Reeb graph path dissimilarity for 3D object matching and retrieval

We introduce a skeletal graph for topological 3D shape representation using Morse theory. The proposed skeletonization algorithm encodes a 3D shape into a topological Reeb graph using a normalized mixture distance function. We also propose a novel graph matching algorithm by comparing the relative shortest paths between the skeleton endpoints. Experimental results demonstrate the feasibility of the proposed topological Reeb graph as a shape signature for 3D object matching and retrieval.     

基于最优子序列双射的骨架树匹配

基于图描述的骨架图匹配大多考虑骨架图的拓扑结构,使得匹配精度受到影响。先通过骨架构造以骨架中心为根节点的骨架树,使用骨架中心到骨架端点测地路径等信息来描述骨架树的叶子节点,利用改进的最优子序列双射时序匹配算法来确定两幅骨架树叶子节点的匹配关系,该算法不考虑骨架树的拓扑结构,只匹配骨架树的叶子节点。通过匹配实验结果和检索实验结果,表明该方法有效地提高了匹配精度。     

基于空间句法和最短路径的图核

针对图模式识别领域中现有图核方法对反映图本身拓扑结构的节点特征挖掘不够充分的问题,提出了基于空间句法和最短路径的图核。借鉴建筑学与城市规划学科中的空间句法理论构造分布于图节点上的拓扑特征的量化描述,基于此提出了可表示、计算,正定、适用范围较广的空间句法核和基于最短路径的空间句法核,进而借助支持向量机实现了非精确图匹配。不同于其他图核方法,该方法对图的拓扑特征表达能力强,通用性较好。实验结果表明,所设计的图核在分类精度方面相较于最短路径核有较显著的改善。     

The exact distance to destination in undirected world

Shortest distance queries are essential not only in graph analysis and graph mining tasks but also in database applications, when a large graph needs to be dealt with. Such shortest distance queries are frequently issued by end-users or requested as a subroutine in real applications. For intensive queries on large graphs, it is impractical to compute shortest distances on-line from scratch, and impractical to materialize all-pairs shortest distances. In the literature, 2-hop distance labeling is proposed to index the all-pairs shortest distances. It assigns distance labels to vertices in a large graph in a pre-computing step off-line and then answers shortest distance queries on-line by making use of such distance labels, which avoids exhaustively traversing the large graph when answering queries. However, the existing algorithms to generate 2-hop distance labels are not scalable to large graphs. Finding an optimal 2-hop distance labeling is NP-hard, and heuristic algorithms may generate large size distance labels while still needing to pre-compute all-pairs shortest paths. In this paper, we propose a multi-hop distance labeling approach, which generates a subset of the 2-hop distance labels as index off-line. We can compute the multi-hop distance labels efficiently by avoiding pre-computing all-pairs shortest paths. In addition, our multi-hop distance labeling is small in size to be stored. To answer a shortest distance query between two vertices, we first generate the query-specific small set of 2-hop distance labels for the two vertices based on our multi-hop distance labels stored and compute the shortest distance between the two vertices based on the 2-hop distance labels generated on-line. We conducted extensive performance studies on large real graphs and confirmed the efficiency of our multi-hop distance labeling scheme.     

并行图论算法研究进展

在这篇综述文章中，我们将重点介绍并行图论处近年来的发展概况及主要成果，并给出一些可能的发展方向。具体内容包括：基于共享存储模型上的图搜索技术、连发支及最小生成树算法、增值并行图论算法、最短路径算法、极大独立集算法、极大匹配与最大匹配算法，图着色算法、求欧拉回路及哈密尔顿回路算法，图同构算法、图Ｋ连通算法以及最大流最小割算法等。     

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.     

Embedding k(n−k) edge-disjoint spanning trees in arrangement graphs

The arrangement graphs are a class of generalized star graphs. In this paper we construct a graph that consists of the maximum number of directed edge-disjoint spanning trees in an arrangement graph. The paths that connect the common root node to any given node through different spanning trees are node-disjoint, and the lengths of these paths differ from the shortest possible lengths by a small additive constant. This graph can be used to derive fault-tolerant algorithms for broadcasting and scattering problems without prior knowledge of the faulty elements of the network.     

基于形状骨架图匹配的文物碎片自动重组方法

为了有效解决文物碎片自动重组中由于断裂部位受损造成几何信息丢失,采用传统几何驱动方法容易失效的问题,本文提出一种基于形状骨架图匹配的文物碎片自动重组方法,将碎片匹配问题转化为碎片表面纹饰中非完整纹元的互补匹配问题.首先,通过提取文物碎片表面特征线得到碎片表面的纹饰信息;然后根据完整纹元的特征确定非完整纹元互补匹配的约束条件,采用视觉骨架剪枝的方法提取完全位于断裂部位的非完整纹元的形状骨架图,基于形状骨架图语法及匹配约束条件判定非完整纹元是否互补匹配;接着,将碎片上非完整纹元的顺序作为上层约束条件,采用基于带剪枝深度优先的搜索方法搜索匹配碎片;最后,以邻接碎片上非完整纹元间公共弦的端点作为邻接约束点,采用最小二乘法计算刚体变换参数得到碎片的初始位置,并采用迭代最近点方法将邻接碎片精确对齐.实验结果表明,该方法能够有效解决断裂部位存在缺损文物碎片的自动重组问题.     

三角网格模型的基本群分割

提出一种有效的三角网格模型分割方法。用Dijkstra算法求出三角网格模型上任意给定一个基点到其余顶点的最短路径树;求出该模型对偶图的最大生成树,且对偶图的边与该最短路径树的边不相交;找出该模型上所有既不属于最短路径树也不和最大生成树相交的边,这些边分别与最短路径树组成的最短环集合就是给定基点处的基本群,沿着这些最短环就可以把网格分割成一个拓扑同胚于圆盘的区域。实验结果表明,该分割方法可以快速、有效地实现网格的分割。     

Estimating graph distance and centrality on shared nothing architectures

We present a parallel toolkit for pairwise distance computation in massive networks. Computing the exact shortest paths between a large number of vertices is a costly operation, and serial algorithms are not practical for billion‐scale graphs. We first describe an efficient parallel method to solve the single source shortest path problem on commodity hardware with no shared memory. Using it as a building block, we introduce a new parallel algorithm to estimate the shortest paths between arbitrary pairs of vertices. Our method exploits data locality, produces highly accurate results, and allows batch computation of shortest paths with 7% average error in graphs that contain billions of edges. The proposed algorithm is up to two orders of magnitude faster than previously suggested algorithms and does not require large amounts of memory or expensive high‐end servers. We further leverage this method to estimate the closeness and betweenness centrality metrics, which involve systems challenges dealing with indexing, joining, and comparing large datasets efficiently. In one experiment, we mined a real‐world Web graph with 700 million nodes and 12 billion edges to identify the most central vertices and calculated more than 63 billion shortest paths in 6 h on a 20‐node commodity cluster. Copyright © 2014 John Wiley & Sons, Ltd.     

Structure and attribute index for approximate graph matching in large graphs

The increasing popularity of graph data in various domains has lead to a renewed interest in developing efficient graph matching techniques, especially for processing large graphs. In this paper, we study the problem of approximate graph matching in a large attributed graph. Given a large attributed graph and a query graph, we compute a subgraph of the large graph that best matches the query graph. We propose a novel structure-aware and attribute-aware index to process approximate graph matching in a large attributed graph. We first construct an index on the similarity of the attributed graph, by partitioning the large search space into smaller subgraphs based on structure similarity and attribute similarity. Then, we construct a connectivity-based index to give a concise representation of inter-partition connections. We use the index to find a set of best matching paths. From these best matching paths, we compute the best matching answer graph using a greedy algorithm. Experimental results on real datasets demonstrate the efficiency of both index construction and query processing. We also show that our approach attains high-quality query answers.     

A comparative study of topological properties of hypercubes andstar graphs

Undertakes a comparative study of two important interconnection network topologies: the star graph and the hypercube, from the graph theory point of view. Topological properties are derived for the star graph and are compared with the corresponding properties of the hypercube. Among other results, the authors determine necessary and sufficient conditions for shortest path routing and characterize maximum-sized families of parallel paths between any two nodes of the star graph. These parallel paths are proven of minimum length within a small additive constant. They also define greedy and asymptotically balanced spanning trees to support broadcasting and personalized communication on the star graph. These results confirm the already claimed topological superiority of the star graph over the hypercube     

Layered data association using graph-theoretic formulation with applications to tennis ball tracking in monocular sequences

In this paper, we propose a multi-layered data association scheme with graph-theoretic formulation for tracking multiple objects that undergo switching dynamics in clutter. The proposed scheme takes as input object candidates detected in each frame. At the object candidate level, "tracklets' are "grown' from sets of candidates that have high probabilities of containing only true positives. At the tracklet level, a directed and weighted graph is constructed, where each node is a tracklet, and the edge weight between two nodes is defined according to the "compatibility' of the two tracklets. The association problem is then formulated as an all-pairs shortest path (APSP) problem in this graph. Finally, at the path level, by analyzing the all-pairs shortest paths, all object trajectories are identified, and track initiation and track termination are automatically dealt with. By exploiting a special topological property of the graph, we have also developed a more efficient APSP algorithm than the general-purpose ones. The proposed data association scheme is applied to tennis sequences to track tennis balls. Experiments show that it works well on sequences where other data association methods perform poorly or fail completely.     

Optimizing graph algorithms for improved cache performance

We develop algorithmic optimizations to improve the cache performance of four fundamental graph algorithms. We present a cache-oblivious implementation of the Floyd-Warshall algorithm for the fundamental graph problem of all-pairs shortest paths by relaxing some dependencies in the iterative version. We show that this implementation achieves the lower bound on processor-memory traffic of /spl Omega/(N/sup 3///spl radic/C), where N and C are the problem size and cache size, respectively. Experimental results show that this cache-oblivious implementation shows more than six times the improvement in real execution time over that of the iterative implementation with the usual row major data layout, on three state-of-the-art architectures. Second, we address Dijkstra's algorithm for the single-source shortest paths problem and Prim's algorithm for minimum spanning tree problem. For these algorithms, we demonstrate up to two times the improvement in real execution time by using a simple cache-friendly graph representation, namely adjacency arrays. Finally, we address the matching algorithm for bipartite graphs. We show performance improvements of two to three times in real execution time by using the technique of making the algorithm initially work on subproblems to generate a suboptimal solution and, then, solving the whole problem using the suboptimal solution as a starting point. Experimental results are shown for the Pentium III, UltraSPARC III, Alpha 21264, and MIPS R12000 machines.     

Topological properties of star graphs

Our purpose in the present paper is to investigate different topological properties of the recently introduced star graphs which are being viewed as attractive alternatives to n-cubes or hypercubes. These properties are interesting by themselves from a graph theory point of view as well as they have direct in generating vertex disjoint paths and minimal paths in a star graph. They can also be readily utilized to design routing algorithms and to compute contention and traffic congestion in networks that use star graphs as the underlying topology.     

Shortest paths on dynamic graphs

Among the variants of the well-known shortest path problem, those that refer to dynamically changing graphs are theoretically interesting, as well as computationally challenging. Application-wise, there is an industrial need for computing point-to-point shortest paths on large-scale road networks whose arcs are weighted with a travelling time that depends on traffic conditions. We survey recent techniques for dynamic graph weights as well as dynamic graph topology.     

Shortest paths between shortest paths

We study the following problem on reconfiguring shortest paths in graphs: Given two shortest s-t paths, what is the minimum number of steps required to transform one into the other, where each intermediate path must also be a shortest s-t path and must differ from the previous one by only one vertex. We prove that the shortest reconfiguration sequence can be exponential in the size of the graph and that it is NP-hard to compute the shortest reconfiguration sequence even when we know that the sequence has polynomial length.     

On certain connectivity properties of the internet topology

We show that random graphs in the preferential connectivity model have constant conductance, and hence have worst-case routing congestion that scales logarithmically with the number of nodes. Another immediate implication is constant spectral gap between the first and second eigenvalues of the random walk matrix associated with these graphs. We also show that the expected frugality (overpayment in the Vickrey–Clarke–Groves mechanism for shortest paths) of a sparse Erdős–Renyi random graph is bounded by a small constant.     

基于路径映射的相似子图匹配算法

迄今为止,相关的图相似性匹配方法通常不考虑节点关系以及边权重的实际意义。提出一种基于路径映射 的相似子图匹配方法,用以更精确地查找具有相似拓扑结构的加权图。其创新之处在于充分利用标签信息,综合考虑 拓扑结构特征,克服了忽略节点结构关系和边权重的意义去分析图相似性的弊端。因此,该方法在很大程度上提高了 图相似性匹配的应用范围和匹配精度。实验表明本方法具有较高的查询质量和效率。