A graph-based mathematical morphology reader

This survey paper aims at providing a "literary" anthology of mathematical morphology on graphs. It describes in the English language many ideas stemming from a large number of different papers, hence providing a unified view of an active and diverse field of research

Recent Advances in Graph Partitioning

We survey recent trends in practical algorithms for balanced graph partitioning together with applications and future research directions

Analysis of Farthest Point Sampling for Approximating Geodesics in a Graph

A standard way to approximate the distance between any two vertices $p$ and $q$ on a mesh is to compute, in the associated graph, a shortest path from $p$ to $q$ that goes through one of $k$ sources, which are well-chosen vertices. Precomputing the distance between each of the $k$ sources to all vertices of the graph yields an efficient computation of approximate distances between any two vertices. One standard method for choosing $k$ sources, which has been used extensively and successfully for isometry-invariant surface processing, is the so-called Farthest Point Sampling (FPS), which starts with a random vertex as the first source, and iteratively selects the farthest vertex from the already selected sources. In this paper, we analyze the stretch factor $\mathcal{F}_{FPS}$ of approximate geodesics computed using FPS, which is the maximum, over all pairs of distinct vertices, of their approximated distance over their geodesic distance in the graph. We show that $\mathcal{F}_{FPS}$ can be bounded in terms of the minimal value $\mathcal{F}^*$ of the stretch factor obtained using an optimal placement of $k$ sources as $\mathcal{F}_{FPS}\leq 2 r_e^2 \mathcal{F}^*+ 2 r_e^2 + 8 r_e + 1$, where $r_e$ is the ratio of the lengths of the longest and the shortest edges of the graph. This provides some evidence explaining why farthest point sampling has been used successfully for isometry-invariant shape processing. Furthermore, we show that it is NP-complete to find $k$ sources that minimize the stretch factor.Comment: 13 pages, 4 figure

On Compact Routing for the Internet

While there exist compact routing schemes designed for grids, trees, and Internet-like topologies that offer routing tables of sizes that scale logarithmically with the network size, we demonstrate in this paper that in view of recent results in compact routing research, such logarithmic scaling on Internet-like topologies is fundamentally impossible in the presence of topology dynamics or topology-independent (flat) addressing. We use analytic arguments to show that the number of routing control messages per topology change cannot scale better than linearly on Internet-like topologies. We also employ simulations to confirm that logarithmic routing table size scaling gets broken by topology-independent addressing, a cornerstone of popular locator-identifier split proposals aiming at improving routing scaling in the presence of network topology dynamics or host mobility. These pessimistic findings lead us to the conclusion that a fundamental re-examination of assumptions behind routing models and abstractions is needed in order to find a routing architecture that would be able to scale ``indefinitely.''Comment: This is a significantly revised, journal version of cs/050802

An efficient method for multiobjective optimal control and optimal control subject to integral constraints

We introduce a new and efficient numerical method for multicriterion optimal control and single criterion optimal control under integral constraints. The approach is based on extending the state space to include information on a "budget" remaining to satisfy each constraint; the augmented Hamilton-Jacobi-Bellman PDE is then solved numerically. The efficiency of our approach hinges on the causality in that PDE, i.e., the monotonicity of characteristic curves in one of the newly added dimensions. A semi-Lagrangian "marching" method is used to approximate the discontinuous viscosity solution efficiently. We compare this to a recently introduced "weighted sum" based algorithm for the same problem. We illustrate our method using examples from flight path planning and robotic navigation in the presence of friendly and adversarial observers.Comment: The final version accepted by J. Comp. Math. : 41 pages, 14 figures. Since the previous version: typos fixed, formatting improved, one mistake in bibliography correcte

Geodesic Paths On 3D Surfaces: Survey and Open Problems

This survey gives a brief overview of theoretically and practically relevant algorithms to compute geodesic paths and distances on three-dimensional surfaces. The survey focuses on polyhedral three-dimensional surfaces

Toward human-like pathfinding with hierarchical approaches and the GPS of the brain theory

Pathfinding for autonomous agents and robots has been traditionally driven by finding optimal paths. Where typically optimality means finding the shortest path between two points in a given environment. However, optimality may not always be strictly necessary. For example, in the case of video games, often computing the paths for non-player characters (NPC) must be done under strict time constraints to guarantee real time simulation. In those cases, performance is more important than finding the shortest path, specially because often a sub-optimal path can be just as convincing from the point of view of the player. When simulating virtual humanoids, pathfinding has also been used with the same goal: finding the shortest path. However, humans very rarely follow precise shortest paths, and thus there are other aspects of human decision making and path planning strategies that should be incorporated in current simulation models. In this thesis we first focus on improving performance optimallity to handle as many virtual agents as possible, and then introduce neuroscience research to propose pathfinding algorithms that attempt to mimic humans in a more realistic manner.In the case of simulating NPCs for video games, one of the main challenges is to compute paths as efficiently as possible for groups of agents. As both the size of the environments and the number of autonomous agents increase, it becomes harder to obtain results in real time under the constraints of memory and computing resources. For this purpose we explored hierarchical approaches for two reasons: (1) they have shown important performance improvements for regular grids and other abstract problems, and (2) humans tend to plan trajectories also following an topbottom abstraction, focusing first on high level location and then refining the path as they move between those high level locations. Therefore, we believe that hierarchical approaches combine the best of our two goals: improving performance for multi-agent pathfinding and achieving more human-like pathfinding. Hierarchical approaches, such as HNA* (Hierarchical A* for Navigation Meshes) can compute paths more efficiently, although only for certain configurations of the hierarchy. For other configurations, the method suffers from a bottleneck in the step that connects the Start and Goal positions with the hierarchy. This bottleneck can drop performance drastically.In this thesis we present different approaches to solve the HNA* bottleneck and thus obtain a performance boost for all hierarchical configurations. The first method relies on further memory storage, and the second one uses parallelism on the GPU. Our comparative evaluation shows that both approaches offer speed-ups as high as 9x faster than A*, and show no limitations based on hierarchical configuration. Then we further exploit the potential of CUDA parallelism, to extend our implementation to HNA* for multi-agent path finding. Our method can now compute paths for over 500K agents simultaneously in real-time, with speed-ups above 15x faster than a parallel multi-agent implementation using A*. We then focus on studying neurosience research to learn about the way that humans build mental maps, in order to propose novel algorithms that take those finding into account when simulating virtual humans. We propose a novel algorithm for path finding that is inspired by neuroscience research on how the brain learns and builds cognitive maps. Our method represents the space as a hexagonal grid, based on the GPS of the brain theory, and fires memory cells as counters. Our path finder then combines a method for exploring unknown environments while building such a cognitive map, with an A* search using a modified heuristic that takes into account the GPS of the brain cognitive map.El problema de Pathfinding para agentes autónomos o robots, ha consistido tradicionalmente en encontrar un camino óptimo, donde por óptimo se entiende el camino de distancia mínima entre dos posiciones de un entorno. Sin embargo, en ocasiones puede que no sea estrictamente necesario encontrar una solución óptima. Para ofrecer la simulación de multitudes de agentes autónomos moviéndose en tiempo real, es necesario calcular sus trayectorias bajo condiciones estrictas de tiempo de computación, pero no es fundamental que las soluciones sean las de distancia mínima ya que, con frecuencia, el observador no apreciará la diferencia entre un camino óptimo y un sub-óptimo. Por tanto, suele ser suficiente con que la solución encontrada sea visualmente creíble para el observado. Cuando se simulan humanoides virtuales en aplicaciones de realidad virtual que requieren avatares que simulen el comportamiento de los humanos, se tiende a emplear los mismos algoritmos que en video juegos, con el objetivo de encontrar caminos de distancia mínima. Pero si realmente queremos que los avatares imiten el comportamiento humano, tenemos que tener en cuenta que, en el mundo real, los humanos rara vez seguimos precisamente el camino más corto, y por tanto se deben considerar otros aspectos que influyen en la toma de decisiones de los humanos y la selección de rutas en el mundo real. En esta tesis nos centraremos primero en mejorar el rendimiento de la búsqueda de caminos para poder simular grandes números de humanoides virtuales autónomos, y a continuación introduciremos conceptos de investigación con mamíferos en neurociencia, para proponer soluciones al problema de pathfinding que intenten imitar con mayor realismo, el modo en el que los humanos navegan el entorno que les rodea. A medida que aumentan tanto el tamaño de los entornos virtuales como el número de agentes autónomos, resulta más difícil obtener soluciones en tiempo real, debido a las limitaciones de memoria y recursos informáticos. Para resolver este problema, en esta tesis exploramos enfoques jerárquicos porque consideramos que combinan dos objetivos fundamentales: mejorar el rendimiento en la búsqueda de caminos para multitudes de agentes y lograr una búsqueda de caminos similar a la de los humanos. El primer método presentado en esta tesis se basa en mejorar el rendimiento del algoritmo HNA* (Hierarchical A* for Navigation Meshes) incrementando almacenamiento de datos en memoria, y el segundo utiliza el paralelismo para mejorar drásticamente el rendimiento. La evaluación cuantitativa realizada en esta tesis, muestra que ambos enfoques ofrecen aceleraciones que pueden llegar a ser hasta 9 veces más rápidas que el algoritmo A* y no presentan limitaciones debidas a la configuración jerárquica. A continuación, aprovechamos aún más el potencial del paralelismo ofrecido por CUDA para extender nuestra implementación de HNA* a sistemas multi-agentes. Nuestro método permite calcular caminos simultáneamente y en tiempo real para más de 500.000 agentes, con una aceleración superior a 15 veces la obtenida por una implementación paralela del algoritmo A*. Por último, en esta tesis nos hemos centrado en estudiar los últimos avances realizados en el ámbito de la neurociencia, para comprender la manera en la que los humanos construyen mapas mentales y poder así proponer nuevos algoritmos que imiten de forma más real el modo en el que navegamos los humanos. Nuestro método representa el espacio como una red hexagonal, basada en la distribución de ¿place cells¿ existente en el cerebro, e imita las activaciones neuronales como contadores en dichas celdas. Nuestro buscador de rutas combina un método para explorar entornos desconocidos mientras construye un mapa cognitivo hexagonal, utilizando una búsqueda A* con una nueva heurística adaptada al mapa cognitivo del cerebro y sus contadores