On Succinct Greedy Drawings of Plane Triangulations and 3-Connected Plane Graphs

Geometric routing by using virtual locations is an elegant way for solving network routing problems. In its simplest form, greedy routing, a message is simply forwarded to a neighbor that is closer to the destination. It has been an open conjecture whether every 3-connected plane graph has a greedy drawing in the Euclidean plane R ² (by Papadimitriou and Ratajczak in Theor. Comp. Sci. 344(1):3–14, 2005). Leighton and Moitra (Discrete Comput. Geom. 44(3):686–705, 2010) recently settled this conjecture positively. One main drawback of this approach is that the coordinates of the virtual locations require Ω(nlogn) bits to represent (the same space usage as traditional routing table approaches). This makes greedy routing infeasible in applications. In this paper, we show that the classical Schnyder drawing in R ² of plane triangulations is greedy with respect to a simple natural metric function H(u,v) over R ² that is equivalent to Euclidean metric D _E (u,v) (in the sense that $D_{E}(u,v) \leq H(u,v) \leq2\sqrt{2}D_{E}(u,v)$ ). The drawing uses two integer coordinates between 0 and 2n?5, which can be represented by logn bits. We also show that the classical Schnyder drawing in R ² of 3-connected plane graphs is weakly greedy with respect to the same metric function H(?,?). The drawing uses two integer coordinates between 0 and f (where f is the number of internal faces of G).     

A Linear-Time Algorithm for Symmetric Convex Drawings of Internally Triconnected Plane Graphs

We study a crossing minimization problem of drawing a bipartite graph with a radial drawing of two orbits. Radial drawings are one of well-known drawing conventions in social network analysis and visualization, in particular, displaying centrality indices of actors (Wasserman and Faust, Social Network Analysis: Methods and Applications. Cambridge University Press, Cambridge, 1994). The main problem in this paper is called the one-sided radial crossing minimization, if the positions of vertices in the outer orbit are fixed. The problem is known to be NP-hard (Bachmaier, IEEE Trans. Vis. Comput. Graph. 13, 583–594, 2007), and a number of heuristics are available (Bachmaier, IEEE Trans. Vis. Comput. Graph. 13, 583–594, 2007). However, there is no approximation algorithm for the crossing minimization problem in radial drawings. We present the first polynomial time constant-factor approximation algorithm for the one-sided radial crossing minimization problem.     

Pruning 2-Connected Graphs

Given an undirected graph G with edge costs and a specified set of terminals, let the density of any subgraph be the ratio of its cost to the number of terminals it contains. If G is 2-connected, does it contain smaller 2-connected subgraphs of density comparable to that of?G? We answer this question in the affirmative by giving an algorithm to pruneG and find such subgraphs of any desired size, incurring only a logarithmic factor increase in density (plus a small additive term). We apply our pruning techniques to give algorithms for two NP-Hard problems on finding large 2-vertex-connected subgraphs of low cost; no previous approximation algorithm was known for either problem. In the k-2VC problem, we are given an undirected graph G with edge costs and an integer k; the goal is to find a minimum-cost 2-vertex-connected subgraph of G containing at least k vertices. In the Budget-2VC problem, we are given a graph G with edge costs, and a budget B; the goal is to find a 2-vertex-connected subgraph H of G with total edge cost at most B that maximizes the number of vertices in H. We describe an O(log?nlog?k) approximation for the k-2VC problem, and a bicriteria approximation for the Budget-2VC problem that gives an $O(\frac{1}{\epsilon}\log^{2} n)$ approximation, while violating the budget by a factor of at most 2+ε.     

The Isomorphism Problem for Planar 3-Connected Graphs Is in Unambiguous Logspace

The isomorphism problem for planar graphs is known to be efficiently solvable. For planar 3-connected graphs, the isomorphism problem can be solved by efficient parallel algorithms, it is in the class AC ¹. In this paper we improve the upper bound for planar 3-connected graphs to unambiguous logspace, in fact to UL∩coUL. As a consequence of our method we get that the isomorphism problem for oriented graphs is in NL. We also show that the problems are hard for L.     

Small Area Drawings of Outerplanar Graphs

We show three linear-time algorithms for constructing planar straight-line grid drawings of outerplanar graphs. The first and the second algorithm are for balanced outerplanar graphs. Both require linear area. The drawings produced by the first algorithm are not outerplanar while those produced by the second algorithm are. On the other hand, the first algorithm constructs drawings with better angular resolution. The third algorithm constructs outerplanar drawings of general outerplanar graphs with O(n ^1.48) area. Further, we study the interplay between the area requirements of the drawings of an outerplanar graph and the area requirements of a special class of drawings of its dual tree. Work partially supported by MUR under Project MAINSTREAM Algorithms for Massive Information Structures and Data Streams.     

Measuring Symmetry in Drawings of Graphs

Layout symmetry is an important and desired feature in graph drawing. While there is a substantial body of work in computer vision around the detection and measurement of symmetry in images, there has been little effort to define and validate meaningful measures of the symmetry of graph drawings. In this paper, we evaluate two algorithms that have been proposed for measuring graph drawing symmetry, comparing their judgments to those of human subjects, and investigating the use of stress as an alternative measure of symmetry. We discuss advantages and disadvantages of these measures, possible ways to improve them, and implications for the design of algorithms that optimize the symmetry in the layout.     

Efficient Orthogonal Drawings of High Degree Graphs

Most of the work that appears in the two-dimensional orthogonal graph drawing literature deals with graphs whose maximum degree is four. In this paper we present an algorithm for orthogonal drawings of simple graphs with degree higher than four. Vertices are represented by rectangular boxes of perimeter less than twice the degree of the vertex. Our algorithm is based on creating groups / pairs of vertices of the graph. The orthogonal drawings produced by our algorithm have area at most (m-1) ( m / 2 +2) . Two important properties of our algorithm are that the drawings exhibit a small total number of bends (less than m ), and that there is at most one bend per edge. Received January 15, 1997; revised February 1, 1998.     

The Convex Hull of Rational Plane Curves

We present an algorithm that computes the convex hull of multiple rational curves in the plane. The problem is reformulated as one of finding the zero-sets of polynomial equations in one or two variables; using these zero-sets we characterize curve segments that belong to the boundary of the convex hull. We also present a preprocessing step that can eliminate many redundant curve segments.     

Finding Maximum Edge Bicliques in Convex Bipartite Graphs

A bipartite graph G=(A,B,E) is convex on B if there exists an ordering of the vertices of B such that for any vertex v??A, vertices adjacent to v are consecutive in?B. A complete bipartite subgraph of a graph G is called a biclique of G. Motivated by an application to analyzing DNA microarray data, we study the problem of finding maximum edge bicliques in convex bipartite graphs. Given a bipartite graph G=(A,B,E) which is convex on B, we present a new algorithm that computes a maximum edge biclique of G in O(nlog?³ nlog?log?n) time and O(n) space, where n=|A|. This improves the current O(n ²) time bound available for the problem. We also show that for two special subclasses of convex bipartite graphs, namely for biconvex graphs and bipartite permutation graphs, a maximum edge biclique can be computed in O(n??(n)) and O(n) time, respectively, where n=min?(|A|,|B|) and ??(n) is the slowly growing inverse of the Ackermann function.     

Algorithms for Maximum Independent Set in Convex Bipartite Graphs

A bipartite graph G=(V,W,E) is convex if there exists an ordering of the vertices of W such that, for each v∈V, the neighbors of v are consecutive in W. We describe both a sequential and a BSP/CGM algorithm to find a maximum independent set in a convex bipartite graph. The sequential algorithm improves over the running time of the previously known algorithm and the BSP/CGM algorithm is a parallel version of the sequential one. The complexity of the algorithms does not depend on |W|. This work was supported by FAPESP (Proc. 98/06327-0). The first author was also supported by FAPESP (Proc. 96/04505–2), and CNPq/MCT/FINEP (PRONEX project 107/97).     

Linear-Time Algorithm for the Paired-Domination Problem in Convex Bipartite Graphs

A bipartite graph G=(U,W,E) with vertex set V=U∪W is convex if there exists an ordering of the vertices of W such that for each u∈U, the neighbors of u are consecutive in W. A compact representation of a convex bipartite graph for specifying such an ordering can be computed in O(|V|+|E|) time. The paired-domination problem on bipartite graphs has been shown to be NP-complete. The complexity of the paired-domination problem on convex bipartite graphs has remained unknown. In this paper, we present an O(|V|) time algorithm to solve the paired-domination problem on convex bipartite graphs given a compact representation. As a byproduct, we show that our algorithm can be directly applied to solve the total domination problem on convex bipartite graphs in the same time bound.     

建筑图三维重建方法

随着AutoCAD等软件的出现和广泛使用,建筑设计人员已经越来越多的使用计算机来设计建筑工程图.但是后期的计算、放样等过程还依赖于人工读图,效率低下.于是建筑工程图的三维重建技术成为我们研究的主要内容.我们对建筑图三维重建方面做了一些研究和探讨.主要有以下内容:采用了分步骤、分层次的自动识别方法,高效的建立构件间的全局关系;分析了国内平面表达法建筑制图新标准中仍存在的建筑物三维信息描述的分散性和多样性,采用工程图的信息提取和整合方法,完成了整体三维重建.     

Schnyder Decompositions for Regular Plane Graphs and Application to Drawing

Schnyder woods are decompositions of simple triangulations into three edge-disjoint spanning trees crossing each other in a specific way. In this article, we generalize the definition of Schnyder woods to d-angulations (plane graphs with faces of degree d) for all d≥3. A Schnyder decomposition is a set of d spanning forests crossing each other in a specific way, and such that each internal edge is part of exactly d?2 of the spanning forests. We show that a Schnyder decomposition exists if and only if the girth of the d-angulation is d. As in the case of Schnyder woods (d=3), there are alternative formulations in terms of orientations (“fractional” orientations when d≥5) and in terms of corner-labellings. Moreover, the set of Schnyder decompositions of a fixed d-angulation of girth d has a natural structure of distributive lattice. We also study the dual of Schnyder decompositions which are defined on d-regular plane graphs of mincut d with a distinguished vertex v ^?: these are sets of d spanning trees rooted at v ^? crossing each other in a specific way and such that each edge not incident to v ^? is used by two trees in opposite directions. Additionally, for even values of d, we show that a subclass of Schnyder decompositions, which are called even, enjoy additional properties that yield a reduced formulation; in the case d=4, these correspond to well-studied structures on simple quadrangulations (2-orientations and partitions into 2 spanning trees). In the case d=4, we obtain straight-line and orthogonal planar drawing algorithms by using the dual of even Schnyder decompositions. For a 4-regular plane graph G of mincut 4 with a distinguished vertex v ^? and n?1 other vertices, our algorithms places the vertices of Gv ^? on a (n?2)×(n?2) grid according to a permutation pattern, and in the orthogonal drawing each of the 2n?4 edges of Gv ^? has exactly one bend. The vertex v ^? can be embedded at the cost of 3 additional rows and columns, and 8 additional bends. We also describe a further compaction step for the drawing algorithms and show that the obtained grid-size is strongly concentrated around 25n/32×25n/32 for a uniformly random instance with n vertices.     

基于有向图的分布式连续时间非光滑耦合约束凸优化分析

研究一类分布式优化问题, 其目标是在满足耦合不等式约束和局部可行集约束的情况下使非光滑全局代价函数值最小. 首先, 对原有的分布式连续时间投影算法进行拓展, 结合线性代数理论分析, 设计一个适用于强连通加权平衡有向通信网络拓扑图的算法. 其次, 在局部代价函数和耦合不等式约束函数是非光滑凸函数的假设条件下, 利用Moreau-Yosida函数正则化使目标函数和约束函数近似光滑可微. 然后, 根据强连通加权平衡有向图的分布式连续时间投影算法构造李雅普诺夫函数, 证明该算法下的平衡解是分布式优化问题最优解, 并对算法进行收敛性分析. 最后, 通过数值仿真验证算法的有效性.     

A Linear-Time Algorithm for Star-Shaped Drawings of Planar Graphs with the Minimum Number of Concave Corners

A star-shaped drawing of a graph is a straight-line drawing such that each inner facial cycle is drawn as a star-shaped polygon, and the outer facial cycle is drawn as a convex polygon. In this paper, we consider the problem of finding a star-shaped drawing of a biconnected planar graph with the minimum number of concave corners. We first show new structural properties of planar graphs to derive a lower bound on the number of concave corners. Based on the lower bound, we prove that the problem can be solved in linear time by presenting a linear-time algorithm for finding a best plane embedding of a biconnected planar graph with the minimum number of concave corners. This is in spite of the fact that a biconnected planar graph may have an exponential number of different plane embeddings.     

A Total Order Heuristic-Based Convex Hull Algorithm for Points in the Plane

Computing the convex hull of a set of points is a fundamental operation in many research fields, including geometric computing, computer graphics, computer vision, robotics, and so forth. This problem is particularly challenging when the number of points goes beyond some millions. In this article, we describe a very fast algorithm that copes with millions of points in a short period of time without using any kind of parallel computing. This has been made possible because the algorithm reduces to a sorting problem of the input point set, what dramatically minimizes the geometric computations (e.g., angles, distances, and so forth) that are typical in other algorithms. When compared with popular convex hull algorithms (namely, Graham’s scan, Andrew’s monotone chain, Jarvis’ gift wrapping, Chan’s, and Quickhull), our algorithm is capable of generating the convex hull of a point set in the plane much faster than those five algorithms without penalties in memory space.     

Continuous-time Distributed Heavy-ball Algorithm for Distributed Convex Optimization over Undirected and Directed Graphs

Machine Intelligence Research - This paper proposes second-order distributed algorithms over multi-agent networks to solve the convex optimization problem by utilizing the gradient tracking...     

基于区域正交化分割的平面点集凸包算法EI北大核心CSCD

为解决实际工程应用中具有超大规模的平面点集的凸包计算问题,提出了一种基于点集所在区域正交化分割的新算法.利用点集几何结构的部分极点对平面点集进行正交化分割,以获取不相干的点集子集簇,再对所有点集子集分别计算其凸包极点,最后合并极点得到凸包点集.在不同层级的正交化分割过程中,根据已知极点的信息,逐层舍去对于凸包极点生成没有贡献的无效点,进而提高算法运行效率.在与目前常用凸包算法的对比实验中,该算法处理超大规模的平面点集时稳定性高且速度更快.