Given a set S of n points in R^d, we show, for fixed d, how to construct in O(n log n) time a data structure we call the Balanced Aspect Ratio (BAR) tree. A BAR tree is a binary space partition tree on S that has O(logn) depth and in which every region is convex and “fat ” (that is, has a bounded aspect ratio). While previous hierarchical data structures, such as k-d trees, quadtrees, octrees, fair-split trees, and balanced box decompositions can guarantee some of these properties, we know of no previous data structure that combines alI of these properties simultaneously. The BAR tree data structure has numerous applications ranging from solving several geometric searching problems in fixed dimensional space to aiding in the visualization of graphs and three-dimensional worlds.

Abstract. We present a novel hierarchical force-directed method for drawing large graphs. The algorithm produces a graph embedding in an Euclidean space E of any dimension. A two or three dimensional drawing of the graph is then obtained by projecting a higher-dimensional embedding into a two or three dimensional subspace of E. Projecting high-dimensional drawings onto two or three dimensions often results in drawings that are “smoother ” and more symmetric. Among the other notable features of our approach are the utilization of a maximal independent set filtration of the set of vertices of a graph, a fast energy function minimization strategy, efficient memory management, and an intelligent initial placement of vertices. Our implementation of the algorithm can draw graphs with tens of thousands of vertices using a negligible amount of memory in less than one minute on a mid-range PC. 1

We describe MGV, an integrated visualization and exploration system for massive multi-digraph navigation.

We present a novel hierarchical force-directed method for drawing large graphs. The algorithm produces a graph embedding in an Euclidean space E of any dimension. A two or three dimensional drawing of the graph is then obtained by projecting a higher-dimensional embedding into a two or three dimensional subspace of E. Projecting high-dimensional drawings onto two or three dimensions often results in drawings that are "smoother" and more symmetric. Among the other notable features of our approach are the utilization of a maximal independent set filtration of the set of vertices of a graph, a fast energy function minimization strategy, e#cient memory management, and an intelligent initial placement of vertices. Our implementation of the algorithm can draw graphs with tens of thousands of vertices using a negligible amount of memory in less than one minute on a mid-range PC. 1 Introduction Graphs are common in many applications, from data structures to networks, from software engineering...

We introduce the tree cross-product problem, which abstracts a data structure common to applications in graph visualization, string matching, and software analysis. We design solutions with a variety of tradeoffs, yielding improvements and new results for these applications.

. A broad spectrum of massive data sets can be modeled as dynamic weighted multi-digraphs with sizes ranging from tens of gigabytes to petabytes. The sheer size of these data repositories brings with it interesting visualization and computational challenges. We introduce the notion of graph surfaces as a metaphor that allows the integration of visualization and computation over these data sets. By using out-ofcore algorithms we build a hierarchy of graph surfaces that represents a virtual geography for the data set. In order to provide the user with navigation control and interactive response, we incorporate a number of geometric techniques from 3D computer graphics like terrain triangulation and mesh simplification. We highlight the main algorithmic ideas behind the tools and formulate some novel mathematical problems that have surfaced along the way. 1 Introduction Massive data sets bring with them a series of special computational challenges. Many of these data sets ca...

Abstract. Compound-fisheye views are introduced as a method for the display and interaction with large graphs. The method relies on a hierarchical clustering of the graph, and a generalization of the traditional fisheye view, together with a treemap representation of the cluster tree. 1

Abstract. In this paper we present a novel approach for cluster-based drawing of large planar graphs that maintains planarity. Our technique works for arbitrary planar graphs and produces a clustering which satisfies the conditions for compound-planarity (c-planarity). Using the clustering, we obtain a representation of the graph as a collection of O(log n) layers, where each succeeding layer represents the graph in an increasing level of detail. At the same time, the difference between two graphs on neighboring layers of the hierarchy is small, thus preserving the viewer's mental map. The overall running time of the algorithm is O(n log n), where n is the number of vertices of graph G.

We describe MGV, an integrated visualization and exploration system for massive multi-digraph navigation. MGV’s only assumption is that the vertex set of the underlying digraph corresponds to the set of leaves of a predetermined tree T. MGV builds an out-of-core graph hierarchy and provides mechanisms to plug in arbitrary visual representations for each graph hierarchy slice. Navigation from one level to another of the hierarchy corresponds to the implementation of a drill-down interface. In order to provide the user with navigation control and interactive response, MGV incorporates a number of visualization techniques like interactive pixel-oriented 2D and 3D maps, statistical displays, multi-linked views, and a zoomable label based interface. This makes the association of geographic information and graph data very natural. MGV follows the clientserver paradigm and it is implemented in C and Java-3D. We highlight the main algorithmic and visualization techniques behind the tools and point out along the way several possible application scenarios. Our techniques are being applied to multi-graphs defined on vertex sets with sizes ranging from 100 million to 250 million vertices.

Abstract. We show that a popular variant of the well known k-d tree data structure satisfies an important packing lemma. This variant is a binary spatial partitioning tree T defined on a set of n points in IR d, for fixed d ≥ 1, using the simple rule of splitting each node’s hyperrectangular region with a hyperplane that cuts the longest side. An interesting consequence of the packing lemma is that standard algorithms for performing approximate nearest-neighbor searching or range searchingqueriesvisitatmostO(log d−1 n)nodesofsuchatreeT in the worst case. Traditionally, many variants of k-d trees have been empirically shown to exhibit polylogarithmic performance, and under certain restrictions in the data distribution some theoretical expected case results have been proven. This result, however, is the first one proving a worst-case polylogarithmic time bound for approximate geometric queries using the simple k-d tree data structure. 1