We consider the minimum Manhattan network problem, which is defined as follows. Given a set of points called terminals in Rd, find a minimum-length network such that each pair of terminals is connected by a set of axis-parallel line segments whose total length is equal to the pair's Manhattan (that is, L1-) distance. The problem is NP-hard in 2D and there is no PTAS for 3D (unless P = NP). Approximation algorithms are known for 2D, but not for 3D. We present, for any fixed dimension d and any ε 0, an O(nε)- approximation. For 3D, we also give a 4(k - 1)-approximation for the case that the terminals are contained in the union of k ≥ 2 parallel planes.

Many Internet mapping projects have used traceroute as a measurement primitive and have generated hundreds of millions of traceroute samples used, among many purposes, for network debugging, troubleshooting and Internet mapping. In this work, we advocate the use of visualization as a means to extract semantics from large sets of traceroute data collected from large-scale traceroute campaigns. Using subsets of these datasets consisting of all traceroutes traversing a particular Internet eXchange Point (IXP), we illustrate how visualization allows us to (1) examine existing large datasets in a different way and (2) present different views of the data that can provide new insights and lead to new questions.

Visualizing graphs using virtual physical models is probably the most heavily used technique for drawing graphs in practice. There are many algorithms that are efficient and produce high-quality layouts. If one requires that the layout also respect a given set of non-uniform edge lengths, however, force-based approaches become problematic while energy-based layouts become intractable. In this paper, we propose a reformulation of the stress function into a two-part convex objective function to which we can apply semi-definite programming SDP. We avoid the high computational cost associated with SDP by a novel, compact re-parameterization of the objective function using the eigenvectors of the graph Laplacian. This sparse representation makes our approach scalable. We provide experimental results to show that this method scales well and produces reasonable layouts while dealing with the edge length constraints.

The many endless rivers of text now available present a serious challenge in the task of gleaning, analyzing and discovering useful information. In this paper, we describe a methodology for visualizing text streams in real time. The approach automatically groups similar messages into "countries," with keyword summaries, using semantic analysis, graph clustering and map generation techniques. It handles the need for visual stability across time by dynamic graph layout and Procrustes projection techniques, enhanced with a novel stable component packing algorithm. The result provides a continuous, succinct view of evolving topics of interest. It can be used in passive mode for overviews and situational awareness, or as an interactive data exploration tool. To make these ideas concrete, we describe their application to an online service called TwitterScope.

We introduce and study a generalization of the well-known region of influence proximity drawings, called (ε1,ε2)-proximity drawings. Intuitively, given a definition of proximity and two real numbers ε1⩾0 and ε2⩾0, an (ε1,ε2)-proximity drawing of a graph is a planar straight-line drawing Γ such that: (i) for every pair of adjacent vertices u, v, their proximity region “shrunk” by the multiplicative factor 11+ε1 does not contain any vertices of Γ; (ii) for every pair of non-adjacent vertices u, v, their proximity region “expanded” by the factor (1+ε2) contains some vertices of Γ other than u and v. In particular, the locations of the vertices in such a drawing do not always completely determine which edges must be present/absent, giving us some freedom of choice. We show that this generalization significantly enlarges the family of representable planar graphs for relevant definitions of proximity drawings, including Gabriel drawings, Delaunay drawings, and β-drawings, even for arbitrarily small values of ε1 and ε2. We also study the extremal case of (0,ε2)-proximity drawings, which generalize the well-known weak proximity drawing paradigm.

In this paper, we consider the problem of representing graphs by triangles whose sides touch. We present linear time algorithms for creating touching triangles representations for outerplanar graphs, square grid graphs, and hexagonal grid graphs. The class of graphs with touching triangles representations is not closed under minors, making characterization difficult. We do show that pairs of vertices can only have a small common neighborhood, and we present a complete characterization of the subclass of biconnected graphs that can be represented as triangulations of some polygon.

In this paper, we consider the problem of representing graphs by polygons whose sides touch. We show that at least six sides per polygon are necessary by constructing a class of planar graphs that cannot be represented by pentagons. We also show that the lower bound of six sides is matched by an upper bound of six sides with a linear time algorithm for representing any planar graph by touching hexagons. Moreover, our algorithm produces convex polygons with edges with slopes 0, 1, -1.

Information visualization is essential in making sense out of large data sets. Often, high-dimensional data are visualized as a collection of points in 2-dimensional space through dimensionality reduction techniques. However, these traditional methods often do not capture well the underlying structural information, clustering, and neighborhoods. In this paper, we describe GMap, a practical algorithm for visualizing relational data with geographic-like maps. We illustrate the effectiveness of this approach with examples from several domains.

When drawing graphs whose nodes contain text or graphics. the non-trivial node sizes must be taken into account. either as part of the initial layout or as a post-processing step. The core problem is to avoid overlaps while retaining the structural information inherent in a layout using little additional area. This paper presents a new node overlap removal algorithm that does well by these measures.

For users, recommendations can sometimes seem odd or counterintuitive. Visualizing recommendations can remove some of this mystery, showing how a recommendation is grouped with other choices. A drawing can also lead a user's eye to other options. Traditional 2D-embeddings of points can be used to create a basic layout, but these methods, by themselves, do not illustrate clusters and neighborhoods very well. In this paper, we propose the use of geographic maps to enhance the definition of clusters and neighborhoods, and consider the effectiveness of this approach in visualizing similarities and recommendations arising from TV shows.

Circular graph layout is a drawing scheme where all nodes are placed on the perimeter of a circle. An inherent issue with circular layouts is that the rigid restriction on node placement often gives rise to long edges and an overall dense drawing. We suggest here three independent, complementary techniques for lowering the density and improving the readability of circular layouts. First, a new algorithm is given for placing the nodes on the circle such that edge lengths are reduced. Second, we enhance the circular drawing style by allowing some of the edges to be routed around the exterior of the circle. This is accomplished with an algorithm for optimally selecting such a set of externally routed edges. The third technique reduces density by coupling groups of edges as bundled splines that share part of their route. Together, these techniques are able to reduce clutter, density and crossings compared with existing methods.

Both the implementation and the user’s view of eXene rely heavily on the concurrency model provided by CML . CML is based on the sequential language SML[MTH90, MT91] and inherits the following good features of SML: functions as first-class values, strong static typing, polymorphism, datatypes and pattern matching, lexical scoping, exception handling and a state-of-the-art module facility. The sequential performance of CML benefits from the quality of the SML/NJ compiler. In addition CML has the following properties:

Graph drawing is a basic visualization tool. For graphs of up to hundreds of nodes and edges, there are many effective techniques available. At greater scale, data density and occlusion problems often negate its effectiveness. Conventional pan-and-zoom, and multiscale and geometric fisheye views are not fully satisfactory solutions to this problem. As an alternative, we describe a topological zooming method. It is based on the precomputation of a hierarchy of coarsened graphs, which are combined on-the-fly into renderings with the level of detail dependent on the distance from one or more foci. We also discuss a related distortion method that allows our technique to achieve constant information density displays.

One of the most popular graph drawing methods is based on achieving graph-theoretic target distances. This method was used by Kamada and Kawai [15], who formulated it as an energy optimization problem. Their energy is known in the multidimensional scaling (MDS) community as the stress function. In this work, we show how to draw graphs by stress majorization, adapting a technique known in the MDS community for more than two decades. It appears that majorization has advantages over the technique of Kamada and Kawai in running time and stability. We also found the majorization-based optimization being essential to a few extensions to the basic energy model. These extensions can improve layout quality and computation speed in practice.

Although routing is a well-studied problem in various contexts, there remain unsolved problems in routing edges for graph layouts. In contrast with techniques from other domains such as VLSI CAD and robotics, where physical constraints play a major role, aesthetics play the more important role in graph layout. For graphs, we seek paths that are easy to follow and add meaning to the layout. We describe a collection of aesthetic attributes applicable to drawing edges in graphs, and present a general approach for routing individual edges subject to these principles. We also give implementation details and survey difficulties that arise in an implementation.

Graphviz is a collection of software for viewing and manipulating abstract graphs. It provides graph visualization for tools and web sites in domains such as software engineering, networking, databases, knowledge representation, and bioinformatics. Hundreds of thousands of copies have been distributed under an open source license.

An active research area in telecommunications concerns how to specify and control the addition of new services, such as call waiting or instant messaging, into existing software. One approach is to rely on a component-based architecture such as Distributed Feature Composition (DFC), by which a new service can be specified as a composition of primitive features over time. Formally, a communication episode is represented by a dynamic graph of software feature boxes, called a usage. This serves as the fundamental model for how services are invoked and how they interact with other services.This paper, after providing some background on DFC, discusses a technique for visualizing the usages which arise through DFC specifications. With the visualization, users can monitor and validate service protocols and feature interactions in real time or through playback logs. The principal display component uses a novel variation of force-directed layouts for undirected graphs. The resulting graphical interface has become a principal tool for developers building services using DFC.

Graphviz is a heterogeneous collection of graph drawing tools containing batch layout programs (dot, neato, fdp, twopi); a platform for incremental layout (Dynagraph); customizable graph editors (dotty, Grappa); a server for including graphs in Web pages (WebDot); support for graphs as COM objects (Montage); utility programs useful in graph visualization; and libraries for attributed graphs. The software is available under an Open Source license. The article[1] provides a detailed description of the package.