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Balanced Aspect Ratio Trees: Combining the Advantages of kd Trees and Octrees
"... 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 as ..."
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Cited by 55 (7 self)
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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 kd trees, quadtrees, octrees, fairsplit 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 threedimensional worlds.
A Multidimensional Approach to ForceDirected Layouts of Large Graphs
, 2000
"... Abstract. We present a novel hierarchical forcedirected 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 higherdimensional embedding into a two or thr ..."
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Cited by 36 (5 self)
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Abstract. We present a novel hierarchical forcedirected 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 higherdimensional embedding into a two or three dimensional subspace of E. Projecting highdimensional 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 midrange PC. 1
MGV: A System for Visualizing Massive MultiDigraphs
 IEEE Transactions on Visualization and Computer Graphics
, 2002
"... We describe MGV, an integrated visualization and exploration system for massive multidigraph navigation. ..."
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Cited by 33 (7 self)
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We describe MGV, an integrated visualization and exploration system for massive multidigraph navigation.
A Fast MultiDimensional Algorithm for Drawing Large Graphs
 In Graph Drawing’00 Conference Proceedings
, 2000
"... We present a novel hierarchical forcedirected 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 higherdimensional embedding into a two or three dimensi ..."
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Cited by 28 (4 self)
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We present a novel hierarchical forcedirected 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 higherdimensional embedding into a two or three dimensional subspace of E. Projecting highdimensional 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 midrange PC. 1 Introduction Graphs are common in many applications, from data structures to networks, from software engineering...
Range searching over tree cross products
 In Proc. 8th European Symposium on Algorithms (ESA
, 2000
"... Abstract. We introduce the tree crossproduct 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. 1 ..."
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Cited by 18 (0 self)
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Abstract. We introduce the tree crossproduct 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. 1
Graph Surfaces
"... A broad spectrum of massive data sets can be modeled as dynamic weighted multidigraphs 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 a ..."
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Cited by 11 (7 self)
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A broad spectrum of massive data sets can be modeled as dynamic weighted multidigraphs 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 outofcore 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 simpli cation. We highlight the main algorithmic ideas behind the tools and formulate some novel mathematical problems that have surfaced along the way.
Planaritypreserving clustering and embedding for large planar graphs
 In Graph Drawing (GD'99
, 1999
"... Abstract. In this paper we present a novel approach for clusterbased drawing of large planar graphs that maintains planarity. Our technique works for arbitrary planar graphs and produces a clustering which satisfies the conditions for compoundplanarity (cplanarity). Using the clustering, we obtai ..."
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Cited by 10 (3 self)
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Abstract. In this paper we present a novel approach for clusterbased drawing of large planar graphs that maintains planarity. Our technique works for arbitrary planar graphs and produces a clustering which satisfies the conditions for compoundplanarity (cplanarity). 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.
KD Trees Are Better when Cut on the Longest Side
 IN LNCS 1879, ESA 2000
, 2000
"... We show that a popular variant of the well known kd 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 h ..."
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Cited by 10 (2 self)
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We show that a popular variant of the well known kd 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 nearestneighbor searching or range searching queries visit at most O(log d−1 n) nodes of such a tree T in the worst case. Traditionally, many variants of kd 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 worstcase polylogarithmic time bound for approximate geometric queries using the simple kd tree data structure.
Visualizing large graphs with compoundfisheye views and treemaps
 In 12th Symposium on Graph Drawing (GD
, 2004
"... Abstract. Compoundfisheye 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 ..."
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Cited by 10 (1 self)
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Abstract. Compoundfisheye 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
Visualizing Massive MultiDigraphs
 In IEEE Proc. Information Visualization
, 2000
"... We describe MGV, an integrated visualization and exploration system for massive multidigraph 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 outofcore graph hierarchy and provides mechanis ..."
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Cited by 9 (5 self)
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We describe MGV, an integrated visualization and exploration system for massive multidigraph 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 outofcore 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 drilldown interface. In order to provide the user with navigation control and interactive response, MGV incorporates a number of visualization techniques like interactive pixeloriented 2D and 3D maps, statistical displays, multilinked 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 Java3D. 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 multigraphs defined on vertex sets with sizes ranging from 100 million to 250 million vertices.