Dissertations / Theses on the topic 'Graph algorithms'
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1
Zhou, Hang. "Graph algorithms : network inference and planar graph optimization." Thesis, Paris, Ecole normale supérieure, 2015. http://www.theses.fr/2015ENSU0016/document.
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Cette thèse porte sur deux sujets d’algorithmique des graphes. Le premier sujet est l’inférence de réseaux. Quelle est la complexité pour déterminer un graphe inconnu à partir de requêtes de plus court chemin entre ses sommets ? Nous supposons que le graphe est de degré borné. Dans le problème de reconstruction, le but est de reconstruire le graphe ; tandis que dans le problème de vérification, le but est de vérifier qu’un graphe donné est correct. Nous développons des algorithmes probabilistes utilisant une décomposition en cellules de Voronoi. Ensuite, nous analysons des algorithmes de type glouton, et montrons qu’ils sont quasi-optimaux. Nous étudions aussi ces problèmes sur des familles particulières de graphes, démontrons des bornes inférieures, et étudions la reconstruction approximative. Le deuxième sujet est l’étude de deux problèmes d’optimisation sur les graphes planaires. Dans le problème de classification par corrélations, l’entrée est un graphe pondéré, où chaque arête a une étiquette h+i ou h-i, indiquant si ses extrémités sont ou non dans la même catégorie. Le but est de trouver une partition des sommets en catégories qui respecte au mieux les étiquettes. Dans le problème d’augmentation 2-arête-connexe, l’entrée est un graphe pondéré et un sous-ensemble R des arêtes. Le but est de trouver un sous-ensemble S des arêtes de poids minimum, tel que pour chaque arête de R, ses extrémités sont dans une composante 2-arête-connexe de l’union de R et S. Pour les graphes planaires, nous réduisons le premier problème au deuxième et montrons que les deux problèmes, bien que NP-durs, ont un schéma d’approximation en temps polynomial. Nous utilisons la technique récente de décomposition en briquesThis thesis focuses on two topics of graph algorithms. The first topic is network inference. How efficiently can we find an unknown graph using shortest path queries between its vertices? We assume that the graph has bounded degree. In the reconstruction problem, the goal is to find the graph; and in the verification problem, the goal is to check whether a given graph is correct. We provide randomized algorithms based on a Voronoi cell decomposition. Next, we analyze greedy algorithms, and show that they are near-optimal. We also study the problems on special graph classes, prove lower bounds, and study the approximate reconstruction. The second topic is optimization in planar graphs. We study two problems. In the correlation clustering problem, the input is a weighted graph, where every edge has a label of h+i or h−i, indicating whether its endpoints are in the same category or in different categories. The goal is to find a partition of the vertices into categories that tries to respect the labels. In the two-edge-connected augmentation problem, the input is a weighted graph and a subset R of edges. The goal is to produce a minimum-weight subset S of edges, such that for every edge in R, its endpoints are two-edge-connected in the union of R and S. For planar graphs, we reduce correlation clustering to two-edge-connected augmentation, and show that both problems, although they are NP-hard, have a polynomial-time approximation scheme. We build on the brick decomposition technique developed recently
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Bui, Thang Nguyen. "Graph bisection algorithms." Thesis, Massachusetts Institute of Technology, 1986. http://hdl.handle.net/1721.1/77680.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1986.MICROFICHE COPY AVAILABLE IN ARCHIVES AND ENGINEERING.
Bibliography: leaves 64-66.
by Thang Nguyen Bui.
Ph.D.
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Larsson, Patrik. "Analyzing and adapting graph algorithms for large persistent graphs." Thesis, Linköping University, Department of Computer and Information Science, 2008. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-15422.
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In this work, the graph database Neo4j developed by Neo Technology is presented together with some of it's functionality when it comes to accessing data as a graph. This type of data access brings the possibility to implement common graph algorithms on top of Neo4j. Examples of such algorithms are presented together with their theoretical backgrounds. These are mainly algorithms for finding shortest paths and algorithms for different graph measures such as centrality measures. The implementations that have been made are presented, as well as complexity analysis and the performance measures performed on them. The conclusions include that Neo4j is well suited for these types of implementations.
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Profiti, Giuseppe <1980>. "Graph algorithms for bioinformatics." Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2015. http://amsdottorato.unibo.it/6914/1/profiti_giuseppe_tesi.pdf.
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Biological data are inherently interconnected: protein sequences are connected to their annotations, the annotations are structured into ontologies, and so on. While protein-protein interactions are already represented by graphs, in this work I am presenting how a graph structure can be used to enrich the annotation of protein sequences thanks to algorithms that analyze the graph topology. We also describe a novel solution to restrict the data generation needed for building such a graph, thanks to constraints on the data and dynamic programming. The proposed algorithm ideally improves the generation time by a factor of 5. The graph representation is then exploited to build a comprehensive database, thanks to the rising technology of graph databases. While graph databases are widely used for other kind of data, from Twitter tweets to recommendation systems, their application to bioinformatics is new. A graph database is proposed, with a structure that can be easily expanded and queried.
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Profiti, Giuseppe <1980>. "Graph algorithms for bioinformatics." Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2015. http://amsdottorato.unibo.it/6914/.
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Biological data are inherently interconnected: protein sequences are connected to their annotations, the annotations are structured into ontologies, and so on. While protein-protein interactions are already represented by graphs, in this work I am presenting how a graph structure can be used to enrich the annotation of protein sequences thanks to algorithms that analyze the graph topology. We also describe a novel solution to restrict the data generation needed for building such a graph, thanks to constraints on the data and dynamic programming. The proposed algorithm ideally improves the generation time by a factor of 5. The graph representation is then exploited to build a comprehensive database, thanks to the rising technology of graph databases. While graph databases are widely used for other kind of data, from Twitter tweets to recommendation systems, their application to bioinformatics is new. A graph database is proposed, with a structure that can be easily expanded and queried.
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Bessy, Stéphane. "Some problems in graph theory and graphs algorithmic theory." Habilitation à diriger des recherches, Université Montpellier II - Sciences et Techniques du Languedoc, 2012. http://tel.archives-ouvertes.fr/tel-00806716.
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This document is a long abstract of my research work, concerning graph theory and algorithms on graphs. It summarizes some results, gives ideas of the proof for some of them and presents the context of the different topics together with some interesting open questions connected to them The first part precises the notations used in the rest of the paper; the second part deals with some problems on cycles in digraphs; the third part is an overview of two graph coloring problems and one problem on structures in colored graphs; finally the fourth part focus on some results in algorithmic graph theory, mainly in parametrized complexity.
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Stewart, Anthony Graham. "Graph algorithms and complexity aspects on special graph classes." Thesis, Durham University, 2017. http://etheses.dur.ac.uk/12144/.
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Graphs are a very flexible tool within mathematics, as such, numerous problems can be solved by formulating them as an instance of a graph. As a result, however, some of the structures found in real world problems may be lost in a more general graph. An example of this is the 4-Colouring problem which, as a graph problem, is NP-complete. However, when a map is converted into a graph, we observe that this graph has structural properties, namely being (K_5, K_{3,3})-minor-free which can be exploited and as such there exist algorithms which can find 4-colourings of maps in polynomial time. This thesis looks at problems which are NP-complete in general and determines the complexity of the problem when various restrictions are placed on the input, both for the purpose of finding tractable solutions for inputs which have certain structures, and to increase our understanding of the point at which a problem becomes NP-complete. This thesis looks at four problems over four chapters, the first being Parallel Knock-Out. This chapter will show that Parallel Knock-Out can be solved in O(n+m) time on P_4-free graphs, also known as cographs, however, remains hard on split graphs, a subclass of P_5-free graphs. From this a dichotomy is shown on $P_k$-free graphs for any fixed integer $k$. The second chapter looks at Minimal Disconnected Cut. Along with some smaller results, the main result in this chapter is another dichotomy theorem which states that Minimal Disconnected Cut is polynomial time solvable for 3-connected planar graphs but NP-hard for 2-connected planar graphs. The third chapter looks at Square Root. Whilst a number of results were found, the work in this thesis focuses on the Square Root problem when restricted to some classes of graphs with low clique number. The final chapter looks at Surjective H-Colouring. This chapter shows that Surjective H-Colouring is NP-complete, for any fixed, non-loop connected graph H with two reflexive vertices and for any fixed graph H’ which can be obtained from H by replacing vertices with true twins. This result enabled us to determine the complexity of Surjective H-Colouring on all fixed graphs H of size at most 4.
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Freeth, S. A. "Compression methods for graph algorithms." Thesis, University of Canterbury. Computer Science, 1985. http://hdl.handle.net/10092/9568.
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Two compression methods for representing graphs are presented, in conjunction with algorithms applying these methods. A decomposition technique for networks that can be generated in O(m) time is presented. The components of the decomposition and the shortest path matrix of the compressed network can be used to find the shortest path between any pair of vertices in the original network in linear time.
A compression method for boolean matrices and a method for applying the compression to boolean matrix multiplication is developed. The algorithms have an expected running time of O(n²*log ₂n). From this compression method a simple heuristic that may be applied to any algorithm for boolean matrix multiplication has been developed. This heuristic will improve the average running time of boolean matrix multiplication algorithms.
An order of magnitude analysis of the results published by Loukakis and Tsouris [1981], on the efficiency of algorithms for finding all maximal independent sets of a graph has been performed. This analysis showed that their conclusions, which are based on a direct comparison of the running times of the algorithms, do not take into account implementation factors.
An average constant factor improvement is developed for the algorithm of Tsukiyama, Ide, Ariyoshi and Shirakawa [1977] for finding all maximal independent sets of a graph.
Analysis of the running time results from the algorithm comparisons presented in this thesis show that the Bron-Kerbosch algorithm has the smallest rate of increase in running time as the size of the graphs increase.
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Ren, Chenghui, and 任成會. "Algorithms for evolving graph analysis." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2014. http://hdl.handle.net/10722/197105.
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In many applications, entities and their relationships are represented by graphs. Examples include social networks (users and friendship), the WWW (web pages and hyperlinks) and bibliographic networks (authors and co-authorship). In a dynamic world, information changes and so the graphs representing the information evolve with time. For example, a Facebook link between two friends is established, or a hyperlink is added to a web page. We propose that historical graph-structured data be archived for analytical processing. We call a historical
evolving graph sequence an EGS.
We study the problem of efficient query processing on an EGS, which finds many applications that lead to interesting evolving graph analysis. To solve the problem, we propose a solution framework called FVF and a cluster-based LU decomposition algorithm called CLUDE, which can evaluate queries efficiently to support EGS analysis.
The Find-Verify-and-Fix (FVF) framework applies to a wide range of queries. We demonstrate how some important graph measures, including shortest-path distance, closeness centrality and graph centrality, can be efficiently computed from EGSs using FVF. Since an EGS generally contains numerous large graphs, we also discuss several compact storage models that support our FVF framework. Through extensive experiments on both real and synthetic datasets, we show that our FVF framework is highly efficient in EGS query processing.
A graph can be conveniently modeled by a matrix from which various quantitative measures are derived like PageRank and SALSA and Personalized PageRank and Random Walk with Restart. To compute these measures, linear systems of the form Ax = b, where A is a matrix that captures a graph's structure, need to be solved. To facilitate solving the linear system, the matrix A is often decomposed into two triangular matrices (L and U). In a dynamic world, the graph that models it changes with time and thus is the matrix A that represents the graph. We consider a sequence of evolving graphs and its associated sequence of evolving matrices. We study how LU-decomposition should be done over the sequence so that (1) the decomposition is efficient and (2) the resulting LU matrices best preserve the sparsity of the matrices A's (i.e., the number of extra non-zero entries introduced in L and U are minimized). We propose a cluster-based algorithm CLUDE for solving the problem. Through an experimental study, we show that CLUDE is about an order of magnitude faster than the traditional incremental update algorithm. The number of extra non-zero entries introduced by CLUDE is also about an order of magnitude fewer than that of the traditional algorithm. CLUDE is thus an efficient algorithm for LU decomposition that produces high-quality LU matrices over an evolving matrix sequence.published_or_final_version
Computer Science
Doctoral
Doctor of Philosophy
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King, David Jonathan. "Functional programming and graph algorithms." Thesis, University of Glasgow, 1996. http://theses.gla.ac.uk/1629/.
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This thesis is an investigation of graph algorithms in the non-strict purely functional language Haskell. Emphasis is placed on the importance of achieving an asymptotic complexity as good as with conventional languages. This is achieved by using the monadic model for including actions on the state. Work on the monadic model was carried out at Glasgow University by Wadler, Peyton Jones, and Launchbury in the early nineties and has opened up many diverse application areas. One area is the ability to express data structures that require sharing. Although graphs are not presented in this style, data structures that graph algorithms use are expressed in this style. Several examples of stateful algorithms are given including union/find for disjoint sets, and the linear time sort binsort. The graph algorithms presented are not new, but are traditional algorithms recast in a functional setting. Examples include strongly connected components, biconnected components, Kruskal's minimum cost spanning tree, and Dijkstra's shortest paths. The presentation is lucid giving more insight than usual. The functional setting allows for complete calculational style correctness proofs - which is demonstrated with many examples. The benefits of using a functional language for expressing graph algorithms are quantified by looking at the issues of execution times, asymptotic complexity, correctness, and clarity, in comparison with traditional approaches. The intention is to be as objective as possible, pointing out both the weaknesses and the strengths of using a functional language.
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He, Dayu. "Algorithms for Graph Drawing Problems." Thesis, State University of New York at Buffalo, 2017. http://pqdtopen.proquest.com/#viewpdf?dispub=10284151.
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A graph G is called planar if it can be drawn on the plan such that no two distinct edges intersect each other but at common endpoints. Such drawing is called a plane embedding of G. A plane graph is a graph with a fixed embedding. A straight-line drawing G of a graph G = (V, E) is a drawing where each vertex of V is drawn as a distinct point on the plane and each edge of G is drawn as a line segment connecting two end vertices. In this thesis, we study a set of planar graph drawing problems.
First, we consider the problem of monotone drawing: A path P in a straight line drawing Γ is monotone if there exists a line l such that the orthogonal projections of the vertices of P on l appear along l in the order they appear in P. We call l a monotone line (or monotone direction) of P. G is called a monotone drawing of G if it contains at least one monotone path Puw between every pair of vertices u,w of G. Monotone drawings were recently introduced by Angelini et al. and represent a new visualization paradigm, and is also closely related to several other important graph drawing problems. As in many graph drawing problems, one of the main concerns of this research is to reduce the drawing size, which is the size of the smallest integer grid such that every graph in the graph class can be drawn in such a grid. We present two approaches for the problem of monotone drawings of trees. Our first approach show that every n-vertex tree T admits a monotone drawing on a grid of size O(n1.205) × O( n1.205) grid. Our second approach further reduces the size of drawing to 12n × 12n, which is asymptotically optimal. Both of our two drawings can be constructed in O(n) time.
We also consider monotone drawings of 3-connected plane graphs. We prove that the classical Schnyder drawing of 3-connected plane graphs is a monotone drawing on a f × f grid, which can be constructed in O(n) time.
Second, we consider the problem of orthogonal drawing. An orthogonal drawing of a plane graph G is a planar drawing of G such that each vertex of G is drawn as a point on the plane, and each edge is drawn as a sequence of horizontal and vertical line segments with no crossings. Orthogonal drawing has attracted much attention due to its various applications in circuit schematics, relationship diagrams, data flow diagrams etc. . Rahman et al. gave a necessary and sufficient condition for a plane graph G of maximum degree 3 to have an orthogonal drawing without bends. An orthogonal drawing D(G) is orthogonally convex if all faces of D(G) are orthogonally convex polygons. Chang et al. gave a necessary and sufficient condition (which strengthens the conditions in the previous result) for a plane graph G of maximum degree 3 to have an orthogonal convex drawing without bends. We further strengthen the results such that if G satisfies the same conditions as in previous papers, it not only has an orthogonally convex drawing, but also a stronger star-shaped orthogonal drawing.
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Mądry, Aleksander. "From graphs to matrices, and back : new techniques for graph algorithms." Thesis, Massachusetts Institute of Technology, 2011. http://hdl.handle.net/1721.1/66014.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2011.Cataloged from PDF version of thesis.
Includes bibliographical references (p. 181-192).
The growing need to deal efficiently with massive computing tasks prompts us to consider the following question: How well can we solve fundamental optimization problems if our algorithms have to run really quickly? The motivation for the research presented in this thesis stems from addressing the above question in the context of algorithmic graph theory. To pursue this direction, we develop a toolkit that combines a diverse set of modern algorithmic techniques, including sparsification, low-stretch spanning trees, the multiplicative-weights-update method, dynamic graph algorithms, fast Laplacian system solvers, and tools of spectral graph theory. Using this toolkit, we obtain improved algorithms for several basic graph problems including: -- The Maximum s-t Flow and Minimum s-t Cut Problems. We develop a new approach to computing (1 - [epsilon])-approximately maximum s-t flow and (1 + [epsilon])-approximately minimum s-t cut in undirected graphs that gives the fastest known algorithms for these tasks. These algorithms are the first ones to improve the long-standing bound of O(n3/2') running time on sparse graphs; -- Multicommodity Flow Problems. We set forth a new method of speeding up the existing approximation algorithms for multicommodity flow problems, and use it to obtain the fastest-known (1 - [epsilon])-approximation algorithms for these problems. These results improve upon the best previously known bounds by a factor of roughly [omega](m/n), and make the resulting running times essentially match the [omega](mn) "flow-decomposition barrier" that is a natural obstacle to all the existing approaches; -- " Undirected (Multi-)Cut-Based Minimization Problems. We develop a general framework for designing fast approximation algorithms for (multi-)cutbased minimization problems in undirected graphs. Applying this framework leads to the first algorithms for several fundamental graph partitioning primitives, such as the (generalized) sparsest cut problem and the balanced separator problem, that run in close to linear time while still providing polylogarithmic approximation guarantees; -- The Asymmetric Traveling Salesman Problem. We design an O( )- approximation algorithm for the classical problem of combinatorial optimization: the asymmetric traveling salesman problem. This is the first asymptotic improvement over the long-standing approximation barrier of e(log n) for this problem; -- Random Spanning Tree Generation. We improve the bound on the time needed to generate an uniform random spanning tree of an undirected graph.
by Aleksander Mądry.
Ph.D.
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Nanongkai, Danupon. "Graph and geometric algorithms on distributed networks and databases." Diss., Georgia Institute of Technology, 2011. http://hdl.handle.net/1853/41056.
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In this thesis, we study the power and limit of algorithms on various models, aiming at applications in distributed networks and databases.
In distributed networks, graph algorithms are fundamental to many applications. We focus on computing random walks which are an important
primitive employed in a wide range of applications but has always been computed naively. We show that a faster solution exists and subsequently
develop faster algorithms by exploiting random walk properties leading to two immediate applications. We also show that this algorithm is optimal.
Our technique in proving a lower bound show the first non-trivial connection between communication complexity and lower bounds of distributed
graph algorithms. We show that this technique has a wide range of applications by proving new lower bounds of many problems. Some of these lower
bounds show that the existing algorithms are tight.
In database searching, we think of the database as a large set of multi-dimensional points stored in a disk and want to help the users to quickly find the most desired point. In this thesis, we develop an algorithm that is significantly faster than previous algorithms both theoretically and experimentally.
The insight is to solve the problem on the streaming model which helps emphasize the benefits of sequential access over random disk access. We also
introduced the randomization technique to the area. The results were complemented with a lower bound. We also initiat a new direction as an attempt to get a better query. We are the first to quantify the output quality using "user satisfaction" which is made possible by borrowing the idea of modeling users by utility functions from game theory and justify our approach through a geometric analysis.
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Pajak, Dominik. "Algorithms for Deterministic Parallel Graph Exploration." Phd thesis, Université Sciences et Technologies - Bordeaux I, 2014. http://tel.archives-ouvertes.fr/tel-01064992.
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Nous étudions dans cette thèse le problème de l'exploration parallèle d'un graphe à l'aide des multiples, synchronisés et mobiles agents. Chaque agent est une entité individuelle qui peut, indépendamment des autres agents, visitez les sommets du graphe ou parcourir ses arêtes. Le but de ensemble des agents est de visiter tous les sommets de graphe. Nous étudions d'abord l'exploration du graphe dans un modèle où chaque agent est équipé de mémoire interne, mais les nœuds n'ont pas de mémoire. Dans ce modèle les agents sont autorisés à communiquer entre eux en échangeant des messages. Nous présentons des algorithmes qui s'exécutent dans un minimum de temps possible pour polynomiale nombre d'agents (polynomiale en nombre de sommets du graphe). Nous étudions aussi quelle est l'impacte de différent méthodes des communications. Nous étudions des algorithmes où les agents peuvent se communiquer à distance arbitraire, mais aussi où communication est possible seulement entre les agents situés dans le même sommet. Dans les deux cas nous présentons des algorithmes efficaces. Nous avons aussi obtenu des limites inférieures qui correspondent bien à la performance des algorithmes. Nous considérons également l'exploration de graphe en supposant que les mouvements des agents sont déterminés par le soi-disant rotor-router mécanisme. Du point de vue d'un sommet fixé, le rotor- router envoie des agents qui visitent les sommet voisins dans un mode round-robin. Nous étudions l'accélération défini comme la proportion entre le pire des cas de l'exploration d'un agent unique et des plusieurs agents. Pour générales graphes, nous montrerons que le gain de vitesse en cas de multi-agent rotor-router est toujours entre fonction logarithmique et linéaire du nombre d'agents. Nous présentons également des résultats optimaux sur l'accélération de multi-agent rotor-router pour cycles, expanseurs, graphes aléatoires, cliques, tores de dimension fixé et une analyse presque optimale pour hypercubes. Finalement nous considérons l'exploration sans collision, où chaque agent doit explorer le graphe de manière indépendante avec la contrainte supplémentaire que deux agents ne peuvent pas occuper le même sommet. Dans le cas où les agents sont donnés le plan de graphe, on présente un algorithme optimal pour les arbres et un algorithme asymptotiquement optimal pour générales graphes. Nous présentons aussi des algorithmes dans le cas de l'exploration sans collision des arbres et des générales graphes dans la situation où les agents ne connaissent pas le graphe. Nous fermons la thèse par des observations finales et une discussion de problèmes ouverts liés dans le domaine de l'exploration des graphes.
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Deel, Troy A. "A statistical study of graph algorithms." Virtual Press, 1985. http://liblink.bsu.edu/uhtbin/catkey/424871.
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The object of this paper is to investigate the behavior of some important graph properties and to statistically analyze the execution times of certain graph are the average degree of a vertex, connectivity of a graph, the existence of Hamilton cycles, Euler tours, and bipartitions in graphs. This study is unique in that it is based on statistical rather than deterministic methods.
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Anderson, Jon K. "Genetic algorithms applied to graph theory." Virtual Press, 1999. http://liblink.bsu.edu/uhtbin/catkey/1136714.
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This thesis proposes two new variations on the genetic algorithm. The first attempts to improve clustering problems by optimizing the structure of a genetic string dynamically during the run of the algorithm. This is done by using a permutation on the allele which is inherited by the next generation. The second is a multiple pool technique which ensures continuing convergence by maintaining unique lineages and merging pools of similar age. These variations will be tested against two well-known graph theory problems, the Traveling Salesman Problem and the Maximum Clique Problem. The results will be analyzed with respect to string rates, child improvement, pool rating resolution, and average string age.Department of Computer Science
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Newman, Alantha. "Algorithms for string and graph layout." Thesis, Massachusetts Institute of Technology, 2004. http://hdl.handle.net/1721.1/28745.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2004.Includes bibliographical references (p. 121-125).
Many graph optimization problems can be viewed as graph layout problems. A layout of a graph is a geometric arrangement of the vertices subject to given constraints. For example, the vertices of a graph can be arranged on a line or a circle, on a two- or three-dimensional lattice, etc. The goal is usually to place all the vertices so as to optimize some specified objective function. We develop combinatorial methods as well as models based on linear and semidefinite programming for graph layout problems. We apply these techniques to some well-known optimization problems. In particular, we give improved approximation algorithms for the string folding problem on the two- and three-dimensional square lattices. This combinatorial graph problem is motivated by the protein folding problem, which is central in computational biology. We then present a new semidefinite programming formulation for the linear ordering problem (also known as the maximum acyclic subgraph problem) and show that it provides an improved bound on the value of an optimal solution for random graphs. This is the first relaxation that improves on the trivial "all edges" bound for random graphs.
by Alantha Newman.
Ph.D.
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Yodpinyanee, Anak. "Sub-linear algorithms for graph problems." Thesis, Massachusetts Institute of Technology, 2018. http://hdl.handle.net/1721.1/120411.
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Thesis: Ph. D., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2018.Cataloged from PDF version of thesis.
Includes bibliographical references (pages 189-199).
In the face of massive data sets, classical algorithmic models, where the algorithm reads the entire input, performs a full computation, then reports the entire output, are rendered infeasible. To handle these data sets, alternative algorithmic models are suggested to solve problems under the restricted, namely sub-linear, resources such as time, memory or randomness. This thesis aims at addressing these limitations on graph problems and combinatorial optimization problems through a number of different models. First, we consider the graph spanner problem in the local computation algorithm (LCA) model. A graph spanner is a subgraph of the input graph that preserves all pairwise distances up to a small multiplicative stretch. Given a query edge from the input graph, the LCA explores a sub-linear portion of the input graph, then decides whether to include this edge in its spanner or not - the answers to all edge queries constitute the output of the LCA. We provide the first LCA constructions for 3 and 5-spanners of general graphs with almost optimal trade-offs between spanner sizes and stretches, and for fixed-stretch spanners of low maximum-degree graphs. Next, we study the set cover problem in the oracle access model. The algorithm accesses a sub-linear portion of the input set system by probing for elements in a set, and for sets containing an element, then computes an approximate minimum set cover: a collection of an approximately-minimum number of sets whose union includes all elements. We provide probe-efficient algorithms for set cover, and complement our results with almost tight lower bound constructions. We further extend our study to the LP-relaxation variants and to the streaming setting, obtaining the first streaming results for the fractional set cover problem. Lastly, we design local-access generators for a collection of fundamental random graph models. We demonstrate how to generate graphs according to the desired probability distribution in an on-the-fly fashion. Our algorithms receive probes about arbitrary parts of the input graph, then construct just enough of the graph to answer these probes, using only polylogarithmic time, additional space and random bits per probe. We also provide the first implementation of random neighbor probes, which is a basic algorithmic building block with applications in various huge graph models.
by Anak Yodpinyanee.
Ph. D.
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Sun, Jiankai. "Directed Graph Analysis: Algorithms and Applications." The Ohio State University, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=osu1565797455907422.
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Ren, Jintong. "Optimization algorithms for graph layout problems." Thesis, Angers, 2020. https://tel.archives-ouvertes.fr/tel-03178385.
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Cette thèse considère deux problèmes de disposition des graphes : le problème de la bande passante cyclique (CBP) et le problème de l’agencement linéaire minimum (MinLA). Le CBP est une extension naturelle du problème de minimisation de la bande passante (BMP) et le MinLA est un problème de somme minimale. Ces problèmes sont largement appliqués dans la vie réelle. Puisqu’ils sont NP-difficile, il est difficile de les résoudre dans le cas général. Par conséquent, cette thèse est consacrée au développement d’algorithmes heuristiques efficaces pour faire face à ces problèmes. Plus précisément, nous introduisons deux algorithmes de recherche locale itétée, un algorithme mémétique avec différents opérateurs de recombinaison pour le CBP et une heuristique de voisinage basée sur un ensemble pour résoudre le MinLA. On montre expérimentalement que pour le CBP, les deux algorithmes de recherche locale itéré pouvaient concurrencer favorablement les méthodes de l’état de l’art, le croisement approprié est identifié pour l’algorithme mémétique. On montre également que pour le MinLA, l’heuristique de voisinage basée sur l’ensemble s’est avérée plus efficace que des algorithmes avec voisinage traditionnel à 2-flipThis thesis considers two graph layout problems: the cyclic bandwidth problem (CBP) and the minimum linear arrangement problem (MinLA). The CBP is a natural extension of the bandwidth minimization problem (BMP) and the MinLA is a min-sum problem. These problems are widely applied in the real life. Since they are NP-hard problems, it is computational difficult to solve them in the general case. Therefore, this thesis is dedicated to developing effective heuristic algorithms to deal with these challenging problems.Specifically, we introduce two iterated local search algorithms, a memetic algorithm with different recombination operators for the CBP and a set based neighborhood heuristic algorithm to solve the MinLA. The two iterated local search algorithms are experimentallydemonstrated to be able to compete favourably with state-of-the-art methods, the feature of a suitable crossover for the memetic algorithm is identified for the CBP and the set based neighborhood heuristic algorithm is proven to be more efficient than the traditional 2-flip neighborhood algorithm
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Neggazi, Brahim. "Self-stabilizing algorithms for graph parameters." Thesis, Lyon 1, 2015. http://www.theses.fr/2015LYO10041/document.
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Le concept d'auto-stabilisation a été introduit par Dijkstra en 1973. Un système distribué est auto-stabilisant s'il peut démarrer de n'importe quelle configuration initiale et retrouver une configuration légitime en un temps fini par lui-même et sans aucune intervention extérieure. La convergence est également garantie lorsque le système est affecté par des fautes transitoires, ce qui en fait une approche élégante, non masquante, pour la tolérance aux pannes. L'auto-stabilisation a été étudiée dans divers domaines des systèmes distribués tels que les problèmes de synchronisation de l'horloge, de la communication et les protocoles de routage. Vu l'importance des paramètres de graphes notamment pour l'organisation et l'optimisation des communications dans les réseaux et les systèmes distribués, plusieurs algorithmes auto-stabilisants pour des paramètres de graphe ont été proposés dans la littérature, tels que les algorithmes autostabilisants permettant de trouver les ensembles dominants minimaux, coloration des graphes, couplage maximal et arbres de recouvrement. Dans cette perspective, nous proposons, dans cette thèse, des algorithmes distribués et autostabilisants pour certains problèmes de graphes bien connus, en particulier pour les décompositions de graphes et les ensembles dominants qui n'ont pas encore été abordés avec le concept de l'autostabilisation. Les quatre problèmes majeurs considérés dans cette thèse sont: partitionnement en triangles, décomposition en p-étoiles, Monitoring des arêtes, fort ensemble dominant et indépendant. Ainsi, le point commun entre ces problèmes, est qu'ils sont tous considérés comme des variantes des problèmes de domination et de couplage dans les graphes et leur traitement se fait d'une manière auto-stabilisanteThe concept of self-stabilization was first introduced by Dijkstra in 1973. A distributed system is self-stabilizing if it can start from any possible configuration and converges to a desired configuration in finite time by itself without using any external intervention. Convergence is also guaranteed when the system is affected by transient faults. This makes self-stabilization an effective approach for non-masking fault-tolerance. The self-stabilization was studied in various fields in distributed systems such as the problems of clock synchronization, communication and routing protocols. Given the importance of graph parameters, especially for organization and communication of networks and distributed systems, several self-stabilizing algorithms for classic graph parameters have been developed in this direction, such as self-stabilizing algorithms for finding minimal dominating sets, coloring, maximal matching, spanning tree and so on. Thence, we propose in this thesis, distributed and self-stabilizing algorithms to some wellknown graphs problems, particularly for graph decompositions and dominating sets problems that have not yet been addressed in a view of self-stabilization. The four major problems considered in this thesis are: the partitioning into triangles, p-star decomposition, edge monitoring set and independent strong dominating set problems. The common point between these four problems is that they are considered as variants of dominating set and matching problems and all propositions deal with the self-stabilization paradigm
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Sun, Wen. "Heuristic Algorithms for Graph Coloring Problems." Thesis, Angers, 2018. http://www.theses.fr/2018ANGE0027/document.
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Cette thèse concerne quatre problèmes de coloration de graphes NPdifficiles, à savoir le problème de coloration (GCP), le problème de coloration équitable (ECP), le problème de coloration des sommets pondérés et le problème de sous-graphe critique (k-VCS). Ces problèmes sont largement étudiés dans la littérature, non seulement pour leur difficulté théorique, mais aussi pour leurs applications réelles dans de nombreux domaines. Étant donné qu'ils appartiennent à la classe de problèmes NP-difficiles, il est difficile de les résoudre dans le cas général de manière exacte. Pour cette raison, cette thèse est consacrée au développement d'approches heuristiques pour aborder ces problèmes complexes. Plus précisément, nous développons un algorithme mémétique de réduction (RMA) pour la coloration des graphes, un algorithme de recherche réalisable et irréalisable (FISA) pour la coloration équitable et un réalisable et irréalisable (AFISA) pour le problème de coloration des sommets pondérés et un algorithme de suppression basé sur le retour en arrière (IBR) pour le problème k-VCS. Tous les algorithmes ont été expérimentalement évalués et comparés aux méthodes de l'état de l'artThis thesis concerns four NP-hard graph coloring problems, namely, graph coloring (GCP), equitable coloring (ECP), weighted vertex coloring (WVCP) and k-vertex-critical subgraphs (k-VCS). These problems are extensively studied in the literature not only for their theoretical intractability, but also for their real-world applications in many domains. Given that they belong to the class of NP-hard problems, it is computationally difficult to solve them exactly in the general case. For this reason, this thesis is devoted to developing effective heuristic approaches to tackle these challenging problems. We develop a reduction memetic algorithm (RMA) for the graph coloring problem, a feasible and infeasible search algorithm (FISA) for the equitable coloring problem, an adaptive feasible and infeasible search algorithm (AFISA) for the weighted vertex coloring problem and an iterated backtrack-based removal (IBR) algorithm for the k-VCS problem. All these algorithms were experimentally evaluated and compared with state-of-the-art methods
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Sivanathan, Gowrishankar. "Sink free orientations in a graph." Diss., Online access via UMI:, 2009.
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Rinke, Sebastian. "Analysis and Adaption of Graph Mapping Algorithms for Regular Graph Topologies." Master's thesis, Universitätsbibliothek Chemnitz, 2009. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200901453.
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The Message Passing Interface (MPI) standard defines virtual topologies that can be applied to systems
of cooperating processes. Among issues regarding a more convenient namespace this may be used to
optimize the placement of MPI processes in order to reduce communication time. That means, the
processes with their main communication paths represent a graph that has to be cost efficiently mapped
onto the graph representing the actual communication network. In this context, this work analyses
and compares state-of-the-art task mapping strategies with respect to running time and their quality
of solutions to the MPI mapping problem. In particular, the focus is on generic strategies that can
be used for arbitrary process/network topologies although, here, the topologies of interest are regular
ones, where the number of processes is greater than the number of processors in the underlying physical
network. Additionally, different measures of mapping quality are discussed and a close correspondence
between the most appropriate, the weighted edge cut, and program execution time is shown. In order
to investigate how mapping quality affects MPI program execution time, some mapping strategies have
been incorporated into Open MPI. Finally, benchmark results prove that optimized process-to-processor
mappings can improve program execution time by up to 60%, compared to the default mapping in many
MPI implementations (linear mapping). The findings in this work can serve as reference not only for MPI
implementors, but also for researchers investigating static process-to-processor mappings, in general.
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Slade, Michael L. "A layout algorithm for hierarchical graphs with constraints /." Online version of thesis, 1994. http://hdl.handle.net/1850/11724.
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Newton, Matthew. "Sequential and parallel algorithms for low-crossing graph drawing." Thesis, Loughborough University, 2007. https://dspace.lboro.ac.uk/2134/12944.
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The one- and two-sided bipartite graph drawing problem alms to find a layout of a bipartite graph, with vertices of the two parts placed on parallel imaginary lines, that has the minimum number of edge-crossings. Vertices of one part are in fixed positions for the one-sided problem, whereas all vertices are free to move along their lines in the two-sided version. Many different heuristics exist for finding approximations to these problems, which are NP-hard. New sequential and parallel methods for producing drawings with low edgecrossings are investigated and compared to existing algorithms, notably Penalty Minimisation and Sifting, the current leaders. For the one-sided problem, new methods that include those based on simple stochastic hillclimbing, simulated annealing and genet.ic algorithms were tested. The new block-crossover genetic algorithm produced very good results with lower crossings than existing methods, although it tended to be slower. However, time was a secondary aim, the priority being to achieve low numbers of crossings. This algorithm can also be seeded with the output of an existing algorithm to improve results; combining with Penalty Minimisation in this way improved both the speed and number of crossings. Four parallel methods for the one-sided problem have been created, although two were abandoned because they gave bad results for even simple graphs. The other two methods, based on stochastic hill-climbing, produced acceptable results in faster times than similar sequential methods. PVM was used as the parallel communication system. Two new heuristics were studied for the two-sided problem, for which the only known existing method is to apply one-sided algorithms iteratively. The first is based on a heuristic for the linear arrangment problem; the second is a method of performing stochastic hill-climbing on two sides. A way of applying anyone-sided algorithm iteratively was also created. The linear arrangement method based on the Koren-Harel multi-scale algorithm achieved the best results, outperforming iterative Barycentre (previously the best method) and iterative Penalty Minimisation. Another area of this work created three new heuristics for the k-planar drawing problem where k > 1. These are the first known practical algorithms to solve this problem. A sequential genetic algorithm based on TimGA is devised to work on k-planar graphs. Two parallel algorithms, one island model and the other a 'mesh' model, are also given. Comparison of results for k = 2 indicate that the parallel island method is better than the other two methods. MPI was used for the parallel communication. Overall, 14 new methods are introduced, of which 10 were developed into working algorithms. For the one-sided bipartite graph drawing problem the new block-crossover genetic algorithm can produce drawings with lower crossings than the current best available algorithms. The parallel methods do not perform as well as the sequential ones, although they generally achieved the same results faster. All of the new two-sided methods worked well; the weighted two-sided swap stochastic hill-climbing method was comparable to the existing best method, iterative Barycentre, and generally produced drawings with lower crossings, although it suffered with needing a good termination condition. The new methods based on the linear arrangement problem consistently produced drawings with lower crossings than iterative Barycentre, although they were nearly always slower. A new parallel algorithm for the k-planar drawing problem, based on the island model, generally created drawings with the lowest edge-crossings, although no algorithms were known to exist to make comparisons.
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Jayachandran, Jayakanth. "Improving resiliency using graph based evolutionary algorithms." Diss., Rolla, Mo. : Missouri University of Science and Technology, 2010. http://scholarsmine.mst.edu/thesis/pdf/Jayachandran_09007dcc807d6ba6.pdf.
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Thesis (M.S.)--Missouri University of Science and Technology, 2010.Vita. The entire thesis text is included in file. Title from title screen of thesis/dissertation PDF file (viewed July 19, 2010) Includes bibliographical references (p. 56-62).
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McConville, Ryan. "Clustering algorithms for large scale graph data." Thesis, Queen's University Belfast, 2017. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.727648.
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Liang, Weifa, and wliang@cs anu edu au. "Designing Efficient Parallel Algorithms for Graph Problems." The Australian National University. Department of Computer Science, 1997. http://thesis.anu.edu.au./public/adt-ANU20010829.114536.
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Graph algorithms are concerned with the algorithmic aspects of solving graph problems. The problems are motivated from and have application to diverse areas of computer science, engineering and other disciplines. Problems arising from these areas of application are good candidates for parallelization since they often have both intense computational needs and stringent response time requirements. Motivated by these concerns, this thesis investigates parallel algorithms for these kinds of graph problems that have at least one of the following properties: the problems involve some type of dynamic updates; the sparsification technique is applicable; or the problems are closely related to communications network issues. The models of parallel computation used in our studies are the Parallel Random Access Machine (PRAM) model and the practical interconnection network models such as meshes and hypercubes.
¶
Consider a communications network which can be represented by a graph G = (V;E), where V is a set of sites (processors), and E is a set of links which are used to connect the sites (processors). In some cases, we also assign weights and/or directions to the edges in E. Associated with this network, there are many problems such as (i) whether the network is k-edge (k-vertex) connected withfixed k; (ii) whether there are k-edge (k-vertex) disjoint paths between u and v for a pair of given vertices u and v after the network is dynamically updated by adding and/or deleting an edge etc; (iii) whether the sites in the network can communicate with each other when some sites and links fail; (iv) identifying the first k edges in the network whose deletion will result in the maximum increase in the routing cost in the resulting network for fixed k; (v) how to augment the network at optimal cost with a given feasible set of weighted edges such that the augmented network is k-edge (k-vertex) connected; (vi) how to route messages through the network efficiently. In this thesis we answer the problems mentioned above by presenting efficient parallel algorithms to solve them. As far as we know, most of the proposed algorithms are the first ones in the parallel setting.
¶
Even though most of the problems concerned in this thesis are related to communications networks, we also study the classic edge-coloring problem. The outstanding difficulty to solve this problem in parallel is that we do not yet know whether or not it is in NC. In this thesis we present an improved parallel algorithm for the problem which needs [bigcircle]([bigtriangleup][superscript 4.5]log [superscript 3] [bigtriangleup] log n + [bigtriangleup][superscript 4] log [superscript 4] n) time using [bigcircle](n[superscript 2][bigtriangleup] + n[bigtriangleup][superscript 3]) processors, where n is the number of vertices and [bigtriangleup] is the maximum vertex degree. Compared with a previously known result on the same model, we improved by an [bigcircle]([bigtriangleup][superscript 1.5]) factor in time. The non-trivial part is to reduce this problem to the edge-coloring update problem. We also generalize this problem to the approximate edge-coloring problem by giving a faster parallel algorithm for the latter case.
¶
Throughout the design and analysis of parallel graph algorithms, we also find a technique called the sparsification technique is very powerful in the design of efficient sequential and parallel algorithms on dense undirected graphs. We believe that this technique may be useful in its own right for guiding the design of efficient sequential and parallel algorithms for problems in other areas as well as in graph theory.
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Ortmann, Mark [Verfasser]. "Combinatorial Algorithms for Graph Sparsification / Mark Ortmann." Konstanz : Bibliothek der Universität Konstanz, 2017. http://d-nb.info/1173616438/34.
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Chong, Ka-wong, and 莊家旺. "Improved algorithms for some classical graph problems." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 1996. http://hub.hku.hk/bib/B31234793.
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Holloway, Nick. "Parallel algorithms in graph theory and algebra." Thesis, University of Warwick, 1994. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.338724.
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Ghaffari, Mohsen. "Improved distributed algorithms for fundamental graph problems." Thesis, Massachusetts Institute of Technology, 2017. http://hdl.handle.net/1721.1/109000.
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Thesis: Ph. D., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2017.Cataloged from PDF version of thesis.
Includes bibliographical references (pages 237-255).
Distributed graph algorithms provide efficient and theoretically sound methods for solving graph problems in distributed settings and more generally for performing distributed computation in networks. These algorithms are applicable in a wide variety of settings, ranging from computer networks to massively parallel computing and beyond. This thesis addresses a number of the central problems of distributed graph algorithms. These problems generally revolve around two of the principal challenges of the area, locality and congestion. The problems include computing maximal independent set, minimum spanning tree, minimum edge cut and minimum vertex cut, graph connectivity decompositions, network information dissemination, minimum-weight connected dominating set, and scheduling distributed protocols. We develop novel techniques, concepts, and tools for these problems, and present algorithms and impossibility results which improve considerably on the state of the art, in several cases resolving or advancing long-standing open problems.
by Mohsen Ghaffari.
Ph. D.
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Nguyen, Huy Ngoc Ph D. Massachusetts Institute of Technology. "Constant time algorithms in sparse graph model." Thesis, Massachusetts Institute of Technology, 2010. http://hdl.handle.net/1721.1/62426.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2010.Cataloged from PDF version of thesis.
Includes bibliographical references (p. 87-91).
We focus on constant-time algorithms for graph problems in bounded degree model. We introduce several techniques to design constant-time approximation algorithms for problems such as Vertex Cover, Maximum Matching, Maximum Weighted Matching, Maximum Independent Set and Set Cover. Some of our techniques can also be applied to design constant-time testers for minor-closed properties. In Chapter 1, we show how to construct a simple oracle that provides query access to a fixed Maximal Independent Set (MIS) of the input graph. More specifically, the oracle gives answers to queries of the form "Is v in the MIS?" for any vertex v in the graph. The oracle runs in constant-time, i.e., the running time for the oracle to answer a single query, is independent to the size of the input graph. Combining this oracle with a simple sampling scheme immediately implies an approximation algorithm for size of the minimum vertex cover. The second technique, called oracle hierarchy, transforms classical approximation algorithms into constant-time algorithms that approximate the size of the optimal solution. The technique is applicable to a certain subclass of algorithms that compute a solution in a constant number of phases. In the transformation, oracle hierarchy uses the MIS oracle to simulates each phase. The problems amenable to these techniques include Maximum Matching, Maximum Weight Matching, Set Cover, and Minimum Dominating Set. For example, for Maximum Matching, we give the first constant-time algorithm that for the class of graphs of degree bounded by d, computes the maximum matching size to within en, for any e > 0, where n is the number of vertices in the graph. The running time of the algorithm is independent of n, and only depends on d and e. In Chapter 2, we introduce a new tool called partitioning oracle which provides query access to a fixed partition of the input graph. In particular, the oracle gives answers to queries of the form "Which part in the fixed partition contains v?" for any vertex v in the graph. We develop methods for constructing a partitioning oracle for any class of bounded-degree graphs with an excluded minor. For any e > 0, our partitioning oracle provides query access to a fixed partition of the input constant-degree minor-free graph, in which every part has size 0(1/ 2 ), and the number of edges removed is at most en. We illustrate the power of this technique by using it to extend and simplify a number of previous approximation and testing results for sparse graphs, as well as to provide new results that were unachievable with existing techniques. For instance: " We give constant-time approximation algorithms for the size of the minimum vertex cover, the minimum dominating set, and the maximum independent set for any class of graphs with an excluded minor. * We show a simple proof that any minor-closed graph property is testable in constant time in the bounded degree model. Finally, in Chapter 3, we construct a more efficient partitioning oracle for graphs with constant treewidth. Although the partitioning oracle in Chapter 2 runs in time independent of the size of the input graph, it has to make 2POlY(1/E)) queries to the input graph to answer a query about the partition. Our new partitioning oracle improves this query complexity to poly(1/E) for graphs with constant treewidth. The new oracle can be used to test constant treewidth in poly(1/E) time in the bounded-degree model. Another application is a poly(1/E)-time algorithm that approximates the maximum matching size, the minimum vertex cover size, and the minimum dominating set size up to an additive en in bounded treewidth graphs.
by Huy Ngoc Nguyen.
Ph.D.
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Hasenplaugh, William Cleaburn. "Parallel algorithms for scheduling data-graph computations." Thesis, Massachusetts Institute of Technology, 2016. http://hdl.handle.net/1721.1/103666.
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Thesis: Ph. D., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2016.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.
Cataloged from student-submitted PDF version of thesis.
Includes bibliographical references (pages 169-182).
A data-graph computation - popularized by such programming systems as Pregel, GraphLab, Galois, Ligra, PowerGraph, and GraphChi - is an algorithm that iteratively performs local updates on the vertices of a graph. During each round of a data-graph computation, a user-supplied update function atomically modifies the data associated with a vertex as a function of the vertex's prior data and that of adjacent vertices. A dynamic data-graph computation updates only an active subset of the vertices during a round, and those updates determine the set of active vertices for the next round. In this thesis, I explore two ways of scheduling deterministic parallel data-graph computations that provide performance guarantees culminating in theoretical contributions to graph theory and practical, high-performance systems. In particular, I describe a system called Prism which processes dynamic and static data-graph computations on arbitrary graphs using a technique called chromatic scheduling. Using a vertex-coloring to identify independent sets of vertices, which may be safely processed in parallel, Prism serializes through the colors and processes the independent sets in parallel, thus executing data-graph computations deterministically and without the use of costly atomic instructions (e.g., Compare-And-Swap). Prism supports dynamic data-graph computations deterministically and work-efficiently through the introduction of multibag and multivector data structures. Prism requires a vertex-coloring, and since graphs are generally not supplied with one, it is necessary to find one as a preprocessing step. Furthermore, the runtime of Prism is linear in the number of colors and thus motivates a study in this thesis of fast parallel coloring algorithms that provide vertex-colorings with few colors in practice. At the core of the analysis of these coloring algorithms lies a new result about the maximum depth of a random priority dag, the dag that results from randomly ordering vertices and directing edges from lower to higher numbered vertices in the order. In particular, when the largest degree [delta] in the graph G = (V,E) is less than ln !V !, I show a tight bound on the longest path: [theta](ln V / ln (e ln V / [delta])) with high probability. When [delta] is greater than ln !V!, the longest path in the dag is simply [theta] (min {[delta], [square root sign]E}) , also with high probability. I also present a system called Laika which processes data-graph computations for the special, but important, case of graphs representing physical simulations. Such graphs typically have vertices with coordinates in 3D space and are connected to other "nearby" vertices. We take advantage of these two properties to execute physical simulations, cast as data-graph computations, that make efficient use of cache resources. I analyze a contrived graph construction - a random cube graph - as a proxy for the mesh graphs that arise in physical simulations: n vertices are uniformly randomly assigned positions in the unit cube and have edges connecting them to any other vertices that are within a distance r = O (V -¹/³) . For such a graph and given a cache sufficiently large to hold M vertices, I improve on previous theory to show that a fraction O(M-¹/³) of edges will connect to vertices not in the cache, whereas previous theory held that this "miss rate" is O(M-¹/⁴). Laika also guarantees linear speedup for any random cube graph G = (V,E) with constant average degree for any number of workers P = O (V= lg² V).
by William Cleaburn Hasenplaugh.
Ph. D.
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Ka-wong, Chong. "Improved algorithms for some classical graph problems /." Hong Kong : University of Hong Kong, 1996. http://sunzi.lib.hku.hk/hkuto/record.jsp?B19668508.
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Soleimanfallah, Arezou. "Fixed-parameter tractable algorithms in graph theory." Thesis, Royal Holloway, University of London, 2010. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.531326.
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CRUCIANI, EMILIO. "Simple Randomized Distributed Algorithms for Graph Clustering." Doctoral thesis, Gran Sasso Science Institute, 2019. http://hdl.handle.net/20.500.12571/9951.
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Label Propagation Algorithms are a class of heuristics for the problem of graph clustering, i.e., the problem of detecting groups of nodes whose connections are dense within each group and sparse between the groups. At the onset, a label is assigned to each node of the graph; then, each node iteratively updates its label according to a function of the labels of its neighbors. Empirical studies show that, after only a few rounds, nodes in the same cluster share the same label while nodes in different clusters have different labels. Although they are widely used in practice given their simplicity, efficiency, and effectiveness, there is no theoretical foundation to explain why such simple algorithms are able to perform such a hard task. The absence of theoretical progress in the analysis of Label Propagation Algorithms is due to the lack of mathematical techniques for handling the interplay between the non-linearity of their update rule and the topology of the underlying graph.
In this thesis we contextualize Label Propagation Algorithms in the framework of computational dynamics, simple dynamical processes on networks whose behavior has been formally characterized on some classes of graphs. The analyses of computational dynamics were mainly focused on graphs with good connectivity properties, such as cliques or expanders, and on the problem of consensus, showing that they naturally converge to a configuration in which all the nodes agree on some value. We move a step forward in this direction by rigorously analyzing two simple dynamics, the 2-Choices dynamics and the Averaging dynamics, reaching a more fine-grained comprehension of their consensus behavior in classes of graphs that exhibit a clustered structure. In particular we formally prove that, with non-negligible probability, the two dynamics quickly bring the graph in a configuration where each cluster reaches an internal consensus on a value that is different among the clusters, and then enters a long metastable phase in which the internal consensus are maintained.
We show how to exploit such metastable behavior to design simple randomized distributed algorithms for graph clustering.
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Nourbakhsh, Farshad <1979>. "Algorithms for graph compression : theory and experiments." Doctoral thesis, Università Ca' Foscari Venezia, 2014. http://hdl.handle.net/10579/5639.
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Graphs and networks are everywhere, from social networks to the World Wide Web. Since the last decade, massive graphs has become the center attention of an intense research activity, both industrial and academic research centers. So these graphs are a new challenge for the storage, geometric visual representation and retrieve information. These information retrieving need a efficient techniques to compress the graph.
Compressing data consists in changing its representation in a way to require fewer bits. Depending on the reversibility of this encoding process we might have a lossy or lossless compression.
In the first part of thesis, we have addressed this problem by a two steps clustering strategy and the solution takes advantage of the strong notion of edge density Regularity introduced by Endre Szemeredi.
In the second chapter, we address the problem of encoding a graph of order n into a graph of order k < n in a way to minimize reconstruction error. This encoding is characterized in terms of a particular factorization of the adjacency matrix of the original graph. The factorization is determined as the solution of a discrete optimization problem, which is for convenience relaxed into a continuous, but equivalent, one. Our formulation does not require to have the full graph, but it can factorize the graph also in the presence of partial information. We propose a multiplicative update rule for the optimization task resembling the ones introduced for nonnegative matrix factorization, and convergence properties are proven. Experiments are conducted to assess the effectiveness of the proposed approach.
Our main contributions are summarized as:
i) We link matrix factorization with graph compression by proposing a factorization that can be used to reduce the order of a graph and can be employed also in the presence of incomplete observations. We show that the same technique can be used to compress a kernel, by retaining a kernel as the reduced representation; Moreover, we consider a general setting, where the observations of the original graph/kernel are incomplete;
ii) We cast the discrete problem of finding the best factorization into a continuous optimization problem for which we formally prove the equivalence between the discrete and continuous formulations;
iii) We provide a novel algorithm to approximately find the proposed factorization, which resembles the NMF algorithm in [57] (under ` L2 divergence) and the Baum-Eagon dynamics [6]. Additionally, we formally prove convergence properties for our algorithm and we believe that this theoretical contribution can be helpful for devising other factorization algorithms working on the domain of stochastic matrices (rather than simply nonnegative matrices);
iv) Finally, we establish a relation between clustering and our graph compression model and show that existing clustering approaches in the literature can be regarded as particular, constrained variants of our matrix factorization.
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Lee, William W. L. (William Wai Lam) Carleton University Dissertation Computer Science. "Tree editing algorithms." Ottawa, 1992.
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Gajewar, Amita Surendra. "Approximate edge 3-coloring of cubic graphs." Thesis, Atlanta, Ga. : Georgia Institute of Technology, 2008. http://hdl.handle.net/1853/29735.
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Thesis (M. S.)--Computing, Georgia Institute of Technology, 2009.Committee Chair: Prof. Richard Lipton; Committee Member: Prof. Dana Randall; Committee Member: Prof. H. Venkateswaran. Part of the SMARTech Electronic Thesis and Dissertation Collection.
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Ediger, David. "Analyzing hybrid architectures for massively parallel graph analysis." Diss., Georgia Institute of Technology, 2013. http://hdl.handle.net/1853/47659.
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The quantity of rich, semi-structured data generated by sensor networks, scientific simulation, business activity, and the Internet grows daily. The objective of this research is to investigate architectural requirements for emerging applications in massive graph analysis. Using emerging hybrid systems, we will map applications to architectures and close the loop between software and hardware design in this application space. Parallel algorithms and specialized machine architectures are necessary to handle the immense size and rate of change of today's graph data. To highlight the impact of this work, we describe a number of relevant application areas ranging from biology to business and cybersecurity. With several proposed architectures for massively parallel graph analysis, we investigate the interplay of hardware, algorithm, data, and programming model through real-world experiments and simulations. We demonstrate techniques for obtaining parallel scaling on multithreaded systems using graph algorithms that are orders of magnitude faster and larger than the state of the art. The outcome of this work is a proposed hybrid architecture for massive-scale analytics that leverages key aspects of data-parallel and highly multithreaded systems. In simulations, the hybrid systems incorporating a mix of multithreaded, shared memory systems and solid state disks performed up to twice as fast as either homogeneous system alone on graphs with as many as 18 trillion edges.
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Warty, Durgesh A. "Development of Graphcards a hypertext system for learning graph theory and graph algorithms." Virtual Press, 1998. http://liblink.bsu.edu/uhtbin/catkey/1101590.
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GraphCards is a research project devoted to the development of a system for learning graph theory and implementing graph algorithms. It contains an information base for learning and referencing graph theory topics, integrated with an experimentation tool set to create and manipulate graphs. Due to the non-linear relationship of the information, its organization is hypertext based. The hypertext system NoteCards 1 is used to develop the application.The contribution of the current project is to complete and improve an existing system by reclassifying and rewriting the textual information into different chunks called "typed cards". This should serve to enhance the organization and make the traversal by the user easier.This project will also contribute to the development of an interface between the Information Base and the Graph Experimentation Tool Set.Department of Computer Science
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Peternek, Fabian Hans Adolf. "Graph compression using graph grammars." Thesis, University of Edinburgh, 2018. http://hdl.handle.net/1842/31094.
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This thesis presents work done on compressed graph representations via hyperedge replacement grammars. It comprises two main parts. Firstly the RePair compression scheme, known for strings and trees, is generalized to graphs using graph grammars. Given an object, the scheme produces a small context-free grammar generating the object (called a “straight-line grammar”). The theoretical foundations of this generalization are presented, followed by a description of a prototype implementation. This implementation is then evaluated on real-world and synthetic graphs. The experiments show that several graphs can be compressed stronger by the new method, than by current state-of-the-art approaches. The second part considers algorithmic questions of straight-line graph grammars. Two algorithms are presented to traverse the graph represented by such a grammar. Both algorithms have advantages and disadvantages: the first one works with any grammar but its runtime per traversal step is dependent on the input grammar. The second algorithm only needs constant time per traversal step, but works for a restricted class of grammars and requires quadratic preprocessing time and space. Finally speed-up algorithms are considered. These are algorithms that can decide specific problems in time depending only on the size of the compressed representation, and might thus be faster than a traditional algorithm would on the decompressed structure. The idea of such algorithms is to reuse computation already done for the rules of the grammar. The possible speed-ups achieved this way is proportional to the compression ratio of the grammar. The main results here are a method to answer “regular path queries”, and to decide whether two grammars generate isomorphic trees.
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Coles, Jonathan. "Algorithms for bounding Folkman numbers /." Online version of thesis, 2005. https://ritdml.rit.edu/dspace/handle/1850/2765.
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Sulong, Ghazali bin. "Algorithms for timetable construction." Thesis, Cardiff University, 1989. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.253664.
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Schwartz, Victor Scott. "Dynamic platform-independent meta-algorithms for graph-partitioning." Thesis, Monterey, Calif. : Springfield, Va. : Naval Postgraduate School ; Available from National Technical Information Service, 1998. http://handle.dtic.mil/100.2/ADA356541.
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Thesis (M.S. in Operations Research) Naval Postgraduate School, September 1998.Thesis advisor(s): Gordon H. Bradley. "September 1998." Includes bibliographical references (p. 99-100). Also available online.
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Preisinger, Timotheus. "Graph-based algorithms for Pareto preference query evaluation." Norderstedt Books on Demand, 2009. http://d-nb.info/1000465993/34.
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Cano, Vila María del Pilar. "Generalized Delaunay triangulations : graph-theoretic properties and algorithms." Doctoral thesis, Universitat Politècnica de Catalunya, 2020. http://hdl.handle.net/10803/669310.
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This thesis studies different generalizations of Delaunay triangulations, both from a combinatorial and algorithmic point of view. The Delaunay triangulation of a point set S, denoted DT(S), has vertex set S. An edge uv is in DT(S) if it satisfies the empty circle property: there exists a circle with u and v on its boundary that does not enclose points of S. Due to different optimization criteria, many generalizations of the DT(S) have been proposed. Several properties are known for DT(S), yet, few are known for its generalizations. The main question we explore is: to what extent can properties of DT(S) be extended for generalized Delaunay graphs?
First, we explore the connectivity of the flip graph of higher order Delaunay triangulations of a point set S in the plane. The order-k flip graph might be disconnected for k = 3, yet, we give upper and lower bounds on the flip distance from one order-k triangulation to another in certain settings.
Later, we show that there exists a length-decreasing sequence of plane spanning trees of S that converges to the minimum spanning tree of S with respect to an arbitrary convex distance function. Each pair of consecutive trees in the sequence is contained in a constrained convex shape Delaunay graph. In addition, we give a linear upper bound and specific bounds when the convex shape is a square.
With focus still on convex distance functions, we study Hamiltonicity in k-order convex shape Delaunay graphs. Depending on the convex shape, we provide several upper bounds for the minimum k for which the k-order convex shape Delaunay graph is always Hamiltonian. In addition, we provide lower bounds when the convex shape is in a set of certain regular polygons.
Finally, we revisit an affine invariant triangulation, which is a special type of convex shape Delaunay triangulation. We show that many properties of the standard Delaunay triangulations carry over to these triangulations. Also, motivated by this affine invariant triangulation, we study different triangulation methods for producing other affine invariant geometric objects.Esta tesis estudia diferentes generalizaciones de la triangulación de Delaunay, tanto desde un punto de vista combinatorio como algorítmico. La triangulación de Delaunay de un conjunto de puntos S, denotada DT(S), tiene como conjunto de vértices a S. Una arista uv está en DT(S) si satisface la propiedad del círculo vacío: existe un círculo con u y v en su frontera que no contiene ningún punto de S en su interior. Debido a distintos criterios de optimización, se han propuesto varias generalizaciones de la DT (S). Hoy en día, se conocen bastantes propiedades de la DT(S), sin embargo, poco se sabe sobre sus generalizaciones. La pregunta principal que exploramos es: ¿Hasta qué punto las propiedades de la DT(S) se pueden extender para generalizaciones de gráficas de Delaunay? Primero, exploramos la conectividad de la gráfica de flips de las triangulaciones de Delaunay de orden alto de un conjunto de puntos S en el plano. La gráfica de flips de triangulaciones de orden k = 3 podría ser disconexa, sin embargo, nosotros damos una cota superior e inferior para la distancia en flips de una triangulación de orden k a alguna otra cuando S cumple con ciertas características. Luego, probamos que existe una secuencia de árboles generadores sin cruces tal que la suma total de la longitud de las aristas con respecto a una distancia convexa arbitraria es decreciente y converge al árbol generador mínimo con respecto a la distancia correspondiente. Cada par de árboles consecutivos en la secuencia se encuentran en una triangulación de Delaunay con restricciones. Adicionalmente, damos una cota superior lineal para la longitud de la secuencia y cotas específicas cuando el conjunto convexo es un cuadrado. Aún concentrados en distancias convexas, estudiamos hamiltonicidad en las gráficas de Delaunay de distancia convexa de k-orden. Dependiendo en la distancia convexa, exhibimos diversas cotas superiores para el mínimo valor de k que satisface que la gráfica de Delaunay de distancia convexa de orden-k es hamiltoniana. También damos cotas inferiores para k cuando el conjunto convexo pertenece a un conjunto de ciertos polígonos regulares. Finalmente, re-visitamos una triangulación afín invariante, la cual es un caso especial de triangulación de Delaunay de distancia convexa. Probamos que muchas propiedades de la triangulación de Delaunay estándar se preservan en estas triangulaciones. Además, motivados por esta triangulación afín invariante, estudiamos diferentes algoritmos que producen otros objetos geométricos afín invariantes.
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Danilenko, Nikita [Verfasser]. "Designing Functional Implementations of Graph Algorithms / Nikita Danilenko." Kiel : Universitätsbibliothek Kiel, 2016. http://d-nb.info/1102933031/34.
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