TopoLayout: multilevel graph layout by topological features

We describe TopoLayout, a feature-based, multilevel algorithm that draws undirected graphs based on the topological features they contain. Topological features are detected recursively inside the graph, and their subgraphs are collapsed into single nodes, forming a graph hierarchy. Each feature is drawn with an algorithm tuned for its topology. As would be expected from a feature-based approach, the runtime and visual quality of TopoLayout depends on the number and types of topological features present in the graph. We show experimental results comparing speed and visual quality for TopoLayout against four other multilevel algorithms on a variety of data sets with a range of connectivities and sizes. TopoLayout frequently improves the results in terms of speed and visual quality on these data sets     

Laplacian spectrum analysis and spanning tree algorithm for circuit partitioning problems

The spectrum of a graph is the set of all eigenvalues of the Laplacian matrix of the graph. There is a closed relationship between the Laplacian spectrum of graphs and some properties of graphs such as connectivity. In the recent years Laplacian spectrum of graphs has been widely applied in many fields. The application of Laplacian spectrum of graphs to circuit partitioning problems is reviewed in this paper. A new criterion of circuit partitioning is proposed and the bounds of the partition ratio for weighted graphs are also presented. Moreover, the deficiency of graph-partitioning algorithms by Laplacian eigenvectors is addressed and an algorithm by means of the minimal spanning tree of a graph is proposed. By virtue of taking the graph structure into consideration this algorithm can fulfill general requirements of circuit partitioning.     

GO: A cluster algorithm for graph visualization

As we are in the big data age, graph data such as user networks in Facebook and Flickr becomes large. How to reduce the visual complexity of a graph layout is a challenging problem. Clustering graphs is regarded as one of effective ways to address this problem. Most of current graph visualization systems, however, directly use existing clustering algorithms that are not originally developed for the visualization purpose. For graph visualization, a clustering algorithm should meet specific requirements such as the sufficient size of clusters, and automatic determination of the number of clusters. After identifying the requirements of clustering graphs for visualization, in this paper we present a new clustering algorithm that is particularly designed for visualization so as to reduce the visual complexity of a layout, together with a strategy for improving the scalability of our algorithm. Experiments have demonstrated that our proposed algorithm is capable of detecting clusters in a way that is required in graph visualization.     

基于图匹配的分层布局算法

针对节点数目较大并且度数比较平均的无向图,根据分层扩展的思想,提出一种基于图匹配的分层布局算法（Graph Matching Hierarchy,GMH）。基于图匹配思想对大图进行递归化简,然后应用FR算法对最粗化图进行布局,最后利用质心布局算法对图进行扩展。实验结果表明,GMH算法能够提高可视化效率,改善布局效果,且分层布局的结果更易于理解。      

Drawing complete binary trees inside rectilinear polygons

Most of graph drawing algorithms draw graphs on unbounded planes. However, there are applications that require graphs to be drawn on the plane inside a given polygon. In this paper, a new algorithm for planar orthogonal drawing of complete binary trees inside rectilinear polygons is presented. Uniform distribution of nodes of graphs on drawing regions is one of the aesthetics criteria in graph drawing. The goal of this paper is to produce planar orthogonal drawings with a relatively uniform node distribution and few edge bends. The proposed algorithm can be considered as a generalization of the H-tree layout method for rectilinear polygons. A new linear time algorithm is also given for bisecting rectilinear polygons into two equi-area rectilinear sub-polygons.     

FMR:基于FR的快速多层次算法

图可视化技术是可视化研究的重要内容,近年来大图的绘制问题一直是图可视化 技术的焦点。为此,提出了一种快速多层次算法用于解决大图绘制问题。采用多层次方法作为 算法的框架,以 FR 力导向算法的变体结合质心算法以及四叉树空间分解等方法对单层布局进 行优化。另外,还使用了约束规范化和能量模型 2 种加速方法。实验表明,该算法具有高效的 性能和良好的布局效果。其效率非常高,在单核 CPU 下,可以在大约 5 s 内很好地绘制出 10 000 个顶点的图。并与几种经典的算法进行了比较,也证明了该算法的有效性和实用性。此外,该 算法易于实现,可被轻易推广到其他布局算法上,以加速其运算。     

Visual Analysis of Large Graphs: State‐of‐the‐Art and Future Research Challenges

The analysis of large graphs plays a prominent role in various fields of research and is relevant in many important application areas. Effective visual analysis of graphs requires appropriate visual presentations in combination with respective user interaction facilities and algorithmic graph analysis methods. How to design appropriate graph analysis systems depends on many factors, including the type of graph describing the data, the analytical task at hand and the applicability of graph analysis methods. The most recent surveys of graph visualization and navigation techniques cover techniques that had been introduced until 2000 or concentrate only on graph layouts published until 2002. Recently, new techniques have been developed covering a broader range of graph types, such as time‐varying graphs. Also, in accordance with ever growing amounts of graph‐structured data becoming available, the inclusion of algorithmic graph analysis and interaction techniques becomes increasingly important. In this State‐of‐the‐Art Report, we survey available techniques for the visual analysis of large graphs. Our review first considers graph visualization techniques according to the type of graphs supported. The visualization techniques form the basis for the presentation of interaction approaches suitable for visual graph exploration. As an important component of visual graph analysis, we discuss various graph algorithmic aspects useful for the different stages of the visual graph analysis process. We also present main open research challenges in this field.     

UML类图层次化自动布图算法

UML类图能够有效地帮助软件工程师理解大规模的软件系统,而优化图元的空间布局可以增强类图的可读性和可理解性.由于类图中继承关系具有明显的层次特性,因此类图自动布局大多采用层次化的布图算法.此外,类图布局需要考虑相关的领域知识以及绘制准则,因而通用嵌套有向图层次化布局算法不能直接用于类图的绘制,它们必须加以扩展.但是,已有的类图层次化方法并没有考虑类图中图元的嵌套关系,这将导致自动布局方法不能处理类图中包与类、接口之间的包含关系.在考虑图绘制美学、UML类图绘制以及软件可视化等相关知识的基础上,选取了一组布     

Learning graph prototypes for shape recognition

This paper presents some new approaches for computing graph prototypes in the context of the design of a structural nearest prototype classifier. Four kinds of prototypes are investigated and compared: set median graphs, generalized median graphs, set discriminative graphs and generalized discriminative graphs. They differ according to (i) the graph space where they are searched for and (ii) the objective function which is used for their computation. The first criterion allows to distinguish set prototypes which are selected in the initial graph training set from generalized prototypes which are generated in an infinite set of graphs. The second criterion allows to distinguish median graphs which minimize the sum of distances to all input graphs of a given class from discriminative graphs, which are computed using classification performance as criterion, taking into account the inter-class distribution. For each kind of prototype, the proposed approach allows to identify one or many prototypes per class, in order to manage the trade-off between the classification accuracy and the classification time.Each graph prototype generation/selection is performed through a genetic algorithm which can be specialized to each case by setting the appropriate encoding scheme, fitness and genetic operators.An experimental study performed on several graph databases shows the superiority of the generation approach over the selection one. On the other hand, discriminative prototypes outperform the generative ones. Moreover, we show that the classification rates are improved while the number of prototypes increases. Finally, we show that discriminative prototypes give better results than the median graph based classifier.     

Exponential Speedup of Fixed-Parameter Algorithms for Classes of Graphs Excluding Single-Crossing Graphs as Minors

We present a fixed-parameter algorithm that constructively solves the $k$-dominating set problem on any class of graphs excluding a single-crossing graph (a graph that can be drawn in the plane with at most one crossing) as a minor in $O(4^{9.55\sqrt{k}}n^{O(1)})$ time. Examples of such graph classes are the $K_{3,3}$-minor-free graphs and the $K_{5}$-minor-free graphs. As a consequence, we extend our results to several other problems such as vertex cover, edge dominating set, independent set, clique-transversal set, kernels in digraphs, feedback vertex set, and a collection of vertex-removal problems. Our work generalizes and extends the recent results of exponential speedup in designing fixed-parameter algorithms on planar graphs due to Alber et al. to other (nonplanar) classes of graphs.     

A constant approximation algorithm for the densest k-subgraph problem on chordal graphs

The densest k-subgraph (DkS) problem asks for a k-vertex subgraph of a given graph with the maximum number of edges. The DkS problem is NP-hard even for special graph classes including bipartite, planar, comparability and chordal graphs, while no constant approximation algorithm is known for any of these classes. In this paper we present a 3-approximation algorithm for the class of chordal graphs. The analysis of our algorithm is based on a graph theoretic lemma of independent interest.     

A user parameter-free approach for mining robust sequential classification rules

Recent years have witnessed extensive studies of graph classification due to the rapid increase in applications involving structural data and complex relationships. To support graph classification, all existing methods require that training graphs should be relevant (or belong) to the target class, but cannot integrate graphs irrelevant to the class of interest into the learning process. In this paper, we study a new universum graph classification framework which leverages additional “non-example” graphs to help improve the graph classification accuracy. We argue that although universum graphs do not belong to the target class, they may contain meaningful structure patterns to help enrich the feature space for graph representation and classification. To support universum graph classification, we propose a mathematical programming algorithm, ugBoost, which integrates discriminative subgraph selection and margin maximization into a unified framework to fully exploit the universum. Because informative subgraph exploration in a universum setting requires the search of a large space, we derive an upper bound discriminative score for each subgraph and employ a branch-and-bound scheme to prune the search space. By using the explored subgraphs, our graph classification model intends to maximize the margin between positive and negative graphs and minimize the loss on the universum graph examples simultaneously. The subgraph exploration and the learning are integrated and performed iteratively so that each can be beneficial to the other. Experimental results and comparisons on real-world dataset demonstrate the performance of our algorithm.     

A distributed low tree-depth decomposition algorithm for bounded expansion classes

We study the distributed low tree-depth decomposition problem for graphs restricted to a bounded expansion class. Low tree-depth decomposition have been introduced in 2006 and have found quite a few applications. For example it yields a linear-time model checking algorithm for graphs in a bounded expansion class. Recall that bounded expansion classes cover classes of graphs of bounded degree, of planar graphs, of graphs of bounded genus, of graphs of bounded treewidth, of graphs that exclude a fixed minor, and many other graphs. There is a sequential algorithm to compute low tree-depth decomposition (with bounded number of colors) in linear time. In this paper, we give the first efficient distributed algorithm for this problem. As it is usual for a symmetry breaking problem, we consider a synchronous model, and as we are interested in a deterministic algorithm, we use the usual assumption that each vertex has a distinct identity number. We consider the distributed message-passing \(\mathcal {CONGEST}_\mathrm{BC}\) model, in which messages have logarithmic length and only local broadcast are allowed. In this model, we present a logarithmic time distributed algorithm for computing a low tree-depth decomposition of graphs in a fixed bounded expansion class. In the sequential centralized case low tree-depth decomposition linear time algorithm are used as a core procedure in several non-trivial linear time algorithms. We believe that, similarly, low tree-depth decomposition could be at the heart of several non-trivial logarithmic time algorithms.     

Linear structure of bipartite permutation graphs and the longest path problem

The class of bipartite permutation graphs is the intersection of two well known graph classes: bipartite graphs and permutation graphs. A complete bipartite decomposition of a bipartite permutation graph is proposed in this note. The decomposition gives a linear structure of bipartite permutation graphs, and it can be obtained in O(n) time, where n is the number of vertices. As an application of the decomposition, we show an O(n) time and space algorithm for finding a longest path in a bipartite permutation graph.     

Data Parallel Implementation of Belief Propagation in Factor Graphs on Multi-core Platforms

We investigate data parallel techniques for belief propagation in acyclic factor graphs on multi-core systems. Belief propagation is a key inference algorithm in factor graph, a probabilistic graphical model that has found applications in many domains. In this paper, we explore data parallelism for basic operations over the potential tables in belief propagation. Data parallel techniques for these table operations are developed for shared memory platforms. We then propose a complete belief propagation algorithm using these table operations to perform exact inference in factor graphs. The proposed algorithms are implemented on state-of-the-art multi-socket multi-core systems with additional NUMA-aware optimizations. Our proposed algorithms exhibit good scalability using a representative set of factor graphs. On a four-socket Intel Westmere-EX system with 40 cores, we achieve 39.5 $\times $ speedup for the table operations and 39 $\times $ speedup for the complete algorithm using factor graphs with large potential tables.     

Algorithm visualization using tree graphs

Recent advances in graphics workstations allow the development of improved visualization tools for algorithm and program development. Algorithm visualization permits better analysis, development, and presentation of the algorithm characteristics. In this paper, we present a simple algorithm visualization technique using tree graphs. The technique is applied to the visualization of three sorting algorithms: the bubble sort, the quicksort, and the merge and sort, and one matrix algorithm, the Gaussian elimination. Key states of the data are displayed on the nodes, while the graph itself represents the underlying structure of the algorithm. All graphics are displayed under the X Window environment using simple graphics and window programming techniques.     

Straight-Line Drawing Algorithms for Hierarchical Graphs and Clustered Graphs

Hierarchical graphs and clustered graphs are useful non-classical graph models for structured relational information. Hierarchical graphs are graphs with layering structures; clustered graphs are graphs with recursive clustering structures. Both have applications in CASE tools, software visualization and VLSI design. Drawing algorithms for hierarchical graphs have been well investigated. However, the problem of planar straight-line representation has not been solved completely. In this paper we answer the question: does every planar hierarchical graph admit a planar straight-line hierarchical drawing? We present an algorithm that constructs such drawings in linear time. Also, we answer a basic question for clustered graphs, that is, does every planar clustered graph admit a planar straight-line drawing with clusters drawn as convex polygons? We provide a method for such drawings based on our algorithm for hierarchical graphs.     

一种新的高效图聚集算法

图聚集是将一个大规模的图用简洁的并能有效反映原始图的结构和属性信息的小规模图来表示的技术.图聚集在图数据管理、分析和可视化中发挥着重要作用.图聚集方面现有研究结果还很少,也很不系统.其主要不足之处是:1)算法依赖于具体应用;2)算法仅考虑了图的某方面信息,如结构信息或属性信息;3)算法对用户提供的交互和反馈信息的约束很强.针对现有图聚集算法存在的主要不足,提出一种有向图新型图聚集算法,该算法采用一种新的聚集图质量函数,全面刻画了聚集图多样性、覆盖性、简洁性和实用性.该算法使用LSH(locality sensitive Hashing)技术和基于熵的划分技术,保证了聚集图的质量.在真实数据集上进行了大量的实验,验证了算法的有效性.     

Fast graph similarity search via hashing and its application on image retrieval

Similarity search in graph databases has been widely investigated. It is worthwhile to develop a fast algorithm to support similarity search in large-scale graph databases. In this paper, we investigate a k-NN (k-Nearest Neighbor) similarity search problem by locality sensitive hashing (LSH). We propose an innovative fast graph search algorithm named LSH-GSS, which first transforms complex graphs into vectorial representations based on prototypes in the database and later accelerates a query in Euclidean space by employing LSH. Because images can be represented as attributed graphs, we propose an approach to transform attributed graphs into n-dimensional vectors and apply LSH-GSS to execute further image retrieval. Experiments on three real graph datasets and two image datasets show that our methods are highly accurate and efficient.     

The Longest Path Problem Is Polynomial on Cocomparability Graphs

The longest path problem is the problem of finding a path of maximum length in a graph. As a generalization of the Hamiltonian path problem, it is NP-complete on general graphs and, in fact, on every class of graphs that the Hamiltonian path problem is NP-complete. Polynomial solutions for the longest path problem have recently been proposed for weighted trees, Ptolemaic graphs, bipartite permutation graphs, interval graphs, and some small classes of graphs. Although the Hamiltonian path problem on cocomparability graphs was proved to be polynomial almost two decades ago, the complexity status of the longest path problem on cocomparability graphs has remained open; actually, the complexity status of the problem has remained open even on the smaller class of permutation graphs. In this paper, we present a polynomial-time algorithm for solving the longest path problem on the class of cocomparability graphs. Our result resolves the open question for the complexity of the problem on such graphs, and since cocomparability graphs form a superclass of both interval and permutation graphs, extends the polynomial solution of the longest path problem on interval graphs and provides polynomial solution to the class of permutation graphs.