Tree Spanners for Bipartite Graphs and Probe Interval Graphs

A tree t-spanner T in a graph G is a spanning tree of G such that the distance between every pair of vertices in T is at most t times their distance in G. The tree t-spanner problem asks whether a graph admits a tree t-spanner, given t. We first substantially strengthen the known results for bipartite graphs. We prove that the tree t-spanner problem is NP-complete even for chordal bipartite graphs for t ≥ 5, and every bipartite ATE-free graph has a tree 3-spanner, which can be found in linear time. The previous best known results were NP-completeness for general bipartite graphs, and that every convex graph has a tree 3-spanner. We next focus on the tree t-spanner problem for probe interval graphs and related graph classes. The graph classes were introduced to deal with the physical mapping of DNA. From a graph theoretical point of view, the classes are natural generalizations of interval graphs. We show that these classes are tree 7-spanner admissible, and a tree 7-spanner can be constructed in (m log n) time.     

Collective Tree Spanners in Graphs with Bounded Parameters

In this paper we study collective additive tree spanners for special families of graphs including planar graphs, graphs with bounded genus, graphs with bounded tree-width, graphs with bounded clique-width, and graphs with bounded chordality. We say that a graph G=(V,E) admits a system of μ collective additive tree r -spanners if there is a system $\mathcal{T}(G)In this paper we study collective additive tree spanners for special families of graphs including planar graphs, graphs with bounded genus, graphs with bounded tree-width, graphs with bounded clique-width, and graphs with bounded chordality. We say that a graph G=(V,E) admits a system of μ collective additive tree r -spanners if there is a system T(G)\mathcal{T}(G) of at most μ spanning trees of G such that for any two vertices x,y of G a spanning tree T ? T(G)T\in\mathcal{T}(G) exists such that d _T (x,y)≤d _G (x,y)+r. We describe a general method for constructing a “small” system of collective additive tree r-spanners with small values of r for “well” decomposable graphs, and as a byproduct show (among other results) that any weighted planar graph admits a system of O(?n)O(\sqrt{n}) collective additive tree 0-spanners, any weighted graph with tree-width at most k−1 admits a system of klog ₂ n collective additive tree 0-spanners, any weighted graph with clique-width at most k admits a system of klog _3/2 n collective additive tree (2w)(2\mathsf{w}) -spanners, and any weighted graph with size of largest induced cycle at most c admits a system of log ₂ n collective additive tree (2?c/2?w)(2\lfloor c/2\rfloor\mathsf{w}) -spanners and a system of 4log ₂ n collective additive tree (2(?c/3?+1)w)(2(\lfloor c/3\rfloor +1)\mathsf {w}) -spanners (here, w\mathsf{w} is the maximum edge weight in G). The latter result is refined for weighted weakly chordal graphs: any such graph admits a system of 4log ₂ n collective additive tree (2w)(2\mathsf{w}) -spanners. Furthermore, based on this collection of trees, we derive a compact and efficient routing scheme for those families of graphs.     

Additive Spanners and Distance and Routing Labeling Schemes for Hyperbolic Graphs

δ-Hyperbolic metric spaces have been defined by M. Gromov in 1987 via a simple 4-point condition: for any four points u,v,w,x, the two larger of the distance sums d(u,v)+d(w,x),d(u,w)+d(v,x),d(u,x)+d(v,w) differ by at most?2δ. They play an important role in geometric group theory, geometry of negatively curved spaces, and have recently become of interest in several domains of computer science, including algorithms and networking. In this paper, we study unweighted δ-hyperbolic graphs. Using the Layering Partition technique, we show that every n-vertex δ-hyperbolic graph with δ≥1/2 has an additive O(δlog?n)-spanner with at most O(δn) edges and provide a simpler, in our opinion, and faster construction of distance approximating trees of δ-hyperbolic graphs with an additive error O(δlog?n). The construction of our tree takes only linear time in the size of the input graph. As a consequence, we show that the family of n-vertex δ-hyperbolic graphs with δ≥1/2 admits a routing labeling scheme with O(δlog?² n) bit labels, O(δlog?n) additive stretch and O(log?₂(4δ)) time routing protocol, and a distance labeling scheme with O(log?² n) bit labels, O(δlog?n) additive error and constant time distance decoder.     

A Browser for Large Directed Graphs

The paper describes an experimental graph browsing tool which provides fast interactive browsing on very large graphs. The particular graph for which the program was developed is based on the distributed tile system of a group of workstations but there is no reason why the same techniques should not be used on graphs from other sources.     

Directed Graphs and Substitutions

A substitution naturally determines a directed graph with an ordering of the edges incident at each vertex. We describe a simple method by which any primitive substitution can be modified (without materially changing the bi-infinite fixed points of the substitution) so that points in the substitution minimal shift are in bijective correspondence with one-sided infinite paths on its associated directed graph. Using this correspondence, we show that primitive substitutive sequences in the substitution minimal shift are precisely those sequences represented by eventually periodic paths. We use directed graphs to show that all measures of cylinders in a substitution minimal shift lie in a finite union of geometric sequences, confirming a conjecture of Boshernitzan. Our methods also yield sufficient conditions for a geometric realization of a primitive substitution to be ``almost injective.' Received January 24, 2000, and in final form August 9, 2001. Online publication November 23, 2001.     

On Coloring Unit Disk Graphs

In this paper the coloring problem for unit disk (UD) graphs is considered. UD graphs are the intersection graphs of equal-sized disks in the plane. Colorings of UD graphs arise in the study of channel assignment problems in broadcast networks. Improving on a result of Clark et al. [2] it is shown that the coloring problem for UD graphs remains NP-complete for any fixed number of colors k≥ 3 . Furthermore, a new 3-approximation algorithm for the problem is presented which is based on network flow and matching techniques. Received February 12, 1996; revised October 9, 1996.     

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 $G$.     

Join-Reachability Problems in Directed Graphs

For a given collection \(\mathcal{G}\) of directed graphs we define the join-reachability graph of \(\mathcal{G}\) , denoted by \(\mathcal{J}(\mathcal{G})\) , as the directed graph that, for any pair of vertices u and v, contains a path from u to v if and only if such a path exists in all graphs of  \(\mathcal{G}\) . Our goal is to compute an efficient representation of  \(\mathcal{J}(\mathcal{G})\) . In particular, we consider two versions of this problem. In the explicit version we wish to construct the smallest join-reachability graph for  \(\mathcal{G}\) . In the implicit version we wish to build an efficient data structure, in terms of space and query time, such that we can report fast the set of vertices that reach a query vertex in all graphs of  \(\mathcal{G}\) . This problem is related to the well-studied reachability problem and is motivated by emerging applications of graph-structured databases and graph algorithms. We consider the construction of join-reachability structures for two graphs and develop techniques that can be applied to both the explicit and the implicit problems. First we present optimal and near-optimal structures for paths and trees. Then, based on these results, we provide efficient structures for planar graphs and general directed graphs.     

Tighter Approximation Bounds for Minimum CDS in Unit Disk Graphs

Connected dominating set (CDS) in unit disk graphs has a wide range of applications in wireless ad hoc networks. A number of approximation algorithms for constructing a small CDS in unit disk graphs have been proposed in the literature. The majority of these algorithms follow a general two-phased approach. The first phase constructs a dominating set, and the second phase selects additional nodes to interconnect the nodes in the dominating set. In the performance analyses of these two-phased algorithms, the relation between the independence number α and the connected domination number γ _c of a unit-disk graph plays the key role. The best-known relation between them is a ￡ 3\frac23g_c+1\alpha\leq3\frac{2}{3}\gamma_{c}+1. In this paper, we prove that α≤3.4306γ _c +4.8185. This relation leads to tighter upper bounds on the approximation ratios of two approximation algorithms proposed in the literature.     

Irredundant Algorithms for Traversing Directed Graphs: The Deterministic Case

Problems of testing program systems modeled by deterministic finite automata are considered. The necessary (and, sometimes, sufficient) component of such testing is a traversal of the graph of the automaton state transitions. The main attention is given to the so-called irredundant traversal algorithms (algorithms for traversing unknown graphs, or on-line algorithms), which do not require an a priori knowledge of the total graph structure.     

Distributed Fixed-time Formation-containment Control for Multiple Euler-Lagrange Systems with Directed Graphs

International Journal of Control, Automation and Systems - In this paper, the distributed fixed-time formation-containment (DFTFC) problem of multiple Euler-Lagrange systems with parametric...     

Formation Reorganization by Primitive Operations on Directed Graphs

In this paper, we study the construction and transformation of 2-D persistent graphs. Persistence is a generalization to directed graphs of the undirected notion of rigidity. Both notions are currently being used in various studies on coordination and control of autonomous multiagent formations. In the context of mobile autonomous agent formations, persistence characterizes the efficacy of a directed formation structure with unilateral distance constraints seeking to preserve the shape of the formation. Analogously to the powerful results about Henneberg sequences in minimal rigidity theory, we propose different types of directed graph operations allowing one to sequentially build any minimally persistent graph (i.e., persistent graph with a minimal number of edges for a given number of vertices), each intermediate graph being also minimally persistent. We also consider the more generic problem of obtaining one minimally persistent graph from another, which corresponds to the online reorganization of the sensing and control architecture of an autonomous agent formation. We prove that we can obtain any minimally persistent formation from any other one by a sequence of elementary local operations such that minimal persistence is preserved throughout the reorganization process. Finally, we briefly explore how such transformations can be performed in a decentralized way.     

Performance and Reliability Analysis Using Directed Acyclic Graphs

A graph-based modeling technique has been developed for the stochastic analysis of systems containing concurrency. The basis of the technique is the use of directed acyclic graphs. These graphs represent event-precedence networks where activities may occur serially, probabilistically, or concurrently. When a set of activities occurs concurrently, the condition for the set of activities to complete is that a specified number of the activities must complete. This includes the special cases that one or all of the activities must complete. The cumulative distribution function associated with an activity is assumed to have exponential polynomial form. Further generality is obtained by allowing these distributions to have a mass at the origin and/or at infinity. The distribution function for the time taken to complete the entire graph is computed symbolically in the time parameter t. The technique allows two or more graphs to be combined hierarchically. Applications of the technique to the evaluation of concurrent program execution time and to the reliability analysis of fault-tolerant systems are discussed.     

Average Convergence for Directed & Undirected Graphs in Distributed Systems

Consensus control of multi-agent systems is an innovative paradigm for the development of intelligent distributed systems. This has fascinated numerous scientific groups for their promising applications as they have the freedom to achieve their local and global goals and make their own decisions. Network communication topologies based on graph and matrix theory are widely used in a various real-time applications ranging from software agents to robotics. Therefore, while sustaining the significance of both directed and undirected graphs, this research emphases on the demonstration of a distributed average consensus algorithm. It uses the harmonic mean in the domain of multi-agent systems with directed and undirected graphs under static topologies based on a control input scheme. The proposed agreement protocol focuses on achieving a constant consensus on directional and undirected graphs using the exchange of information between neighbors to update their status values and to be able to calculate the total number of agents that contribute to the communication network at the same time. The proposed method is implemented for the identical networks that are considered under the directional and non-directional communication links. Two different scenarios are simulated and it is concluded that the undirected approach has an advantage over directed graph communication in terms of processing time and the total number of iterations required to achieve convergence. The same network parameters are introduced for both orientations of the communication graphs. In addition, the results of the simulation and the calculation of various matrices are provided at the end to validate the effectiveness of the proposed algorithm to achieve consensus.     

Acyclic Directed Graphs for Automatic Image Analysis of Lung Parenchymal Geometry

We have developed a simple method for encoding pulmonary alveolar geometry suitable for morphometric analysis and modeling. The approach involves the use of 8-bit video microscopic images of lung which are thresholded to produce binary images. Each alveolar wall was thinned to its midline to produce a skeletonized image, We present an algorithm for converting such an image to an acyclic directed graph. Such a representation is quite suitable for mathematical treatment, lending itself to problems involving simulation or modeling. We present as an example application, the measurement of dihedral angles in rat lung using conventional histological sections.     

一种改进的有向无环图支持向量机

构造合理的有向无环图是有向无环图支持向量机亟需解决的一个关键问题。本文提出一种改进的有向无环图支持向量机,根据超球支持向量机获得类的最小包围球,根据该最小包围球计算类与类之间的最短距离,根据该最短距离形成最短距离矩阵,根据该最短距离矩阵来构造有向无环图。实验结果表明,该改进算法较传统有向无环图支持向量机分类精度有明显提高。