An <i>Interval Routing Scheme (IRS)</i> represents the routing tables in a network in a space-efficient way by labeling each vertex with an unique integer address and the outgoing edges at each vertex with disjoint subintervals of these addresses. An IRS that has at most <i>k</i> intervals per edge label is called a <i>k-IRS</i>. In this thesis, we propose a new type of interval routing scheme, called an <i>Ordered Interval Routing Scheme (OIRS)</i>, that uses an ordering of the outgoing edges at each vertex and allows nondisjoint intervals in the labels of those edges. Our results on a number of graphs show that using an OIRS instead of an IRS reduces the size of the routing tables in the case of <i>optimal</i> routing, i. e. , routing along shortest paths. We show that optimal routing in any <i>k</i>-tree is possible using an OIRS with at most 2^k-1 intervals per edge label, although the best known result for an IRS is 2^k+1 intervals per edge label. Any torus has an optimal 1-OIRS, although it may not have an optimal 1-IRS. We present similar results for the Petersen graph, <i>k</i>-garland graphs and a few other graphs.

An <i>Interval Routing Scheme (IRS)</i> represents the routing tables in a network in a space-efficient way by labeling each vertex with an unique integer address and the outgoing edges at each vertex with disjoint subintervals of these addresses. An IRS that has at most <i>k</i> intervals per edge label is called a <i>k-IRS</i>. In this thesis, we propose a new type of interval routing scheme, called an <i>Ordered Interval Routing Scheme (OIRS)</i>, that uses an ordering of the outgoing edges at each vertex and allows nondisjoint intervals in the labels of those edges. Our results on a number of graphs show that using an OIRS instead of an IRS reduces the size of the routing tables in the case of <i>optimal</i> routing, i. e. , routing along shortest paths. We show that optimal routing in any <i>k</i>-tree is possible using an OIRS with at most 2^k-1 intervals per edge label, although the best known result for an IRS is 2^k+1 intervals per edge label. Any torus has an optimal 1-OIRS, although it may not have an optimal 1-IRS. We present similar results for the Petersen graph, <i>k</i>-garland graphs and a few other graphs.

Successful tracking of a mobile target with a sensor network requires effective answers to the challenges of uncertainty in the measured data, small latency in acquiring and reporting the tracking information, and compliance with the stringent constraints imposed by the scarce resources available on each sensor node: limited available power, restricted availability of the inter-node communication links, relatively moderate computational power. This thesis introduces the architecture of a hierarchical, self-organizing, two-tier, mission-specific sensor network, composed of sensors and routers, to track the trajectory and velocity of a single mobile target in a two-dimensional convex sensor field. A query-driven approach is proposed to input configuration parameters to the network, which allow sensors to self-configure into regions, and routers into tree-like structures, with the common goal of sensing and tracking the target in an energy-aware manner, and communicating this tracking data to a base station node incurring low-overhead responses, respectively. The proposed algorithms to define and organize the sensor regions, establish the data routing scheme, and create the data stream representing the real-time location/velocity of a target, are heuristic, distributed, and represent localized node collaborations. Node behaviours have been modeled using state diagrams and inter-node collaborations have been designed using straightforward messaging schemes. This work has attempted to establish that by using a query-driven approach to track a target, high-level knowledge can be injected to the sensor network self-organization processes and its following operation, which allows the implementation of an energy-efficient, low-overhead tracking scheme. The resulting system, although built upon simple components and interactions, is complex in extension, and not directly available for exact evaluation. However, it provides intuitively advantageous behaviours. / Thesis (Master, Computing) -- Queen's University, 2009-03-04 11:18:14.392

Dans cette thèse, nous étudions les décompositions arborescentes qui satisfont certaines contraintes supplémentaires et nous proposons des algorithmes pour les calculer dans certaines classes de graphes. Finalement, nous résolvons des problèmes liés au routage en utilisant ces décompositions ainsi que des propriétés structurelles des graphes. Cette thèse est divisée en deux parties. Dans la première partie, nous étudions les décompositions arborescentes satisfaisant des propriétés spécifiques. Dans le Chapitre 2, nous étudions les décompositions de taille minimum, c’est-À-Dire avec un nombre minimum de sacs. Etant donné une entier k 4 fixé, nous prouvons que le problème de calculer une décomposition arborescente de largeur au plus k et de taille minimum est NP-Complet dans les graphes de largeur arborescente au plus 4. Nous décrivons ensuite des algorithmes qui calculent des décompositions de taille minimum dans certaines classes de graphes de largeur arborescente au plus 3. Ces résultats ont été présentés au workshop international ICGT 2014. Dans le Chapitre 3, nous étudions la cordalité des graphes et nous introduisons la notion de k-Good décomposition arborescente. Nous étudions tout d’abord les jeux de Gendarmes et Voleur dans les graphes sans long cycle induit. Notre résultat principal est un algorithme polynomial qui, étant donné un graphe G, soit trouve un cycle induit de longueur au moins k+1, ou calcule une k-Good décomposition de G. Ces résultats ont été publiés à la conférence internationale ICALP’12 et dans la revue internationale Algorithmica. Dans la seconde partie de la thèse, nous nous concentrons sur des problèmes de routage. / A tree decomposition of a graph is a way to represent it as a tree by preserving some connectivity properties of the initial graph. Tree decompositions have been widely studied for their algorithmic applications, in particular using dynamic programming approach. In this thesis, we study tree decompositions satisfying various constraints and design algorithms to compute them in some graph classes. We then use tree decompositions or specific graph properties to solve several problems related to routing. The thesis is divided into two parts. In the first part, we study tree decompositions satisfying some properties. In Chapter 2, we investigate minimum size tree decompositions, i.e., with minimum number of bags. Given a fixed k 4, we prove it is NP-Hard to compute a minimum size decomposition with width at most k in the class of graphs with treewidth at least 4. We design polynomial time algorithms to compute minimum size tree decompositions in some classes of graphs with treewidth at most 3 (including trees). Part of these results will be presented in ICGT 2014. In Chapter 3, we study the chordality (longest induced cycle) of graphs and introduce the notion of good tree decomposition (where each bag must satisfy some particular structure). Precisely, we study the Cops and Robber games in graphs with no long induced cycles. Our main result is the design of a polynomial-Time algorithm that either returns an induced cycle of length at least k+1 of a graph G or compute a k-Good tree decomposition of G. These results have been published in ICALP 2012 and Algorithmica. In the second part of the thesis, we focus on routing problems.