7 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
Abstraction in Model Checking Multi-Agent Systems
This thesis presents existential abstraction techniques for multi-agent systems preserving temporal-epistemic
specifications. Multi-agent systems, defined in the interpreted system frameworks,
are abstracted by collapsing the local states and actions of each agent. The goal of abstraction
is to reduce the state space of the system under investigation in order to cope with the state
explosion problem that impedes the verification of very large state space systems. Theoretical
results show that the resulting abstract system simulates the concrete one. Preservation
and correctness theorems are proved in this thesis. These theorems assure that if a temporal-epistemic
formula holds on the abstract system, then the formula also holds on the concrete
one. These results permit to verify temporal-epistemic formulas in abstract systems instead of
the concrete ones, therefore saving time and space in the verification process.
In order to test the applicability, usefulness, suitability, power and effectiveness of the abstraction
method presented, two different implementations are presented: a tool for data-abstraction
and one for variable-abstraction. The first technique achieves a state space reduction by collapsing
the values of the domains of the system variables. The second technique performs a
reduction on the size of the model by collapsing groups of two or more variables. Therefore, the
abstract system has a reduced number of variables. Each new variable in the abstract system
takes values belonging to a new domain built automatically by the tool. Both implementations
perform abstraction in a fully automatic way. They operate on multi agents models specified
in a formal language, called ISPL (Interpreted System Programming Language). This is the
input language for MCMAS, a model checker for multi-agent systems. The output is an ISPL
file as well (with a reduced state space).
This thesis also presents several suitable temporal-epistemic examples to evaluate both techniques.
The experiments show good results and point to the attractiveness of the temporal-epistemic
abstraction techniques developed in this thesis. In particular, the contributions of
the thesis are the following ones:
âą We produced correctness and preservation theoretical results for existential abstraction.
âą We introduced two algorithms to perform data-abstraction and variable-abstraction on
multi-agent systems.
âą We developed two software toolkits for automatic abstraction on multi-agent scenarios:
one tool performing data-abstraction and the second performing variable-abstraction.
âą We evaluated the methodologies introduced in this thesis by running experiments on
several multi-agent system examples
Algorithmik und KomplexitÀt OBDD-reprÀsentierter Graphen
Ordered Binary Decision Diagrams (OBDDs) werden in vielen praktischen Anwendungsgebieten erfolgreich als Datenstruktur zur kompakten ReprĂ€sentation boolescher Funktionen eingesetzt. Auch sehr groĂe Graphen werden in Bereichen wie CAD und Model Checking oft implizit durch boolesche Funktionen und OBDDs dargestellt. Diese Dissertation behandelt grundlegende graphtheoretische Probleme auf OBDD-reprĂ€sentierten Graphen und lotet die
Möglichkeiten entsprechender OBDD-basierter Algorithmen aus. Zum einen werden neue Algorithmen vorgestellt und ihre Eigenschaften im Hinblick auf das Entwurfsziel sublinearer Heuristiken analysiert. Zum anderen werden Grenzen des Ansatzes durch komplexitÀtstheoretische HÀrteresultate und konkrete untere Schranken aufgezeigt
The complexity of problems on implicitly represented inputs
Abstract. Highly regular data can be represented succinctly by various kinds of implicit data structures. Many problems in P are known to be hard if their input is given as circuit or Ordered Binary Decision Diagram (OBDD). Nevertheless, in practical areas like CAD and Model Checking, symbolic algorithms using functional operations on OBDD-represented data are well-established. Their theoretical analysis has mostly been restricted to the number of functional operations yet. We show that Pcomplete problems have no symbolic algorithms using a polylogarithmic number of functional operations, unless P=NC. Moreover, we complement PSPACE-hardness results for problems on OBDD-represented inputs by fixed-parameter intractability results, where the OBDD width serves as the fixed parameter.
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