Minimising the Number of Bends and Volume in 3-Dimensional Orthogonal Graph Drawings with a Diagonal Vertex Layout

A 3-dimensional orthogonal drawing of a graph with maximum degree at most 6, positions the vertices at grid-points in the 3-dimensional orthogonal grid, and routes edges along grid-lines such that edge routes only intersect at common end-vertices. Minimising the number of bends and the volume of 3-dimensional orthogonal drawings are established criteria for measuring the aesthetic quality of a given drawing. In this paper we present two algorithms for producing 3-dimensional orthogonal graph drawings with the vertices positioned along the main diagonal of a cube, so-called diagonal drawings. This vertex-layout strategy was introduced in the 3-BENDS algorithm of Eades et al. [Discrete Applied Math. 103:55–87, 2000]. We show that minimising the number of bends in a diagonal drawing of a given graph is NP-hard. Our first algorithm minimises the total number of bends for a fixed ordering of the vertices along the diagonal in linear time. Using two heuristics for determining this vertex-ordering we obtain upper bounds on the number of bends. Our second algorithm, which is a variation of the above-mentioned 3-BENDS algorithm, produces 3-bend drawings with n³+o(n³) volume, which is the best known upper bound for the volume of 3-dimensional orthogonal graph drawings with at most three bends per edge.     

On-line Animated Visualization of Huge Graphs using a Modified Spring Algorithm

On-line graph drawing deals with huge graphs which are partially unknown. At any time, a tiny part of the graph is displayed on the screen. Examples include web graphs and graphs of links in distributed file systems. This paper discusses issues arising in the presentation of such graphs. The paper describes a system for dealing with web graphs using on-line graph drawing.     

Techniques for Edge Stratification of Complex Graph Drawings

We propose an approach that allows a user (e.g., an analyst) to explore a layout produced by any graph drawing algorithm, in order to reduce the visual complexity and clarify its presentation. Our approach is based on stratifying the drawing into layers with desired properties; to this aim, heuristics are presented. The produced layers can be explored and combined by the user to gradually acquire details. We present a user study to test the effectiveness of our approach. Furthermore, we performed an experimental analysis on popular force-directed graph drawing algorithms, in order to evaluate what is the algorithm that produces the smallest number of layers and if there is any correlation between the number of crossings and the number of layers of a graph layout. The proposed approach is useful to explore graph layouts, as confirmed by the presented user study. Furthermore, interesting considerations arise from the experimental evaluation, in particular, our results suggest that the number of layers of a graph layout may represent a reliable measure of its visual complexity. The algorithms presented in this paper can be effectively applied to graph layouts with a few hundreds of edges and vertices. For larger drawings that contain lots of crossings, the time complexity of our algorithms grows quadratically in the number of edges and more efficient techniques need to be devised. The proposed approach takes as input a layout produced by any graph drawing algorithm, therefore it can be applied in a variety of application domains. Several research directions can be explored to extend our framework and to devise new visualization paradigms to effectively present stratified drawings.     

Three-Dimensional Orthogonal Graph Drawing with Optimal Volume

An orthogonal drawing of a graph is an embedding of the graph in the rectangular grid, with vertices represented by axis-aligned boxes, and edges represented by paths in the grid that only possibly intersect at common endpoints. In this paper we study three-dimensional orthogonal drawings and provide lower bounds for three scenarios: (1) drawings where the vertices have a bounded aspect ratio, (2) drawings where the surfaces of vertices are proportional to their degrees, and (3) drawings without any such restrictions. Then we show that these lower bounds are asymptotically optimal, by providing constructions that in all scenarios match the lower bounds within a constant factor.     

Information visualization; assisting low spatial individuals with information access tasks through the use of visual mediators

This study investigated the use of visual mediators to facilitate information access by low spatial individuals. Based on theories of adaptive learning and field-dependence, two human-computer interfaces were developed which were intended to compensate for the inability of low spatial individuals to readily construct visual mental models of a menu system's structure. The two compensatory interfaces included: a 2D visual hierarchy and a linear structure. The information search performance of high and low spatial individuals was compared on the two compensatory interfaces and a third challenge match interface, which challenged individuals to construct a mental model of a hierarchical menu system in order to perform efficiently. The visual mediators were successful in accommodating low spatial individuals, as indicated by the lack of any significant performance differences being detected between the high and low spatial groups on the two compensatory interfaces. High spatial individuals outperformed low spatial individuals only when information search tasks required the use of spatial ability in mentally constructing a model of the organization and structure of embedded task information. The key factor in the accommodation process was the elimination of the need to mentally visualize the structure of embedded task information. These results indicate that visualization techniques can be successfully used to enhance the information search performance of low spatial individuals.     

A layout algorithm for undirected compound graphs

We present an algorithm for the layout of undirected compound graphs, relaxing restrictions of previously known algorithms in regards to topology and geometry. The algorithm is based on the traditional force-directed layout scheme with extensions to handle multi-level nesting, edges between nodes of arbitrary nesting levels, varying node sizes, and other possible application-specific constraints. Experimental results show that the execution time and quality of the produced drawings with respect to commonly accepted layout criteria are quite satisfactory. The algorithm has also been successfully implemented as part of a pathway integration and analysis toolkit named PATIKA, for drawing complicated biological pathways with compartmental constraints and arbitrary nesting relations to represent molecular complexes and various types of pathway abstractions.     

Space Efficient Dynamic Orthogonal Range Reporting

In this paper we present new space efficient dynamic data structures for orthogonal range reporting. The described data structures support planar range reporting queries in time O(log n+klog log (4n/(k+1))) and space O(nlog log n), or in time O(log n+k) and space O(nlog  ^ε n) for any ε>0. Both data structures can be constructed in O(nlog n) time and support insert and delete operations in amortized time O(log ² n) and O(log nlog log n) respectively. These results match the corresponding upper space bounds of Chazelle (SIAM J. Comput. 17, 427–462, 1988) for the static case. We also present a dynamic data structure for d-dimensional range reporting with search time O(log  ^d−1 n+k), update time O(log  ^d n), and space O(nlog  ^d−2+ε n) for any ε>0. The model of computation used in our paper is a unit cost RAM with word size log n. A preliminary version of this paper appeared in the Proceedings of the 21st Annual ACM Symposium on Computational Geometry 2005. Work partially supported by IST grant 14036 (RAND-APX).     

A tabu search based approach for graph layout

This paper describes an automated tabu search based method for drawing general graph layouts with straight lines. To our knowledge, this is the first time tabu methods have been applied to graph drawing. We formulated the task as a multi-criteria optimization problem with a number of metrics which are used in a weighted fitness function to measure the aesthetic quality of the graph layout. The main goal of this work is to speed up the graph layout process without sacrificing layout quality. To achieve this, we use a tabu search based method that goes through a predefined number of iterations to minimize the value of the fitness function. Tabu search always chooses the best solution in the neighbourhood. This may lead to cycling, so a tabu list is used to store moves that are not permitted, meaning that the algorithm does not choose previous solutions for a set period of time. We evaluate the method according to the time spent to draw a graph and the quality of the drawn graphs. We give experimental results applied on random graphs and we provide statistical evidence that our method outperforms a fast search-based drawing method (hill climbing) in execution time while it produces comparably good graph layouts. We also demonstrate the method on real world graph datasets to show that we can reproduce similar results in a real world setting.     

Pathfinder: Visual Analysis of Paths in Graphs

The analysis of paths in graphs is highly relevant in many domains. Typically, path‐related tasks are performed in node‐link layouts. Unfortunately, graph layouts often do not scale to the size of many real world networks. Also, many networks are multivariate, i.e., contain rich attribute sets associated with the nodes and edges. These attributes are often critical in judging paths, but directly visualizing attributes in a graph layout exacerbates the scalability problem. In this paper, we present visual analysis solutions dedicated to path‐related tasks in large and highly multivariate graphs. We show that by focusing on paths, we can address the scalability problem of multivariate graph visualization, equipping analysts with a powerful tool to explore large graphs. We introduce Pathfinder, a technique that provides visual methods to query paths, while considering various constraints. The resulting set of paths is visualized in both a ranked list and as a node‐link diagram. For the paths in the list, we display rich attribute data associated with nodes and edges, and the node‐link diagram provides topological context. The paths can be ranked based on topological properties, such as path length or average node degree, and scores derived from attribute data. Pathfinder is designed to scale to graphs with tens of thousands of nodes and edges by employing strategies such as incremental query results. We demonstrate Pathfinder's fitness for use in scenarios with data from a coauthor network and biological pathways.     

Property Testing in Bounded Degree Graphs

Abstract. We further develop the study of testing graph properties as initiated by Goldreich, Goldwasser and Ron. Loosely speaking, given an oracle access to a graph, we wish to distinguish the case when the graph has a pre-determined property from the case when it is ``far' from having this property. Whereas they view graphs as represented by their adjacency matrix and measure the distance between graphs as a fraction of all possible vertex pairs, we view graphs as represented by bounded-length incidence lists and measure the distance between graphs as a fraction of the maximum possible number of edges. Thus, while the previous model is most appropriate for the study of dense graphs, our model is most appropriate for the study of bounded-degree graphs. In particular, we present randomized algorithms for testing whether an unknown bounded-degree graph is connected, k -connected (for k>1 ), cycle-free and Eulerian. Our algorithms work in time polynomial in 1/ɛ , always accept the graph when it has the tested property, and reject with high probability if the graph is ɛ -far from having the property. For example, the 2-connectivity algorithm rejects (with high probability) any N -vertex d -degree graph for which more than ɛ dN edges need to be added in order to make the graph 2-edge-connected. In addition we prove lower bounds of Ω(\sqrt N ) on the query complexity of testing algorithms for the bipartite and expander properties.     

An Algorithm for Straight-Line Drawing of Planar Graphs

We present a new algorithm for drawing planar graphs on the plane. It can be viewed as a generalization of the algorithm of Chrobak and Payne, which, in turn, is based on an algorithm by de Fraysseix, Pach, and Pollack. Our algorithm improves the previous ones in that it does not require a preliminary triangulation step; triangulation proves problematic in drawing graphs ``nicely,' as it has the tendency to ruin the structure of the input graph. The new algorithm retains the positive features of the previous algorithms: it embeds a biconnected graph of n vertices on a grid of size (2n-4) x (n-2) in linear time. We have implemented the algorithm as part of a software system for drawing graphs nicely. Received September 21, 1995; revised March 15, 1996.     

Improving the running time of embedded upward planarity testing

We consider the standard algorithm by Bertolazzi et al. to test the upward planarity of embedded digraphs. We show how to improve its running time from O(n+r²) to , where r is the number of sources and sinks in the digraph. We also discuss an application of this technique: improving the running time of getting a quasi-upward planar drawing for an embedded digraph with minimum number of bends.     

The Readability of Path‐Preserving Clusterings of Graphs

Graph visualization systems often exploit opaque metanodes to reduce visual clutter and improve the readability of large graphs. This filtering can be done in a path‐preserving way based on attribute values associated with the nodes of the graph. Despite extensive use of these representations, as far as we know, no formal experimentation exists to evaluate if they improve the readability of graphs. In this paper, we present the results of a user study that formally evaluates how such representations affect the readability of graphs. We also explore the effect of graph size and connectivity in terms of this primary research question. Overall, for our tasks, we did not find a significant difference when this clustering is used. However, if the graph is highly connected, these clusterings can improve performance. Also, if the graph is large enough and can be simplified into a few metanodes, benefits in performance on global tasks are realized. Under these same conditions, however, performance of local attribute tasks may be reduced.     

Tablorer – An Interactive Tree Visualization System for Tablet PCs

A variety of mobile devices are available today, but there is no dominant tree visualization system in the devices. This paper proposes Tablorer, a novel interactive tree visualization system for medium‐sized mobile devices, especially for tablet PCs. The system shows the hierarchical information with a compact way using an expandable table format. For efficient navigation, the system provides an integrated view of context and focus information. The experimental results show that Tablorer can reduce the search time by about 22%.     

Disk Embeddings of Planar Graphs

Given a planar graph $G=(V,E)$ and a rooted forest ${\FF}=(V_{\FF}, A_{\FF})$ with leaf set $V$, we wish to decide whether $G$ has a plane embedding $\GG$ satisfying the following condition: There are $|V_{\FF}|-|V|$ pairwise noncrossing Jordan curves in the plane one-to-one corresponding to the nonleaf vertices of ${\FF}$ such that for every nonleaf vertex $f$ of ${\FF}$, the interior of the curve $\JJ_f$ corresponding to $f$ contains all the leaf descendants of $f$ in ${\FF}$ but contains no other leaves of ${\FF}$. This problem arises from theoretical studies in geographic database systems. It is unknown whether this problem can be solved in polynomial time. This paper presents an almost linear-time algorithm for a nontrivial special case where the set of leaf descendants of each nonleaf vertex $f$ in ${\FF}$ induces a connected subgraph of $G$.     

Visual exploration and retrieval of XML document collections with the generic system X2

This article reports on the XML retrieval system x² that has been developed at the University of Munich over the last 5 years. In a typical session with x², the user first browses a structural summary of the XML database in order to select interesting elements and keywords occurring in documents. Using this intermediate result, queries combining structure and textual references are composed semiautomatically. After query evaluation, the full set of answers is presented in a visual and structured way. x² largely exploits the structure found in documents, queries and answers to enable new interactive visualization and exploration techniques that support mixed IR and database-oriented querying, thus bridging the gap between these three views on the data to be retrieved. Another salient characteristic of x² that distinguishes it from other visual query systems for XML is that it supports various degrees of detailedness in the presentation of answers, as well as techniques for dynamically reordering, grouping and ranking retrieved elements once the complete answer set has been computed.