Floyd-Warshall Algorithm 1

Abstract: There are several applications in VLSI technology that require high-speed shortest-path computations. The shortest path is a path between two nodes (or points) in a graph such that the sum of the weights of its constituent edges is minimum. Floyd-Warshall algorithm provides fastest computation of shortest path between all pair of nodes present in the graph. With rapid advances in VLSI technology, Field Programmable Gate Arrays (FPGAs) are receiving the attention of the Parallel and High Performance Computing community. This paper gives implementation outcome of Floyd-Warshall algorithm to solve the all pairs shortest-paths problem for directed graph in Verilog

Time-Dependent Shortest Path Queries Among Growing Discs

The determination of time-dependent collision-free shortest paths has received a fair amount of attention. Here, we study the problem of computing a time-dependent shortest path among growing discs which has been previously studied for the instance where the departure times are fixed. We address a more general setting: For two given points $s$ and $d$, we wish to determine the function $\mathcal{A}(t)$ which is the minimum arrival time at $d$ for any departure time $t$ at $s$. We present a $(1+\epsilon)$-approximation algorithm for computing $\mathcal{A}(t)$. As part of preprocessing, we execute $O({1 \over \epsilon} \log({\mathcal{V}_{r} \over \mathcal{V}_{c}}))$ shortest path computations for fixed departure times, where $\mathcal{V}_{r}$ is the maximum speed of the robot and $\mathcal{V}_{c}$ is the minimum growth rate of the discs. For any query departure time $t \geq 0$ from $s$, we can approximate the minimum arrival time at the destination in $O(\log ({1 \over \epsilon}) + \log\log({\mathcal{V}_{r} \over \mathcal{V}_{c}}))$ time, within a factor of $1+\epsilon$ of optimal. Since we treat the shortest path computations as black-box functions, for different settings of growing discs, we can plug-in different shortest path algorithms. Thus, the exact time complexity of our algorithm is determined by the running time of the shortest path computations.Comment: 16 pages, 9 figures, abridged version submitted to CCCG 201

Dynamic Shortest Path Algorithms for Hypergraphs

A hypergraph is a set V of vertices and a set of non-empty subsets of V, called hyperedges. Unlike graphs, hypergraphs can capture higher-order interactions in social and communication networks that go beyond a simple union of pairwise relationships. In this paper, we consider the shortest path problem in hypergraphs. We develop two algorithms for finding and maintaining the shortest hyperpaths in a dynamic network with both weight and topological changes. These two algorithms are the first to address the fully dynamic shortest path problem in a general hypergraph. They complement each other by partitioning the application space based on the nature of the change dynamics and the type of the hypergraph

Modeling dendritic shapes - using path planning

Dendritic shapes are commonplace in the natural world such as trees, lichens, coral and lightning. Models of dendritic shapes are widely needed in many areas. Because of their branching fractal and erratic structures modeling dendritic shapes is a tricky task. Existing methods for modeling dendritic shapes are slow and complicated.In this thesis we present a procedural algorithm of using path planning to model dendritic shapes. We generate a dendrite by finding the least-cost paths from multiple endpoints to a common generator and use the dendrite to build the geometric model. With the control handles of endpoint placement, fractal shape, edge weights distribution and path width, we create different shapes of dendrites that simulate different kinds of dendritic shapes very well. Compared with some existing methods, our algorithm is fast and simple

Desenvolupament d’una eina de planificació de vol per UAVs

Aquest document té com a finalitat el desenvolupament d’una eina de planificació de vol per Aeronaus no Tripulades (UAV). Per fer-ho, primerament es realitza una introducció general al programa ideat i una explicació dels seus modes de funcionament. Posteriorment es descriu la teoria i l’implementació de cada bloc que forma el programa per separat. En el capítol següent es parla dels softwares utilitzats i es descriu el codi breument, classe a classe. Per acabar es proposen tres casos pràctics sobre el funcionament dels diferents modes de l’eina. És en aquesta part del treball on es posa en evidència el funcionament en conjunt de tots els diferents blocs que formen el programa. L’eina de planificació de vol per UAVs té com a objectiu principal crear les bases per realitzar un pla de vol de manera automàtica a partir d’unes dades reduïdes d’entrada, que depenen del mode de funcionament de la mateixa eina. No obstant, els tres modes es basen en el mateix procediment de funcionament: A partir d’uns obstacles i una altura de vol definida per l’usuari o per la pròpia eina es crea un diagrama de rutes segures al voltant dels obstacles de tal manera que, seguint les arestes d’aquest diagrama es poden unir dos punts sense colisionar amb cap obstacle. Posteriorment, es cerca la ruta òptima entre aquests dos punts i es representen els resultats obtinguts mitjançant Google Earth. Aquesta eina ha estat implementada mitjançant el llenguatge C-Sharp, de la plataforma .NET

Geometric Path-Planning Algorithm in Cluttered 2D Environments Using Convex Hulls

Routing or path planning is the problem of finding a collision-free path in an environment usually scattered with multiple objects. Finding the shortest route in a planar (2D) or spatial (3D) environment has a variety of applications such as robot motion planning, navigating autonomous vehicles, routing of cables, wires, and harnesses in vehicles, routing of pipes in chemical process plants, etc. The problem often times is decomposed into two main sub-problems: modeling and representation of the workspace geometrically and optimization of the path. Geometric modeling and representation of the workspace are paramount in any path planning problem since it builds the data structures and provides the means for solving the optimization problem. The optimization aspect of the path planning involves satisfying some constraints, the most important of which is to avoid intersections with the interior of any object and optimizing one or more criteria. The most common criterion in path planning problems is to minimize the length of the path between a source and a destination point of the workspace while other criteria such as minimizing the number of links or curves could also be taken into account. Planar path planning is mainly about modeling the workspace of the problem as a collision-free graph. The graph is, later on, searched for the optimal path using network optimization techniques such as branch-and-bound or search algorithms such as Dijkstra\u27s. Previous methods developed to construct the collision-free graph explore the entire workspace of the problem which usually results in some unnecessary information that has no value but to increase the time complexity of the algorithm, hence, affecting the efficiency significantly. For example, the fastest known algorithm to construct the visibility graph, which is the most common method of modeling the collision-free space, in a workspace with a total of n vertices has a time complexity of order O(n2). In this research, first, the 2D workspace of the problem is modeled using the tessellated format of the objects in a CAD software which facilitates handling of any free-form object. Then, an algorithm is developed to construct the collision-free graph of the workspace using the convex hulls of the intersecting obstacles. The proposed algorithm focuses only on a portion of the workspace involved in the straight line connecting the source and destination points. Considering the worst case that all the objects of the workspace are intersecting, the algorithm yields a time complexity of O(nlog(n/f)), with n being the total number of vertices and f being the number of objects. The collision-free graph is later searched for the shortest path between the two given nodes using a search algorithm known as Dijkstra\u27s

Feature-rich distance-based terrain synthesis

This thesis describes a novel terrain synthesis method based on distances in a weighted graph. The method begins with a regular lattice with arbitrary edge weights; heights are determined by path cost from a set of generator nodes. The shapes of individual terrain features, such as mountains, hills, and craters, are specified by a monotonically decreasing profile describing the cross-sectional shape of a feature, while the locations of features in the terrain are specified by placing the generators. Pathing places ridges whose initial location have a dendritic shape. The method is robust and easy to control, making it possible to create pareidolia effects. It can produce a wide range of realistic synthetic terrains such as mountain ranges, craters, faults, cinder cones, and hills. The algorithm incorporates random graph edge weights, permits the inclusion of multiple topography profiles, and allows precise control over placement of terrain features and their heights. These properties all allow the artist to create highly heterogeneous terrains that compare quite favorably to existing methods

Mitigation of cascade failures in complex networks: theory and application

Complex networks such as transportation networks, the Internet, and electrical power grids are fundamental parts of modern life, and their robustness under any attack or fault has always been a concern. Failure and intentional removal of components in complex networks might affect the flow of information and change balance of flows in the network. This phenomenon may require load redistribution all over the network. Component overloaded can act as a trigger for a chain of overload failures. This overload, could, for example, increase the amount of information a router must transmit and ultimately make internet congestion. One of the major applications of complex network theory is to study power systems. Power systems are the most complex human-made infrastructures, and almost every individual's life is dependent on electrical energy and resilient functioning of power systems. Recently, there have been many reports about massive power outages leaving vast areas without power that sometimes takes a few days to have the power back. One of the most critical areas in the power system is the root cause analysis of such catastrophes and trying to resolve them. From an electrical engineering point of view, these power outages occur following an initial failure due to problems, such as generators tripping, transformers overheating, faulty power generation units, damage to the transmission system, substations or distribution systems, or overloading of the power system. A faulty protection relay or malicious attack to control centres can also trigger it. In any of these cases, the failed component will be out of service immediately and to keep the robust power delivery to all customers, their loads should be redistributed across the power system, and henceforth some of them might become overloaded as well, and accordingly get out of service. This chain of failures can be propagated all over the system and lead to a catastrophic blackout. This thesis conducts a full study on how to mitigate cascade failures in complex networks. First, cascade depth is applied to quantify nodes criticality for cascade failures. Then, a wide range of node centrality parameters is considered to find out the relationship between the node vitality and these centralities. To discover the structure of cascade propagation in complex networks, the edge geodesic distance is considered for computing the structural distance between two arbitrary edges in the network. Then, starting with the single edge removal events, the route that cascade tends to spread is studied. In the next step, the impact of two or three concurrent edge removals on the way the cascade spreads are examined. Besides, the power system vulnerability is studied using the maximum flow algorithm based on Ford-Fulkerson method and critical capacity parameters are identified. A synthetic model with the same properties as a real power system is generated and examined. For a power line, to be overloaded, a new method is developed to overpass across the network and shortlist the busbars for load reduction. Next, a novel sensitivity method is formulated based on AC load flow analysis to rank the loads according to their effect on the lines power flow