Layered Graph Research Articles

An SDP approach to multi-level crossing minimization

We present an approach based on semidefinite programs (SDP) to tackle the multi-level crossing minimization problem. We are given a layered graph (i.e., the graph's vertices are assigned to multiple parallel levels) and are asked for an ordering of the nodes on each level such that, when drawing the graph with straight lines, the resulting number of crossings is minimized. Solving this step is crucial in what is probably the most widely used graph drawing scheme, the Sugiyama framework. The problem has received a lot of attention in both the fields of heuristics and exact methods. For a long time, integer linear programming (ILP) approaches were the only exact algorithms applicable, at least for small graphs. Recently, SDP formulations for the special case of two levels were proposed and dominated the ILP for dense instances. In this article, we present a new SDP formulation for the general multi-level version that, for two levels, is even stronger than the aforementioned specialized SDP. As a by-product, we also obtain an SDP-based heuristic, which in practice always gives (near-)optimal solutions. We conduct a large set of experiments, both on randomized and on real-world instances, and compare our approach to a state-of-the-art ILP-based branch-and-cut implementation. The SDP clearly dominates for denser graphs, while the ILP approach is usually faster for sparse instances. However, even for such sparse graphs, the SDP solves more instances to optimality than the ILP. In fact, there is no single instance that the ILP solved that the SDP did not. Overall, our experiments reveal that, for sparse graphs, one should usually try to find an optimal solution with the ILP first. If this approach does not solve the instance to optimality within reasonable time, the SDP still has a good chance to do so. Being able to solve larger real-world instances than reported before, we are also able to evaluate heuristics for this problem. In this article, we do so for the traditional barycenter-heuristic (showing that it leaves a large gap to the true optimum) and the state-of-the-art upward-planarization method (showing that it is usually close to the optimum).

Developing and Evaluating Quilts for the Depiction of Large Layered Graphs

Traditional layered graph depictions such as flow charts are in wide use. Yet as graphs grow more complex, these depictions can become difficult to understand. Quilts are matrix-based depictions for layered graphs designed to address this problem. In this research, we first improve Quilts by developing three design alternatives, and then compare the best of these alternatives to better-known node-link and matrix depictions. A primary weakness in Quilts is their depiction of skip links, links that do not simply connect to a succeeding layer. Therefore in our first study, we compare Quilts using color-only, text-only, and mixed (color and text) skip link depictions, finding that path finding with the color-only depiction is significantly slower and less accurate, and that in certain cases, the mixed depiction offers an advantage over the text-only depiction. In our second study, we compare Quilts using the mixed depiction to node-link diagrams and centered matrices. Overall results show that users can find paths through graphs significantly faster with Quilts (46.6 secs) than with node-link (58.3 secs) or matrix (71.2 secs) diagrams. This speed advantage is still greater in large graphs (e.g. in 200 node graphs, 55.4 secs vs. 71.1 secs for node-link and 84.2 secs for matrix depictions).

Chromatic numbers of layered graphs with a bounded maximal clique

A graph is called n-layered if the set of its vertices is a union of pairwise nonintersecting n-cliques. We estimatechromatic numbers of n-layered graphs without (n + 1)-cliques. Bibliography: 10 titles.

Energy-Efficient Traffic Grooming in WDM Networks With Scheduled Time Traffic

In this paper, we investigate both static and dynamic traffic grooming problems in a wavelength routing network, so as to minimize the total energy consumption of the core network, with the additional consideration of the holding times of the lightpaths and connection requests. In the static case, all connection requests with their setup and tear-down times are known in advance. We formulate an integer linear programming to minimize the energy consumption. In the dynamic case, we adopt a layered graph model called grooming graph and propose a new traffic grooming heuristics called time-aware traffic grooming, which takes the holding time of a new arrival connection request and the remaining holding time of existing lightpaths into consideration. We compare different traffic grooming policies, in terms of the energy efficiency, under various traffic loads. The results provide implications to select the most energy-efficient traffic grooming policies under various scenarios.

Efficient discovery of influential nodes for SIS models in social networks

We address the problem of discovering the influential nodes in a social network under the susceptible/infected/susceptible model that allows multiple activation of the same node, by defining two influence maximization problems: final-time and integral-time. We solve this problem by constructing a layered graph from the original network with each layer added on top as the time proceeds and applying the bond percolation with two effective control strategies: pruning and burnout. We experimentally demonstrate that the proposed method gives much better solutions than the conventional methods that are based solely on the notion of centrality using two real-world networks. The pruning is most effective when searching for a single influential node, but burnout is more powerful in searching for multiple nodes which together are influential. We further show that the computational complexity is much smaller than the naive probabilistic simulation both by theory and experiment. The influential nodes discovered are substantially different from those identified by the centrality measures. We further note that the solutions of the two optimization problems are also substantially different, indicating the importance of distinguishing these two problem characteristics and using the right objective function that best suits the task in hand.

The ring-star problem: A new integer programming formulation and a branch-and-cut algorithm

A ring star in a graph is a subgraph that can be decomposed into a cycle (or ring) and a set of edges with exactly one vertex in the cycle. In the minimum ring-star problem ( mrsp) the cost of a ring star is given by the sum of the costs of its edges, which vary, depending on whether the edge is part of the ring or not. The goal is to find a ring-star spanning subgraph minimizing the sum of all ring and assignment costs. In this paper we show that the mrsp can be reduced to a minimum (constrained) Steiner arborescence problem on a layered graph. This reduction is used to introduce a new integer programming formulation for the mrsp. We prove that the dual bound generated by the linear relaxation of this formulation always dominates the one provided by an early model from the literature. Based on our new formulation, we developed a branch-and-cut algorithm for the mrsp. On the primal side, we devised a grasp heuristic to generate good upper bounds for the problem. Computational tests with these algorithms were conducted on a benchmark of public domain. In these experiments both our exact and heuristics algorithms had excellent performances, noticeably in dealing with instances whose optimal solution has few vertices in the ring. In addition, we also investigate the minimum spanning caterpillar problem ( mscp) which has the same input as the mrsp and admits feasible solutions that can be viewed as ring stars with paths in the place of rings. We present an easy reduction of the mscp to the mrsp, which makes it possible to solve to optimality instances of the former problem too. Experiments carried out with the mscp revealed that our branch-and-cut algorithm is capable to solve to optimality instances with up to 200 vertices in reasonable time.

Layered Graph Matching with Composite Cluster Sampling

This paper presents a framework of layered graph matching for integrating graph partition and matching. The objective is to find an unknown number of corresponding graph structures in two images. We extract discriminative local primitives from both images and construct a candidacy graph whose vertices are matching candidates (i.e., a pair of primitives) and whose edges are either negative for mutual exclusion or positive for mutual consistence. Then we pose layered graph matching as a multicoloring problem on the candidacy graph and solve it using a composite cluster sampling algorithm. This algorithm assigns some vertices into a number of colors, each being a matched layer, and turns off all the remaining candidates. The algorithm iterates two steps: 1) Sampling the positive and negative edges probabilistically to form a composite cluster, which consists of a few mutually conflicting connected components (CCPs) in different colors and 2) assigning new colors to these CCPs with consistence and exclusion relations maintained, and the assignments are accepted by the Markov Chain Monte Carlo (MCMC) mechanism to preserve detailed balance. This framework demonstrates state-of-the-art performance on several applications, such as multi-object matching with large motion, shape matching and retrieval, and object localization in cluttered background.

Modelling the Hop Constrained Connected Facility Location Problem on Layered Graphs

Abstract Gouveia et al. [Gouveia, L., L. Simonetti and E. Uchoa, Modelling hop-constrained and diameter-constrained minimum spanning tree problems as Steiner tree problems over layered graphs , Mathematical Programming (2010). URL http://dx.doi.org/10.1007/s10107-009-0297-2 ] show how to model the Hop Constrained Minimum Spanning tree problem as Steiner tree problem on a layered graph. Following their ideas, we provide three possibilities to model the Hop Constrained (HC) Connected Facility Location problem (ConFL) as ConFL on layered graphs. We show that on all three layered graphs the respective LP relaxations of two cut based models are of the same quality. In our computational study we compare a compact hop-indexed tree model against the two cut based models on the simplest layered graph. We provide results for instances with up to 1300 nodes and 115000 arcs.

Survivable routing algorithm with mixed shared protection in fault‐tolerant GMPLS multi‐layered optical networks

AbstractIn this paper, we studied survivability in general multi‐protocol label switching (GMPLS) multi‐layered optical networks and proposed a novel mixed shared protection algorithm (MSPA) based on the integrated layered graph (ILG) to tolerate single‐fibre link failure. The ILG is composed of one virtual topology layer (VTL), corresponding to IP/MPLS layer; multiple wavelength‐plane layers (WPLs), corresponding to the wavelength‐division multiplexing (WDM) optical layer; and some virtual links that connect the VTL to each WPL in order to solve the problem on routing, wavelength assignment and low‐speed traffic grooming. For every connection request, the MSPA first computes a single‐hop or multi‐hop label switching path (LSP) pair, involving a working LSP and a physical link‐disjoint backup LSP in the VTL. If the LSP pair is not found in the VTL, the MSPA computes the new LSP pair in the WPL. Conversely, if the new LSP pair is not detected in the WPL, the MSPA computes the hybrid multi‐hop LSP pair in the ILG. In the MSPA, if the two working LSPs are in physical link‐disjoint, their corresponding backup LSPs can share the backup bandwidth. In addition, the working LSPs and the backup LSPs can also share the same mixed of wavelength, in which the different bandwidths can be assigned to different working and backup LSPs. Simulation results show that, compared to previous algorithm, the MSPA can obtain a better bandwidth utilisation ratio (BUR) with a lower blocking probability. Copyright © 2010 John Wiley & Sons, Ltd.

Reassortment Networks for Investigating the Evolution of Segmented Viruses

Many viruses of interest, such as influenza A, have distinct segments in their genome. The evolution of these viruses involves mutation and reassortment, where segments are interchanged between viruses that coinfect a host. Phylogenetic trees can be constructed to investigate the mutation-driven evolution of individual viral segments. However, reassortment events among viral genomes are not well depicted in such bifurcating trees. We propose the concept of reassortment networks to analyze the evolution of segmented viruses. These are layered graphs in which the layers represent evolutionary stages such as a temporal series of seasons in which influenza viruses are isolated. Nodes represent viral isolates and reassortment events between pairs of isolates. Edges represent evolutionary steps, while weights on edges represent edit costs of reassortment and mutation events. Paths represent possible transformation series among viruses. The length of each path is the sum edit cost of the events required to transform one virus into another. In order to analyze tau stages of evolution of n viruses with segments of maximum length m, we first compute the pairwise distances between all corresponding segments of all viruses in O(m2n2) time using dynamic programming. The reassortment network, with O(taun2) nodes, is then constructed using these distances. The ancestors and descendents of a specific virus can be traced via shortest paths in this network, which can be found in O(taun3) time.

Noncommutative Koszul algebras from combinatorial topology

Associated to any uniform finite layered graph Gamma there is a noncommutative graded quadratic algebra A(Gamma) given by a construction due to Gelfand, Retakh, Serconek and Wilson. It is natural to ask when these algebras are Koszul. Unfortunately, a mistake in the literature states that all such algebras are Koszul. That is not the case and the theorem was recently retracted. We analyze the Koszul property of these algebras for two large classes of graphs associated to finite regular CW complexes, X. Our methods are primarily topological. We solve the Koszul problem by introducing new cohomology groups H_X(n,k), generalizing the usual cohomology groups H^n(X). Along with several other results, our methods give a new and primarily topological proof of a result of Serconek and Wilson and of Piontkovski.

J2.5dPathway: A 2.5D Visualization Tool to Display Selected Nodes in Biological Pathways, in Parallel Planes

Abstract The characteristics of metabolic pathways make them particularly amenable to layered graph drawing me-thods. This paper presents a visual Java-based tool for drawing and annotating biological pathways in two- and-a-half dimensions (2.5D) as an alternative to three- dimensional (3D) visualizations. Such visualization allows user to display different groups of clustered nodes, in different parallel planes, and to see a detailed view of a group of objects in focus and its place in the context of the whole system. This tool is an extended version of J2dPathway.Availability: An executable demo version is available to download from Sourceforge.net (http://sourceforge.net/ projects/j3dpathway/), with most of the metabolic path-ways included. A hands-on tutorial (in Korean) is also provided. However, our system still awaits full testing of its entire functionalities for practical usage. The current release lacks in some features, however will constantly updated. For academic purposes, contact neo@ewh-a.ac.kr for source codes, for possible collaboration.Keywords: metabolic pathways, pathway editor, visual-ization

A NOVEL APPROACH TO VERIFY GRAPH SCHEMA-BASED SOFTWARE SYSTEMS

Graph Grammars have recently become more and more popular as a general formal modeling language. Behavioral modeling of dynamic systems and model to model transformations are a few well-known examples in which graphs have proven their usefulness in software engineering. A special type of graph transformation systems is layered graphs. Layered graphs are a suitable formalism for modeling hierarchical systems. However, most of the research so far concentrated on graph transformation systems as a modeling means, without considering the need for suitable analysis tools. In this paper we concentrate on how to analyze these models. We will describe our approach to show how one can verify the designed graph transformation systems. To verify graph transformation systems we use a novel approach: using Bogor model checker to verify graph transformation systems. The AGG-like graph transformation systems are translated to BIR — the input language of Bogor — and Bogor verifies that model against some properties defined by combining LTL and special purpose graph rules. Supporting schema-based and layered graphs characterize our approach among existing solutions for verification of graph transformation systems.

Modeling hop-constrained and diameter-constrained minimum spanning tree problems as Steiner tree problems over layered graphs

The hop-constrained minimum spanning tree problem (HMSTP) is an NP-hard problem arising in the design of centralized telecommunication networks with quality of service constraints. We show that the HMSTP is equivalent to a Steiner tree problem (STP) in an appropriate layered graph. We prove that the directed cut model for the STP defined in the layered graph, dominates the best previously known models for the HMSTP. We also show that the Steiner directed cuts in the extended layered graph space can be viewed as being a stronger version of some previously known HMSTP cuts in the original design space. Moreover, we show that these strengthened cuts can be combined and projected into new families of cuts, including facet defining ones, in the original design space. We also adapt the proposed approach to the diameter-constrained minimum spanning tree problem (DMSTP). Computational results with a branch-and-cut algorithm show that the proposed method is significantly better than previously known methods on both problems.

Automated 3-D Intraretinal Layer Segmentation of Macular Spectral-Domain Optical Coherence Tomography Images

With the introduction of spectral-domain optical coherence tomography (OCT), much larger image datasets are routinely acquired compared to what was possible using the previous generation of time-domain OCT. Thus, the need for 3-D segmentation methods for processing such data is becoming increasingly important. We report a graph-theoretic segmentation method for the simultaneous segmentation of multiple 3-D surfaces that is guaranteed to be optimal with respect to the cost function and that is directly applicable to the segmentation of 3-D spectral OCT image data. We present two extensions to the general layered graph segmentation method: the ability to incorporate varying feasibility constraints and the ability to incorporate true regional information. Appropriate feasibility constraints and cost functions were learned from a training set of 13 spectral-domain OCT images from 13 subjects. After training, our approach was tested on a test set of 28 images from 14 subjects. An overall mean unsigned border positioning error of 5.69+/-2.41 microm was achieved when segmenting seven surfaces (six layers) and using the average of the manual tracings of two ophthalmologists as the reference standard. This result is very comparable to the measured interobserver variability of 5.71+/-1.98 microm.

Towards automated verification of layered graph transformation specifications

Graph transformation systems have recently become more and more popular as a general formal modelling language. It is a suitable formalism for modelling different systems like distributed and complex systems. However, modelling must be complemented with proper analysis capabilities to let the user understand how designed models behave and whether stated requirements are fulfilled and model checking has proven to be a viable solution for this purpose. The authors propose an efficient solution for model checking attributed typed and layered graph transformation systems. Layered graph transformation systems are a powerful formalism to formally model different systems like hierarchical systems. In our proposal, AGG layered graph transformation specifications are translated to Bandera intermediate representation (BIR) - the input language of a Bogor model checker - and then Bogor verifies the model against some interesting properties defined by combining LTL (linear temporal logic) and special graph rules. The experimental results are encouraging and show that in most cases our proposal improves existing approaches, in terms of both performance and expressiveness.

A genetic algorithm for solving virtual source placement problem on WDM networks

As WDM technology matures and multicast applications become increasingly popular, supporting multicast at the WDM layer becomes an important and yet challenging topic. In order to reduce the number of wavelength channels to achieve the multicast effectively, switching nodes with light-splitting and wavelength-converting capabilities denoted as virtual source ( VS) nodes are developed. In this paper, given a WDM network, a positive integer k and a set of multicast requests, the VS placement (VSP) problem on WDM networks is studied; the goal is to determine the locations of the VS nodes, the multicast routing and assigned wavelengths of multicast requests so as to minimize the number of used wavelength channels. Since the VSP problem is a hard problem, a genetic algorithm (GA) is proposed to solve it. In the proposed GA, a binary-bit array is used to represent the locations of the VS nodes on network. For a given locations of VS nodes, three multicast routing methods: core-based tree (CBT), link-disjoint CBT (LDCBT), and layered graph (LG) are proposed and used to construct the shared tree for multicast requests. In the CBT and LDCBT methods, a multicast tree constructing (MTC) algorithm is used to construct the multicast tree of a given multicast, and a segment-based wavelength assignment (SBWA) algorithm is proposed and used to determine the assigned wavelength of the multicast tree. Moreover, in the proposed GA, several crossover and mutation operators are developed and used to generate offspring. Simulation results show that the proposed GA together with LG or CBT multicast method can get better results.

Stacked Graphs – Geometry & Aesthetics

In February 2008, the New York Times published an unusual chart of box office revenues for 7500 movies over 21 years. The chart was based on a similar visualization, developed by the first author, that displayed trends in music listening. This paper describes the design decisions and algorithms behind these graphics, and discusses the reaction on the Web. We suggest that this type of complex layered graph is effective for displaying large data sets to a mass audience. We provide a mathematical analysis of how this layered graph relates to traditional stacked graphs and to techniques such as ThemeRiver, showing how each method is optimizing a different "energy function". Finally, we discuss techniques for coloring and ordering the layers of such graphs. Throughout the paper, we emphasize the interplay between considerations of aesthetics and legibility.

Protecting light forest in survivable WDM mesh networks with sparse light splitting

As multicast applications become popular, provisioning survivable multicast connections in wavelength-division-multiplexing (WDM) networks is an important issue. To reduce the network construction cost, the nodes capable of light splitting are always sparsely placed in a WDM network. In this paper, we study the problem of multicast protection in sparse splitting WDM networks, and propose an efficient protection algorithm called sparse splitting constrained multicast protection (SSMP) algorithm. Differing from previous works, the backup paths derived by SSMP can share wavelength channels with primary tree in sparse splitting WDM networks. To achieve wavelength sharing between primary tree and backup paths, a layered graph model is developed. Simulation results show that SSMP can achieve better performance in terms of average network cost and blocking probability than existing algorithms.

Algebras associated to acyclic directed graphs

We construct and study a class of algebras associated to generalized layered graphs, i.e. directed graphs with a ranking function on their vertices and edges. Each finite acyclic directed graph admits countably many structures of a generalized layered graph. We construct linear bases in such algebras and compute their Hilbert series. Our interest to generalized layered graphs and algebras associated to those graphs is motivated by their relations to factorizations of polynomials over noncommutative rings.