41 research outputs found
OBDD-Based Representation of Interval Graphs
A graph can be described by the characteristic function of the
edge set which maps a pair of binary encoded nodes to 1 iff the nodes
are adjacent. Using \emph{Ordered Binary Decision Diagrams} (OBDDs) to store
can lead to a (hopefully) compact representation. Given the OBDD as an
input, symbolic/implicit OBDD-based graph algorithms can solve optimization
problems by mainly using functional operations, e.g. quantification or binary
synthesis. While the OBDD representation size can not be small in general, it
can be provable small for special graph classes and then also lead to fast
algorithms. In this paper, we show that the OBDD size of unit interval graphs
is and the OBDD size of interval graphs is $O(\
| V \ | \log \ | V \ |)\Omega(\ | V \ | \log
\ | V \ |)O(\log \ | V \ |)O(\log^2 \ | V \ |)$ operations and
evaluate the algorithms empirically.Comment: 29 pages, accepted for 39th International Workshop on Graph-Theoretic
Concepts 201
On graph algorithms for large-scale graphs
Die Anforderungen an Algorithmen hat sich in den letzten Jahren grundlegend geĂ€ndert. Die DatengröĂe der zu verarbeitenden Daten wĂ€chst schneller als die zur VerfĂŒgung stehende Rechengeschwindigkeit. Daher sind neue Algorithmen auf sehr groĂen Graphen wie z.B. soziale Netzwerke, Computernetzwerke oder ZustandsĂŒbergangsgraphen entwickelt worden, um das Problem der immer gröĂer werdenden Daten zu bewĂ€ltigen. Diese Arbeit beschĂ€ftigt sich mit zwei Herangehensweisen fĂŒr dieses Problem.
Implizite Algorithmen benutzten eine verlustfreie Kompression der Daten, um die DatengröĂe zu reduzieren, und arbeiten direkt mit den komprimierten Daten, um Optimierungsprobleme zu lösen. Graphen werden hier anhand der charakteristischen Funktion der Kantenmenge dargestellt, welche mit Hilfe von Ordered Binary Decision Diagrams (OBDDs) â eine bekannte Datenstruktur fĂŒr Boolesche Funktionen - reprĂ€sentiert werden können. Wir entwickeln in dieser Arbeit neue Techniken, um die OBDD-GröĂe von Graphen zu bestimmen, und wenden diese Technik fĂŒr mehrere Klassen von Graphen an und erhalten damit (fast) optimale Schranken fĂŒr die OBDD-GröĂen. Kleine Eingabe-OBDDs sind essenziell fĂŒr eine schnelle Verarbeitung, aber wir brauchen auch Algorithmen, die groĂe Zwischenergebnisse wĂ€hrend der AusfĂŒhrung vermeiden. HierfĂŒr entwickeln wir Algorithmen fĂŒr bestimme Graphklassen, die die Kodierung der Knoten ausnutzt, die wir fĂŒr die Resultate der OBDD-GröĂe benutzt haben. ZusĂ€tzlich legen wir die Grundlage fĂŒr die Betrachtung von randomisierten OBDD-basierten Algorithmen, indem wir untersuchen, welche Art von Zufall wir hier verwenden und wie wir damit Algorithmen entwerfen können. Im Zuge dessen geben wir zwei randomisierte Algorithmen an, die ihre entsprechenden deterministischen Algorithmen in einer experimentellen Auswertung ĂŒberlegen sind.
Datenstromalgoritmen sind eine weitere Möglichkeit fĂŒr die Bearbeitung von groĂen Graphen. In diesem Modell wird der Graph anhand eines Datenstroms von KanteneinfĂŒgungen reprĂ€sentiert und den Algorithmen steht nur eine begrenzte Menge von Speicher zur VerfĂŒgung. Lösungen fĂŒr Graphoptimierungsprobleme benötigen hĂ€ufig eine lineare GröĂe bzgl. der Anzahl der Knoten, was eine triviale untere Schranke fĂŒr die Streamingalgorithmen fĂŒr diese Probleme impliziert. Die Berechnung eines Matching ist so ein Beispiel, was aber in letzter Zeit viel Aufmerksamkeit in der Streaming-Community auf sich gezogen hat. Ein Matching ist eine Menge von Kanten, so dass keine zwei Kanten einen gemeinsamen Knoten besitzen. Wenn wir nur an der GröĂe oder dem Gewicht (im Falle von gewichteten Graphen) eines Matching interessiert sind, ist es mögliche diese lineare untere Schranke zu durchbrechen. Wir konzentrieren uns in dieser Arbeit auf dynamische Datenströme, wo auch Kanten gelöscht werden können. Wir reduzieren das Problem, einen SchĂ€tzer fĂŒr ein gewichtsoptimales Matching zu finden, auf das Problem, die GröĂe von Matchings zu approximieren, wobei wir einen kleinen Verlust bzgl. der ApproximationsgĂŒte in Kauf nehmen mĂŒssen. AuĂerdem prĂ€sentieren wir den ersten dynamischen Streamingalgorithmus, der die GröĂe von Matchings in lokal spĂ€rlichen Graphen approximiert. FĂŒr kleine Approximationsfaktoren zeigen wir eine untere Schranke fĂŒr den Platzbedarf von Streamingalgorithmen, die die MatchinggröĂe approximieren.The algorithmic challenges have changed in the last decade due to the rapid growth of the
data set sizes that need to be processed. New types of algorithms on large graphs like social
graphs, computer networks, or state transition graphs have emerged to overcome the problem of ever-increasing data sets. In this thesis, we investigate two approaches to this problem.
Implicit algorithms utilize lossless compression of data to reduce the size and to directly
work on this compressed representation to solve optimization problems. In the case of graphs
we are dealing with the characteristic function of the edge set which can be represented
by Ordered Binary Decision Diagrams (OBDDs), a well-known data structure for Boolean
functions. We develop a new technique to prove upper and lower bounds on the size of OBDDs representing graphs and apply this technique to several graph classes to obtain (almost) optimal bounds. A small input OBDD size is absolutely essential for dealing with large graphs but we also need algorithms that avoid large intermediate results during the computation. For this purpose, we design algorithms for a specific graph class that exploit the encoding of the nodes that we use for the results on the OBDD sizes. In addition, we lay the foundation on the theory of randomization in OBDD-based algorithms by investigating what kind of randomness is feasible and how to design algorithms with it. As a result, we present two randomized algorithms that outperform known deterministic algorithms on many input instances.
Streaming algorithms are another approach for dealing with large graphs. In this model, the
graph is presented one-by-one in a stream of edge insertions or deletions and the algorithms
are permitted to use only a limited amount of memory. Often, the solution to an optimization
problem on graphs can require up to a linear amount of space with respect to the number of
nodes, resulting in a trivial lower bound for the space requirement of any streaming algorithm
for those problems. Computing a matching, i. e., a subset of edges where no two edges are
incident to a common node, is an example which has recently attracted a lot of attention in
the streaming setting. If we are interested in the size (or weight in case of weighted graphs)
of a matching, it is possible to break this linear bound. We focus on so-called dynamic graph
streams where edges can be inserted and deleted and reduce the problem of estimating the
weight of a matching to the problem of estimating the size of a maximum matching with a
small loss in the approximation factor. In addition, we present the first dynamic graph stream
algorithm for estimating the size of a matching in graphs which are locally sparse. On the
negative side, we prove a space lower bound of streaming algorithms that estimate the size of
a maximum matching with a small approximation factor
Pengembangan Algoritma Prim untuk Menentukan Minimum Spanning Forest
Minimum spanning tree (MST) merupakan salah satu permasalahan dalam teori graph. MST dari graph G adalah spanning tree dengan total bobot sisi terkecil pada suatu graph berbobot G yang terhubung. Terdapat dua algoritma klasik di dalam MST yakni algoritma Kruskal dan Prim. Kedua algoritma tersebut dapat menghasilkan sebuah MST. Forest merupakan graph yang terdiri dari beberapa tree. Spanning forest dari graph tak terhubung G merupakan forest yang dibangun dari graph G. Minimum spanning forest (MSF) dari graph G merupakan spanning forest dengan total bobot sisi terkecil atas semua spanning forest pada graph G. Pada hasil, penulis menyajikan sebuah algoritma MSF yang dikembangkan dari algoritma Prim
Probabilistic Inference Using Partitioned Bayesian Networks:Introducing a Compositional Framework
Probability theory offers an intuitive and formally sound way to reason in situations that involve uncertainty. The automation of probabilistic reasoning has many applications such as predicting future events or prognostics, providing decision support, action planning under uncertainty, dealing with multiple uncertain measurements, making a diagnosis, and so forth. Bayesian networks in particular have been used to represent probability distributions that model the various applications of uncertainty reasoning. However, present-day automated reasoning approaches involving uncertainty struggle when models increase in size and complexity to fit real-world applications.In this thesis, we explore and extend a state-of-the-art automated reasoning method, called inference by Weighted Model Counting (WMC), when applied to increasingly complex Bayesian network models. WMC is comprised of two distinct phases: compilation and inference. The computational cost of compilation has limited the applicability of WMC. To overcome this limitation we have proposed theoretical and practical solutions that have been tested extensively in empirical studies using real-world Bayesian network models.We have proposed a weighted variant of OBDDs, called Weighted Positive Binary Decision Diagrams (WPBDD), which in turn is based on the new notion of positive Shannon decomposition. WPBDDs are particularly well suited to represent discrete probabilistic models. The conciseness of WPBDDs leads to a reduction in the cost of probabilistic inference.We have introduced Compositional Weighted Model Counting (CWMC), a language-agnostic framework for probabilistic inference that partitions a Bayesian network into subproblems. These subproblems are then compiled and subsequently composed in order to perform inference. This approach significantly reduces the cost of compilation, yet increases the cost of inference. The best results are obtained by seeking a partitioning that allows compilation to (barely) become feasible, but no more, as compilation cost can be amortized over multiple inference queries.Theoretical concepts have been implemented in a readily available open-source tool called ParaGnosis. Further implementational improvements have been found through parallelism, by exploiting independencies that are introduced by CWMC. The proposed methods combined push the boundaries of WMC, allowing this state-of-the-art method to be used on much larger models than before
A contribution to the evaluation and optimization of networks reliability
LâĂ©valuation de la fiabilitĂ© des rĂ©seaux est un problĂšme combinatoire trĂšs complexe qui nĂ©cessite des moyens de calcul trĂšs puissants. Plusieurs mĂ©thodes ont Ă©tĂ© proposĂ©es dans la littĂ©rature pour apporter des solutions. Certaines ont Ă©tĂ© programmĂ©es dont notamment les mĂ©thodes dâĂ©numĂ©ration des ensembles minimaux et la factorisation, et dâautres sont restĂ©es Ă lâĂ©tat de simples thĂ©ories. Cette thĂšse traite le cas de lâĂ©valuation et lâoptimisation de la fiabilitĂ© des rĂ©seaux. Plusieurs problĂšmes ont Ă©tĂ© abordĂ©s dont notamment la mise au point dâune mĂ©thodologie pour la modĂ©lisation des rĂ©seaux en vue de lâĂ©valuation de leur fiabilitĂ©s. Cette mĂ©thodologie a Ă©tĂ© validĂ©e dans le cadre dâun rĂ©seau de radio communication Ă©tendu implantĂ© rĂ©cemment pour couvrir les besoins de toute la province quĂ©bĂ©coise. Plusieurs algorithmes ont aussi Ă©tĂ© Ă©tablis pour gĂ©nĂ©rer les chemins et les coupes minimales pour un rĂ©seau donnĂ©. La gĂ©nĂ©ration des chemins et des coupes constitue une contribution importante dans le processus dâĂ©valuation et dâoptimisation de la fiabilitĂ©. Ces algorithmes ont permis de traiter de maniĂšre rapide et efficace plusieurs rĂ©seaux tests ainsi que le rĂ©seau de radio communication provincial. Ils ont Ă©tĂ© par la suite exploitĂ©s pour Ă©valuer la fiabilitĂ© grĂące Ă une mĂ©thode basĂ©e sur les diagrammes de dĂ©cision binaire. Plusieurs contributions thĂ©oriques ont aussi permis de mettre en place une solution exacte de la fiabilitĂ© des rĂ©seaux stochastiques imparfaits dans le cadre des mĂ©thodes de factorisation. A partir de cette recherche plusieurs outils ont Ă©tĂ© programmĂ©s pour Ă©valuer et optimiser la fiabilitĂ© des rĂ©seaux. Les rĂ©sultats obtenus montrent clairement un gain significatif en temps dâexĂ©cution et en espace de mĂ©moire utilisĂ© par rapport Ă beaucoup dâautres implĂ©mentations. Mots-clĂ©s: FiabilitĂ©, rĂ©seaux, optimisation, diagrammes de dĂ©cision binaire, ensembles des chemins et coupes minimales, algorithmes, indicateur de Birnbaum, systĂšmes de radio tĂ©lĂ©communication, programmes.Efficient computation of systems reliability is required in many sensitive networks. Despite the increased efficiency of computers and the proliferation of algorithms, the problem of finding good and quickly solutions in the case of large systems remains open. Recently, efficient computation techniques have been recognized as significant advances to solve the problem during a reasonable period of time. However, they are applicable to a special category of networks and more efforts still necessary to generalize a unified method giving exact solution. Assessing the reliability of networks is a very complex combinatorial problem which requires powerful computing resources. Several methods have been proposed in the literature. Some have been implemented including minimal sets enumeration and factoring methods, and others remained as simple theories. This thesis treats the case of networks reliability evaluation and optimization. Several issues were discussed including the development of a methodology for modeling networks and evaluating their reliabilities. This methodology was validated as part of a radio communication network project. In this work, some algorithms have been developed to generate minimal paths and cuts for a given network. The generation of paths and cuts is an important contribution in the process of networks reliability and optimization. These algorithms have been subsequently used to assess reliability by a method based on binary decision diagrams. Several theoretical contributions have been proposed and helped to establish an exact solution of the stochastic networks reliability in which edges and nodes are subject to failure using factoring decomposition theorem. From this research activity, several tools have been implemented and results clearly show a significant gain in time execution and memory space used by comparison to many other implementations. Key-words: Reliability, Networks, optimization, binary decision diagrams, minimal paths set and cuts set, algorithms, Birnbaum performance index, Networks, radio-telecommunication systems, programs
Probabilistic representation and manipulation of Boolean functions using free Boolean diagrams
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1994.Includes bibliographical references (p. 145-149).by Amelia Huimin Shen.Ph.D