an open or closed &beta;-drawing, and, if so, such a drawing can be computed in linear time in the real RAM model. Finally, a complete characterization of all graphs which admit closed strip drawings is given.

Presenting large amounts of information in a limited screen space is a significant challenge in the field of Information Visualization. With the rapid development and growing use of small handheld devices such as PDAs this issue has become more important. Many Focus+Context techniques have been developed to address it but very few of them would effectively aid visualization applications for small handheld devices.