Local Boxicity, Local Dimension, and Maximum Degree

In this paper, we focus on two recently introduced parameters in the literature, namely `local boxicity' (a parameter on graphs) and `local dimension' (a parameter on partially ordered sets). We give an `almost linear' upper bound for both the parameters in terms of the maximum degree of a graph (for local dimension we consider the comparability graph of a poset). Further, we give an $O(n\Delta^2)$ time deterministic algorithm to compute a local box representation of dimension at most $3\Delta$ for a claw-free graph, where $n$ and $\Delta$ denote the number of vertices and the maximum degree, respectively, of the graph under consideration. We also prove two other upper bounds for the local boxicity of a graph, one in terms of the number of vertices and the other in terms of the number of edges. Finally, we show that the local boxicity of a graph is upper bounded by its `product dimension'.Comment: 11 page

Zero Divisor Graphs and Poset Decomposition

A graph is associated to any commutative ring R where the vertices are the non-zero zero divisors of R with two vertices adjacent if x · y = 0. The zero-divisor graph has also been studied for various algebraic stuctures such as semigroups and partially ordered sets. In this paper, we will discuss some known results on zero-divisor graphs of posets as well as the concept of compactness as it relates to zero-divisor graphs. We will dicuss equivalence class graphs defined on the elements of various algebraic structures and also the reduced graph defined on the vertices of a compact graph. After introducing and discussing some known results on poset dimension, we will show that poset decomposition can be directly related to the equivalence classes represented in a reduced graph. Using this decomposition, we can build a poset of a compact graph with any dimension in a specified interval. Thus we have a device which gives us the ability to study the dimension of a poset of a zero-divisor graph

Ramsey numbers for partially-ordered sets

We present a refinement of Ramsey numbers by considering graphs with a partial ordering on their vertices. This is a natural extension of the ordered Ramsey numbers. We formalize situations in which we can use arbitrary families of partially-ordered sets to form host graphs for Ramsey problems. We explore connections to well studied Tur\'an-type problems in partially-ordered sets, particularly those in the Boolean lattice. We find a strong difference between Ramsey numbers on the Boolean lattice and ordered Ramsey numbers when the partial ordering on the graphs have large antichains.Comment: 18 pages, 3 figures, 1 tabl

A survey on algorithmic aspects of modular decomposition

The modular decomposition is a technique that applies but is not restricted to graphs. The notion of module naturally appears in the proofs of many graph theoretical theorems. Computing the modular decomposition tree is an important preprocessing step to solve a large number of combinatorial optimization problems. Since the first polynomial time algorithm in the early 70's, the algorithmic of the modular decomposition has known an important development. This paper survey the ideas and techniques that arose from this line of research

Automatic Retrieval of Skeletal Structures of Trees from Terrestrial Laser Scanner Data

Research on forest ecosystems receives high attention, especially nowadays with regard to sustainable management of renewable resources and the climate change. In particular, accurate information on the 3D structure of a tree is important for forest science and bioclimatology, but also in the scope of commercial applications. Conventional methods to measure geometric plant features are labor- and time-intensive. For detailed analysis, trees have to be cut down, which is often undesirable. Here, Terrestrial Laser Scanning (TLS) provides a particularly attractive tool because of its contactless measurement technique. The object geometry is reproduced as a 3D point cloud. The objective of this thesis is the automatic retrieval of the spatial structure of trees from TLS data. We focus on forest scenes with comparably high stand density and with many occlusions resulting from it. The varying level of detail of TLS data poses a big challenge. We present two fully automatic methods to obtain skeletal structures from scanned trees that have complementary properties. First, we explain a method that retrieves the entire tree skeleton from 3D data of co-registered scans. The branching structure is obtained from a voxel space representation by searching paths from branch tips to the trunk. The trunk is determined in advance from the 3D points. The skeleton of a tree is generated as a 3D line graph. Besides 3D coordinates and range, a scan provides 2D indices from the intensity image for each measurement. This is exploited in the second method that processes individual scans. Furthermore, we introduce a novel concept to manage TLS data that facilitated the researchwork. Initially, the range image is segmented into connected components. We describe a procedure to retrieve the boundary of a component that is capable of tracing inner depth discontinuities. A 2D skeleton is generated from the boundary information and used to decompose the component into sub components. A Principal Curve is computed from the 3D point set that is associated with a sub component. The skeletal structure of a connected component is summarized as a set of polylines. Objective evaluation of the results remains an open problem because the task itself is ill-defined: There exists no clear definition of what the true skeleton should be w.r.t. a given point set. Consequently, we are not able to assess the correctness of the methods quantitatively, but have to rely on visual assessment of results and provide a thorough discussion of the particularities of both methods. We present experiment results of both methods. The first method efficiently retrieves full skeletons of trees, which approximate the branching structure. The level of detail is mainly governed by the voxel space and therefore, smaller branches are reproduced inadequately. The second method retrieves partial skeletons of a tree with high reproduction accuracy. The method is sensitive to noise in the boundary, but the results are very promising. There are plenty of possibilities to enhance the method’s robustness. The combination of the strengths of both presented methods needs to be investigated further and may lead to a robust way to obtain complete tree skeletons from TLS data automatically.Die Erforschung des ÖkosystemsWald spielt gerade heutzutage im Hinblick auf den nachhaltigen Umgang mit nachwachsenden Rohstoffen und den Klimawandel eine große Rolle. Insbesondere die exakte Beschreibung der dreidimensionalen Struktur eines Baumes ist wichtig für die Forstwissenschaften und Bioklimatologie, aber auch im Rahmen kommerzieller Anwendungen. Die konventionellen Methoden um geometrische Pflanzenmerkmale zu messen sind arbeitsintensiv und zeitaufwändig. Für eine genaue Analyse müssen Bäume gefällt werden, was oft unerwünscht ist. Hierbei bietet sich das Terrestrische Laserscanning (TLS) als besonders attraktives Werkzeug aufgrund seines kontaktlosen Messprinzips an. Die Objektgeometrie wird als 3D-Punktwolke wiedergegeben. Basierend darauf ist das Ziel der Arbeit die automatische Bestimmung der räumlichen Baumstruktur aus TLS-Daten. Der Fokus liegt dabei auf Waldszenen mit vergleichsweise hoher Bestandesdichte und mit zahlreichen daraus resultierenden Verdeckungen. Die Auswertung dieser TLS-Daten, die einen unterschiedlichen Grad an Detailreichtum aufweisen, stellt eine große Herausforderung dar. Zwei vollautomatische Methoden zur Generierung von Skelettstrukturen von gescannten Bäumen, welche komplementäre Eigenschaften besitzen, werden vorgestellt. Bei der ersten Methode wird das Gesamtskelett eines Baumes aus 3D-Daten von registrierten Scans bestimmt. Die Aststruktur wird von einer Voxelraum-Repräsentation abgeleitet indem Pfade von Astspitzen zum Stamm gesucht werden. Der Stamm wird im Voraus aus den 3D-Punkten rekonstruiert. Das Baumskelett wird als 3D-Liniengraph erzeugt. Für jeden gemessenen Punkt stellt ein Scan neben 3D-Koordinaten und Distanzwerten auch 2D-Indizes zur Verfügung, die sich aus dem Intensitätsbild ergeben. Bei der zweiten Methode, die auf Einzelscans arbeitet, wird dies ausgenutzt. Außerdem wird ein neuartiges Konzept zum Management von TLS-Daten beschrieben, welches die Forschungsarbeit erleichtert hat. Zunächst wird das Tiefenbild in Komponenten aufgeteilt. Es wird eine Prozedur zur Bestimmung von Komponentenkonturen vorgestellt, die in der Lage ist innere Tiefendiskontinuitäten zu verfolgen. Von der Konturinformation wird ein 2D-Skelett generiert, welches benutzt wird um die Komponente in Teilkomponenten zu zerlegen. Von der 3D-Punktmenge, die mit einer Teilkomponente assoziiert ist, wird eine Principal Curve berechnet. Die Skelettstruktur einer Komponente im Tiefenbild wird als Menge von Polylinien zusammengefasst. Die objektive Evaluation der Resultate stellt weiterhin ein ungelöstes Problem dar, weil die Aufgabe selbst nicht klar erfassbar ist: Es existiert keine eindeutige Definition davon was das wahre Skelett in Bezug auf eine gegebene Punktmenge sein sollte. Die Korrektheit der Methoden kann daher nicht quantitativ beschrieben werden. Aus diesem Grund, können die Ergebnisse nur visuell beurteiltwerden. Weiterhinwerden die Charakteristiken beider Methoden eingehend diskutiert. Es werden Experimentresultate beider Methoden vorgestellt. Die erste Methode bestimmt effizient das Skelett eines Baumes, welches die Aststruktur approximiert. Der Detaillierungsgrad wird hauptsächlich durch den Voxelraum bestimmt, weshalb kleinere Äste nicht angemessen reproduziert werden. Die zweite Methode rekonstruiert Teilskelette eines Baums mit hoher Detailtreue. Die Methode reagiert sensibel auf Rauschen in der Kontur, dennoch sind die Ergebnisse vielversprechend. Es gibt eine Vielzahl von Möglichkeiten die Robustheit der Methode zu verbessern. Die Kombination der Stärken von beiden präsentierten Methoden sollte weiter untersucht werden und kann zu einem robusteren Ansatz führen um vollständige Baumskelette automatisch aus TLS-Daten zu generieren

Cluster-Based Radio Resource Management for D2D-Supported Safety-Critical V2X Communications

Deploying direct device-to-device (D2D) links is a promising technology for vehicle-to-X (V2X) applications. However, intracell interference, along with stringent requirements on latency and reliability, are challenging issues. In this paper, we study the radio resource management problem for D2D-based safety-critical V2X communications. We first transform the V2X requirements into the constraints that are computable using slowly varying channel state information only. Secondly, we formulate an optimization problem, taking into account the requirements of both vehicular users (V-UEs) and cellular users (C-UEs), where resource sharing can take place not only between a V-UE and a C-UE but also among different V-UEs. The NP-hardness of the problem is rigorously proved. Moreover, a heuristic algorithm, called Cluster-based Resource block sharing and pOWer allocatioN (CROWN), is proposed to solve this problem. Finally, simulation results indicate promising performance of the CROWN scheme

Sharp Bounds on Davenport-Schinzel Sequences of Every Order

One of the longest-standing open problems in computational geometry is to bound the lower envelope of $n$ univariate functions, each pair of which crosses at most $s$ times, for some fixed $s$. This problem is known to be equivalent to bounding the length of an order-$s$ Davenport-Schinzel sequence, namely a sequence over an $n$-letter alphabet that avoids alternating subsequences of the form $a \cdots b \cdots a \cdots b \cdots$ with length $s+2$. These sequences were introduced by Davenport and Schinzel in 1965 to model a certain problem in differential equations and have since been applied to bounding the running times of geometric algorithms, data structures, and the combinatorial complexity of geometric arrangements. Let $\lambda_s(n)$ be the maximum length of an order-$s$ DS sequence over $n$ letters. What is $\lambda_s$ asymptotically? This question has been answered satisfactorily (by Hart and Sharir, Agarwal, Sharir, and Shor, Klazar, and Nivasch) when $s$ is even or $s\le 3$. However, since the work of Agarwal, Sharir, and Shor in the mid-1980s there has been a persistent gap in our understanding of the odd orders. In this work we effectively close the problem by establishing sharp bounds on Davenport-Schinzel sequences of every order $s$. Our results reveal that, contrary to one's intuition, $\lambda_s(n)$ behaves essentially like $\lambda_{s-1}(n)$ when $s$ is odd. This refutes conjectures due to Alon et al. (2008) and Nivasch (2010).Comment: A 10-page extended abstract will appear in the Proceedings of the Symposium on Computational Geometry, 201

Teak: A Novel Computational And Gui Software Pipeline For Reconstructing Biological Networks, Detecting Activated Biological Subnetworks, And Querying Biological Networks.

As high-throughput gene expression data becomes cheaper and cheaper, researchers are faced with a deluge of data from which biological insights need to be extracted and mined since the rate of data accumulation far exceeds the rate of data analysis. There is a need for computational frameworks to bridge the gap and assist researchers in their tasks. The Topology Enrichment Analysis frameworK (TEAK) is an open source GUI and software pipeline that seeks to be one of many tools that fills in this gap and consists of three major modules. The first module, the Gene Set Cultural Algorithm, de novo infers biological networks from gene sets using the KEGG pathways as prior knowledge. The second and third modules query against the KEGG pathways using molecular profiling data and query graphs, respectively. In particular, the second module, also called TEAK, is a network partitioning module that partitions the KEGG pathways into both linear and nonlinear subpathways. In conjunction with molecular profiling data, the subpathways are ranked and displayed to the user within the TEAK GUI. Using a public microarray yeast data set, previously unreported fitness defects for dpl1 delta and lag1 delta mutants under conditions of nitrogen limitation were found using TEAK. Finally, the third module, the Query Structure Enrichment Analysis framework, is a network query module that allows researchers to query their biological hypotheses in the form of Directed Acyclic Graphs against the KEGG pathways