Many graphs arising in practice can be represented in a concise and intuitive way that conveys their structure. For example: A planar graph can be represented in the plane with points for vertices and non-crossing curves for edges. An interval graph can be represented on the real line with intervals for vertices and intersection of intervals representing edges. The concept of ``simultaneity'' applies for several types of graphs: the idea is to find representations for two graphs that share some common vertices and edges, and ensure that the common vertices and edges are represented the same way. Simultaneous representation problems arise in any situation where two related graphs should be represented consistently. A main instance is for temporal relationships, where an old graph and a new graph share some common parts. Pairs of related graphs arise in many other situations. For example, two social networks that share some members; two schedules that share some events, overlap graphs of DNA fragments of two similar organisms, circuit graphs of two adjacent layers on a computer chip etc. In this thesis, we study the simultaneous representation problem for several graph classes. For planar graphs the problem is defined as follows. Let G1 and G2 be two graphs sharing some vertices and edges. The simultaneous planar embedding problem asks whether there exist planar embeddings (or drawings) for G1 and G2 such that every vertex shared by the two graphs is mapped to the same point and every shared edge is mapped to the same curve in both embeddings. Over the last few years there has been a lot of work on simultaneous planar embeddings, which have been called `simultaneous embeddings with fixed edges'. A major open question is whether simultaneous planarity for two graphs can be tested in polynomial time. We give a linear-time algorithm for testing the simultaneous planarity of any two graphs that share a 2-connected subgraph. Our algorithm also extends to the case of k planar graphs, where each vertex [edge] is either common to all graphs or belongs to exactly one of them. Next we introduce a new notion of simultaneity for intersection graph classes (interval graphs, chordal graphs etc.) and for comparability graphs. For interval graphs, the problem is defined as follows. Let G1 and G2 be two interval graphs sharing some vertices I and the edges induced by I. G1 and G2 are said to be `simultaneous interval graphs' if there exist interval representations of G1 and G2 such that any vertex of I is assigned to the same interval in both the representations. The `simultaneous representation problem' for interval graphs asks whether G1 and G2 are simultaneous interval graphs. The problem is defined in a similar way for other intersection graph classes. For comparability graphs and any intersection graph class, we show that the simultaneous representation problem for the graph class is equivalent to a graph augmentation problem: given graphs G1 and G2, sharing vertices I and the corresponding induced edges, do there exist edges E' between G1-I and G2-I such that the graph G1 U G_2 U E' belongs to the graph class. This equivalence implies that the simultaneous representation problem is closely related to other well-studied classes in the literature, namely, sandwich graphs and probe graphs. We give efficient algorithms for solving the simultaneous representation problem for interval graphs, chordal graphs, comparability graphs and permutation graphs. Further, our algorithms for comparability and permutation graphs solve a more general version of the problem when there are multiple graphs, any two of which share the same common graph. This version of the problem also generalizes probe graphs.

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A distinguishing colouring of a graph G is a labelling of the vertices of G with colours such that no non-trivial automorphism of G preserves all colours. The distinguishing number of G is the minimum number of colours in a distinguishing colouring. This thesis presents a survey of the history of distinguishing colouring problems and proves new bounds and computational results about distinguishability. An algorithm to generate all labellings of a graph up to isomorphism is presented and compared to a previously published algorithm. The new algorithm is shown to have performance competitive with the existing algorithm, as well as being able to process automorphism groups far larger than the previous limit. A specialization of the algorithm is used to generate all minimal distinguishing colourings of a set of graphs with large automorphism groups and compute their distinguishing numbers. / Graduate / 0984 / 0405 / bbird@uvic.ca

Many graphs arising in practice can be represented in a concise and intuitive way that conveys their structure. For example: A planar graph can be represented in the plane with points for vertices and non-crossing curves for edges. An interval graph can be represented on the real line with intervals for vertices and intersection of intervals representing edges. The concept of ``simultaneity'' applies for several types of graphs: the idea is to find representations for two graphs that share some common vertices and edges, and ensure that the common vertices and edges are represented the same way. Simultaneous representation problems arise in any situation where two related graphs should be represented consistently. A main instance is for temporal relationships, where an old graph and a new graph share some common parts. Pairs of related graphs arise in many other situations. For example, two social networks that share some members; two schedules that share some events, overlap graphs of DNA fragments of two similar organisms, circuit graphs of two adjacent layers on a computer chip etc. In this thesis, we study the simultaneous representation problem for several graph classes. For planar graphs the problem is defined as follows. Let G1 and G2 be two graphs sharing some vertices and edges. The simultaneous planar embedding problem asks whether there exist planar embeddings (or drawings) for G1 and G2 such that every vertex shared by the two graphs is mapped to the same point and every shared edge is mapped to the same curve in both embeddings. Over the last few years there has been a lot of work on simultaneous planar embeddings, which have been called `simultaneous embeddings with fixed edges'. A major open question is whether simultaneous planarity for two graphs can be tested in polynomial time. We give a linear-time algorithm for testing the simultaneous planarity of any two graphs that share a 2-connected subgraph. Our algorithm also extends to the case of k planar graphs, where each vertex [edge] is either common to all graphs or belongs to exactly one of them. Next we introduce a new notion of simultaneity for intersection graph classes (interval graphs, chordal graphs etc.) and for comparability graphs. For interval graphs, the problem is defined as follows. Let G1 and G2 be two interval graphs sharing some vertices I and the edges induced by I. G1 and G2 are said to be `simultaneous interval graphs' if there exist interval representations of G1 and G2 such that any vertex of I is assigned to the same interval in both the representations. The `simultaneous representation problem' for interval graphs asks whether G1 and G2 are simultaneous interval graphs. The problem is defined in a similar way for other intersection graph classes. For comparability graphs and any intersection graph class, we show that the simultaneous representation problem for the graph class is equivalent to a graph augmentation problem: given graphs G1 and G2, sharing vertices I and the corresponding induced edges, do there exist edges E' between G1-I and G2-I such that the graph G1 U G_2 U E' belongs to the graph class. This equivalence implies that the simultaneous representation problem is closely related to other well-studied classes in the literature, namely, sandwich graphs and probe graphs. We give efficient algorithms for solving the simultaneous representation problem for interval graphs, chordal graphs, comparability graphs and permutation graphs. Further, our algorithms for comparability and permutation graphs solve a more general version of the problem when there are multiple graphs, any two of which share the same common graph. This version of the problem also generalizes probe graphs.

Graphs model relationships across real-world entities in web graphs, social network graphs, and road network graphs. Graph algorithms analyze and transform a graph to discover graph properties or to apply a computation. For instance, a pagerank algorithm computes a rank for each page in a webgraph, and a community detection algorithm discovers likely communities in a social network, while a shortest path algorithm computes the quickest way to reach a place from another, in a road network. In Domains such as social information systems, the number of edges can be in billions or trillions. Such large graphs are processed on distributed computer systems or clusters. Graph algorithms can be executed on multi-core CPUs, GPUs with thousands of cores, multi-GPU devices, and CPU+GPU clusters, depending on the size of the graph object. While programming such algorithms on heterogeneous targets, a programmer is required to deal with parallelism and and also manage explicit data communication between distributed devices. This implies that a programmer is required to learn CUDA, OpenMP, MPI, etc., and also the details of the hardware architecture. Such codes are error prone and di cult to debug. A Domain Speci c Language (DSL) which hides all the hardware details and lets the programmer concentrate only the algorithmic logic will be very useful. With this as the research goal, Falcon, graph DSL and its compiler have been developed. Falcon programs are explicitly parallel and Falcon hides all the hardware details from the programmer. Large graphs that do not t into the memory of a single device are automatically partitioned by the Falcon compiler. Another feature of Falcon is that it supports mutation of graph objects and thus enables programming dynamic graph algorithms. The Falcon compiler converts a single DSL code to heterogeneous targets such as multi-core CPUs, GPUs, multi-GPU devices, and CPU+GPU clusters. Compiled codes of Falcon match or outperform state-of-the-art graph frameworks for di erent target platforms and benchmarks.

Graph algorithms are ubiquitously used across domains. They exhibit parallelism, which can be exploited on parallel architectures, such as multi-core processors and accelerators. However, real world graphs are massive in size and cannot fit into the memory of a single machine. Such large graphs are partitioned and processed in a distributed cluster environment which consists of multiple GPUs and CPUs. Existing frameworks that facilitate large scale graph processing in the distributed cluster have their own style of programming and require extensive involvement by the user in communication and synchronization aspects. Adaptation of these frameworks appears to be an overhead for a programmer. Furthermore, these frameworks have been developed to target only CPU clusters and lack the ability to harness the GPU architecture. We provide a back-end framework to the graph Domain Specific Language, Falcon, for large scale graph processing on CPU and GPU clusters. The Motivation behind choosing this DSL as a front-end is its shared-memory based imperative programmability feature. Our framework generates Giraph code for CPU clusters. Giraph code runs on the Hadoop cluster and is known for scalable and fault-tolerant graph processing. For GPU cluster, Our framework applies a set of optimizations to reduce computation and communication latency, and generates efficient CUDA code coupled with MPI. Experimental evaluations show the scalability and performance of our framework for both CPU and GPU clusters. The performance of the framework generated code is comparable to the manual implementations of various algorithms in distributed environments.

Dans cette thèse, nous étudions les aspects structurels et algorithmiques de différents problèmes de théorie des graphes. Rappelons qu’un graphe est un ensemble de sommets éventuellement reliés par des arêtes. Deux sommets sont adjacents s’ils sont reliés par une arête.<p>Tout d’abord, nous considérons les deux problèmes suivants :le problème de vertex cover et celui de dominating set, deux cas particuliers du problème de hitting set. Un vertex cover est un ensemble de sommets qui rencontrent toutes les arêtes alors qu’un dominating set est un ensemble X de sommets tel que chaque sommet n’appartenant pas à X est adjacent à un sommet de X. La version connexe de ces problèmes demande que les sommets choisis forment un sous-graphe connexe. Pour les deux problèmes précédents, nous examinons le prix de la connexité, défini comme étant le rapport entre la taille minimum d’un ensemble répondant à la version connexe du problème et celle d’un ensemble du problème originel. Nous prouvons la difficulté du calcul du prix de la connexité d’un graphe. Cependant, lorsqu’on exige que le prix de la connexité d’un graphe ainsi que de tous ses sous-graphes induits soit borné par une constante fixée, la situation change complètement. En effet, pour les problèmes de vertex cover et de dominating set, nous avons pu caractériser ces classes de graphes pour de petites constantes.<p>Ensuite, nous caractérisons en termes de dominating sets connexes les graphes Pk- free, graphes n’ayant pas de sous-graphes induits isomorphes à un chemin sur k sommets. Beaucoup de problèmes sur les graphes sont étudiés lorsqu’ils sont restreints à cette classe de graphes. De plus, nous appliquons cette caractérisation à la 2-coloration dans les hypergraphes. Pour certains hypergraphes, nous prouvons que ce problème peut être résolu en temps polynomial.<p>Finalement, nous travaillons sur le problème de Pk-hitting set. Un Pk-hitting set est un ensemble de sommets qui rencontrent tous les chemins sur k sommets. Nous développons un algorithme d’approximation avec un facteur de performance de 3. Notre algorithme, basé sur la méthode primal-dual, fournit un Pk-hitting set dont la taille est au plus 3 fois la taille minimum d’un Pk-hitting set. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

La spectrométrie de masse, initialement développée pour de petites molécules, a permis au cours de la dernière écoulée d’étudier en phase gazeuse des assemblages macro-moléculaires intacts, posant nombre de questions algorithmiques difficiles, dont trois sont étudiées dans cette thèse. La première contribution concerne la détermination de stoichiométrie (SD), et vise à trouver le nombre de copies de chaque constituant dans un assemblage. On étudie le cas où la masse cible se trouve dans un intervalle dont les bornes rendent compte des incertitudes des mesures des masses. Nous présentons un algorithme de taille mémoire constante (DIOPHANTINE), et un algorithme de complexité sensible à la sortie (DP++), plus performants que l’état de l’art, pour des masses en nombre entier ou flottant. La seconde contribution traite de l’inférence de connectivité à partir d’une liste d’oligomères dont la composition en termes de sous-unités est connue. On introduit le problème d’inférence de connectivité minimale (MCI) et présente deux algorithmes pour le résoudre. On montre aussi un accord excellent entre les contacts trouvés et ceux détermines expérimentalement. La troisième contribution aborde le problème d’inférence de connectivité de poids minimal, lorsque chaque contact potentiel a un poids reflétant sa probabilité d’occurrence. On présente en particulier un algorithme de bootstrap permettant de trouver un ensemble d’arêtes de sensitivité et spécificité meilleures que celles obtenues pour les solutions du problème MCI. / Mass spectrometry (MS), an analytical technique initially invented to deal with small molecules, has emerged over the past decade as a key approach in structural biology. The recent advances have made it possible to transfer large macromolecular assemblies into the vacuum without their dissociation, raising challenging algorithmic problems. This thesis makes contributions to three such problems. The first contribution deals with stoichiometry determination (SD), namely the problem of determining the number of copies of each subunit of an assembly, from mass measurements. We deal with the interval SD problem, where the target mass belongs to an interval accounting for mass measurement uncertainties. We present a constant memory space algorithm (DIOPHANTINE), and an output sensitive dynamic programming based algorithm (DP++), outperforming state-of-the-art methods both for integer type and float type problems. The second contribution deals with the inference of pairwise contacts between subunits, using a list of sub-complexes whose composition is known. We introduce the Minimum Connectivity Inference problem (MCI) and present two algorithms solving it. We also show an excellent agreement between the contacts reported by these algorithms and those determined experimentally. The third contribution deals with Minimum Weight Connectivity Inference (MWCI), a problem where weights on candidate edges are available, reflecting their likelihood. We present in particular a bootstrap algorithm allowing one to report a set of edges with improved sensitivity and specificity with respect to those obtaining upon solving MCI.

This work refers existing solutions and options for the task registration of photos to 3D model based on the previous knowledge of the geographic position of the camera. The contribution of the work are new ways and possibilities of the solution with the usage of graph algorithms. In this area, the work interests are useful points of interest detection in input data, a construction of graphs and graph matching possibilities.

This thesis deals with theoretical basics of risk modelling in transportation and optimization using aggregated traffic data. In this thesis is suggested the procedure and implemented the application solving network problem of shortest path between geographical points. The thesis includes method for special paths evaluation depending on the frequency of traffic incidents based on real historical data. The thesis also includes a~graphical interface for presentation of the achieved results.