This thesis is devoted to the analysis of a class of iterative probabilistic algorithms in regular graphs, called locally greedy algorithms, which will provide bounds for graph functions in regular graphs with large girth. This class is useful because, by conveniently setting the parameters associated with it, we may derive algorithms for some well-known graph problems, such as algorithms to find a large independent set, a large induced forest, or even a small dominating set in an input graph G. The name ``locally greedy" comes from the fact that, in an algorithm of this class, the probability associated with the random selection of a vertex v is determined by the current state of the vertices within some fixed distance of v. Given r > 2 and an r-regular graph G, we determine the expected performance of a locally greedy algorithm in G, depending on the girth g of the input and on the degree r of its vertices. When the girth of the graph is sufficiently large, this analysis leads to new lower bounds on the independence number of G and on the maximum number of vertices in an induced forest in G, which, in both cases, improve the bounds previously known. It also implies bounds on the same functions in graphs with large girth and maximum degree r and in random regular graphs. As a matter of fact, the asymptotic lower bounds on the cardinality of a maximum induced forest in a random regular graph improve earlier bounds, while, for independent sets, our bounds coincide with asymptotic lower bounds first obtained by Wormald. Our result provides an alternative proof of these bounds which avoids sharp concentration arguments. The main contribution of this work lies in the method presented rather than in these particular new bounds. This method allows us, in some sense, to directly analyse prioritised algorithms in regular graphs, so that the class of locally greedy algorithms, or slight modifications thereof, may be applied to a wider range of problems in regular graphs with large girth.

This thesis is devoted to the analysis of a class of iterative probabilistic algorithms in regular graphs, called locally greedy algorithms, which will provide bounds for graph functions in regular graphs with large girth. This class is useful because, by conveniently setting the parameters associated with it, we may derive algorithms for some well-known graph problems, such as algorithms to find a large independent set, a large induced forest, or even a small dominating set in an input graph G. The name ``locally greedy" comes from the fact that, in an algorithm of this class, the probability associated with the random selection of a vertex v is determined by the current state of the vertices within some fixed distance of v. Given r > 2 and an r-regular graph G, we determine the expected performance of a locally greedy algorithm in G, depending on the girth g of the input and on the degree r of its vertices. When the girth of the graph is sufficiently large, this analysis leads to new lower bounds on the independence number of G and on the maximum number of vertices in an induced forest in G, which, in both cases, improve the bounds previously known. It also implies bounds on the same functions in graphs with large girth and maximum degree r and in random regular graphs. As a matter of fact, the asymptotic lower bounds on the cardinality of a maximum induced forest in a random regular graph improve earlier bounds, while, for independent sets, our bounds coincide with asymptotic lower bounds first obtained by Wormald. Our result provides an alternative proof of these bounds which avoids sharp concentration arguments. The main contribution of this work lies in the method presented rather than in these particular new bounds. This method allows us, in some sense, to directly analyse prioritised algorithms in regular graphs, so that the class of locally greedy algorithms, or slight modifications thereof, may be applied to a wider range of problems in regular graphs with large girth.

Reachability routing is a newly emerging paradigm in networking, where the goal is to determine all paths between a sender and a receiver. It is becoming relevant with the changing dynamics of the Internet and the emergence of low-bandwidth wireless/ad hoc networks. This thesis presents the case for reinforcement learning (RL) as the framework of choice to realize reachability routing, within the confines of the current Internet backbone infrastructure. The setting of the reinforcement learning problem offers several advantages, including loop resolution, multi-path forwarding capability, cost-sensitive routing, and minimizing state overhead, while maintaining the incremental spirit of the current backbone routing algorithms. We present the design and implementation of a new reachability algorithm that uses a model-based approach to achieve cost-sensitive multi-path forwarding. Performance assessment of the algorithm in various troublesome topologies shows consistently superior performance over classical reinforcement learning algorithms. Evaluations of the algorithm based on different criteria on many types of randomly generated networks as well as realistic topologies are presented. / Master of Science

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Random sampling is an appealing approach to build synopses of large data streams because random samples can be used for a broad spectrum of analytical tasks. Users are often interested in analyzing only the most recent fraction of the data stream in order to avoid outdated results. In this paper, we focus on sampling schemes that sample from a sliding window over a recent time interval; such windows are a popular and highly comprehensible method to model recency. In this setting, the main challenge is to guarantee an upper bound on the space consumption of the sample while using the allotted space efficiently at the same time. The difficulty arises from the fact that the number of items in the window is unknown in advance and may vary significantly over time, so that the sampling fraction has to be adjusted dynamically. We consider uniform sampling schemes, which produce each sample of the same size with equal probability, and stratified sampling schemes, in which the window is divided into smaller strata and a uniform sample is maintained per stratum. For uniform sampling, we prove that it is impossible to guarantee a minimum sample size in bounded space. We then introduce a novel sampling scheme called bounded priority sampling (BPS), which requires only bounded space. We derive a lower bound on the expected sample size and show that BPS quickly adapts to changing data rates. For stratified sampling, we propose a merge-based stratification scheme (MBS), which maintains strata of approximately equal size. Compared to naive stratification, MBS has the advantage that the sample is evenly distributed across the window, so that no part of the window is over- or underrepresented. We conclude the paper with a feasibility study of our algorithms on large real-world datasets.

Thesis describes library for algorithm vizualization. It helps to create user interface for application with algorithms. It’s usage is demonstrated on some pathfinding algorithms.These applications are presented on web page.

This paper deals with the problem of object path planning in 3D space. The goal is to create program which allows users to create a scene used for path planning, perform the planning and finally visualize path in the scene. Work is focused on probabilistic algorithms that are described in the theoretical part. The practical part describes the design and implementation of application. Finally, several experiments are performed to compare the performance of different algorithms and demonstrate the functionality of the program.