Routing Games over Time with FIFO policy

We study atomic routing games where every agent travels both along its decided edges and through time. The agents arriving on an edge are first lined up in a \emph{first-in-first-out} queue and may wait: an edge is associated with a capacity, which defines how many agents-per-time-step can pop from the queue's head and enter the edge, to transit for a fixed delay. We show that the best-response optimization problem is not approximable, and that deciding the existence of a Nash equilibrium is complete for the second level of the polynomial hierarchy. Then, we drop the rationality assumption, introduce a behavioral concept based on GPS navigation, and study its worst-case efficiency ratio to coordination.Comment: Submission to WINE-2017 Deadline was August 2nd AoE, 201

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

Efficient motion planning for problems lacking optimal substructure

We consider the motion-planning problem of planning a collision-free path of a robot in the presence of risk zones. The robot is allowed to travel in these zones but is penalized in a super-linear fashion for consecutive accumulative time spent there. We suggest a natural cost function that balances path length and risk-exposure time. Specifically, we consider the discrete setting where we are given a graph, or a roadmap, and we wish to compute the minimal-cost path under this cost function. Interestingly, paths defined using our cost function do not have an optimal substructure. Namely, subpaths of an optimal path are not necessarily optimal. Thus, the Bellman condition is not satisfied and standard graph-search algorithms such as Dijkstra cannot be used. We present a path-finding algorithm, which can be seen as a natural generalization of Dijkstra's algorithm. Our algorithm runs in $O\left((n_B\cdot n) \log( n_B\cdot n) + n_B\cdot m\right)$ time, where~$n$ and $m$ are the number of vertices and edges of the graph, respectively, and $n_B$ is the number of intersections between edges and the boundary of the risk zone. We present simulations on robotic platforms demonstrating both the natural paths produced by our cost function and the computational efficiency of our algorithm

Can local single-pass methods solve any stationary Hamilton-Jacobi-Bellman equation?

The use of local single-pass methods (like, e.g., the Fast Marching method) has become popular in the solution of some Hamilton-Jacobi equations. The prototype of these equations is the eikonal equation, for which the methods can be applied saving CPU time and possibly memory allocation. Then, some natural questions arise: can local single-pass methods solve any Hamilton-Jacobi equation? If not, where the limit should be set? This paper tries to answer these questions. In order to give a complete picture, we present an overview of some fast methods available in literature and we briefly analyze their main features. We also introduce some numerical tools and provide several numerical tests which are intended to exhibit the limitations of the methods. We show that the construction of a local single-pass method for general Hamilton-Jacobi equations is very hard, if not impossible. Nevertheless, some special classes of problems can be actually solved, making local single-pass methods very useful from the practical point of view.Comment: 19 page

Influence-based motion planning algorithms for games

In games, motion planning has to do with the motion of non-player characters (NPCs) from one place to another in the game world. In today’s video games there are two major approaches for motion planning, namely, path-finding and influence fields. Path-finding algorithms deal with the problem of finding a path in a weighted search graph, whose nodes represent locations of a game world, and in which the connections among nodes (edges) have an associated cost/weight. In video games, the most employed pathfinders are A* and its variants, namely, Dijkstra’s algorithm and best-first search. As further will be addressed in detail, the former pathfinders cannot simulate or mimic the natural movement of humans, which is usually without discontinuities, i.e., smooth, even when there are sudden changes in direction. Additionally, there is another problem with the former pathfinders, namely, their lack of adaptivity when changes to the environment occur. Therefore, such pathfinders are not adaptive, i.e., they cannot handle with search graph modifications during path search as a consequence of an event that happened in the game (e.g., when a bridge connecting two graph nodes is destroyed by a missile). On the other hand, influence fields are a motion planning technique that does not suffer from the two problems above, i.e., they can provide smooth human-like movement and are adaptive. As seen further ahead, we will resort to a differentiable real function to represent the influence field associated with a game map as a summation of functions equally differentiable, each associated to a repeller or an attractor. The differentiability ensures that there are no abrupt changes in the influence field, consequently, the movement of any NPC will be smooth, regardless if the NPC walks in the game world in the growing sense of the function or not. Thus, it is enough to have a spline curve that interpolates the path nodes to mimic the smooth human-like movement. Moreover, given the nature of the differentiable real functions that represent an influence field, the removal or addition of a repeller/attractor (as the result of the destruction or the construction of a bridge) does not alter the differentiability of the global function associated with the map of a game. That is to say that, an influence field is adaptive, in that it adapts to changes in the virtual world during the gameplay. In spite of being able to solve the two problems of pathfinders, an influence field may still have local extrema, which, if reached, will prevent an NPC from fleeing from that location. The local extremum problem never occurs in pathfinders because the goal node is the sole global minimum of the cost function. Therefore, by conjugating the cost function with the influence function, the NPC will never be detained at any local extremum of the influence function, because the minimization of the cost function ensures that it will always walk in the direction of the goal node. That is, the conjugation between pathfinders and influence fields results in movement planning algorithms which, simultaneously, solve the problems of pathfinders and influence fields. As will be demonstrated throughout this thesis, it is possible to combine influence fields and A*, Dijkstra’s, and best-first search algorithms, so that we get hybrid algorithms that are adaptive. Besides, these algorithms can generate smooth paths that resemble the ones traveled by human beings, though path smoothness is not the main focus of this thesis. Nevertheless, it is not always possible to perform this conjugation between influence fields and pathfinders; an example of such a pathfinder is the fringe search algorithm, as well as the new pathfinder which is proposed in this thesis, designated as best neighbor first search.Em jogos de vídeo, o planeamento de movimento tem que ver com o movimento de NPCs (“Non-Player Characters”, do inglês) de um lugar para outro do mundo virtual de um jogo. Existem duas abordagens principais para o planeamento de movimento, nomeadamente descoberta de caminhos e campos de influência. Os algoritmos de descoberta de caminhos lidam com o problema de encontrar um caminho num grafo de pesquisa pesado, cujos nós representam localizações de um mapa de um jogo, e cujas ligações (arestas) entre nós têm um custo/peso associado. Os algoritmos de descoberta de caminhos mais utilizados em jogos são o A* e as suas variantes, nomeadamente, o algoritmo de Dijkstra e o algoritmo de pesquisa do melhor primeiro (“best-first search”, do inglês). Como se verá mais adiante, os algoritmos de descoberta de caminhos referidos não permitem simular ou imitar o movimento natural dos seres humanos, que geralmente não possui descontinuidades, i.e., o movimento é suave mesmo quando há mudanças repentinas de direcção. A juntar a este problema, existe um outro que afeta os algoritmos de descoberta de caminhos acima referidos, que tem que ver com a falta de adaptatividade destes algoritmos face a alterações ao mapa de um jogo. Ou seja, estes algoritmos não são adaptativos, pelo que não permitem lidar com alterações ao grafo durante a pesquisa de um caminho em resultado de algum evento ocorrido no jogo (e.g., uma ponte que ligava dois nós de um grafo foi destruída por um míssil). Por outro lado, os campos de influência são uma técnica de planeamento de movimento que não padece dos dois problemas acima referidos, i.e., os campos possibilitam um movimento suave semelhante ao realizado pelo ser humano e são adaptativos. Como se verá mais adiante, iremos recorrer a uma função real diferenciável para representar o campo de influência associado a um mapa de um jogo como um somatório de funções igualmente diferenciáveis, em que cada função está associada a um repulsor ou a um atractor. A diferenciabilidade garante que não existem alterações abruptas ao campo de influência; consequentemente, o movimento de qualquer NPC será suave, independentemente de o NPC caminhar no mapa de um jogo no sentido crescente ou no sentido decrescente da função. Assim, basta ter uma curva spline que interpola os nós do caminho de forma a simular o movimento suave de um ser humano. Além disso, dada a natureza das funções reais diferenciáveis que representam um campo de influência, a remoção ou adição de um repulsor/atractor (como resultado da destruição ou construção de uma ponte) não altera a diferenciabilidade da função global associada ao mapa de um jogo. Ou seja, um campo de influência é adaptativo, na medida em que se adapta a alterações que ocorram num mundo virtual durante uma sessão de jogo. Apesar de ser capaz de resolver os dois problemas dos algoritmos de descoberta de caminhos, um campo de influência ainda pode ter extremos locais, que, se alcançados, impedirão um NPC de fugir desse local. O problema do extremo local nunca ocorre nos algoritmos de descoberta de caminhos porque o nó de destino é o único mínimo global da função de custo. Portanto, ao conjugar a função de custo com a função de influência, o NPC nunca será retido num qualquer extremo local da função de influência, porque a minimização da função de custo garante que ele caminhe sempre na direção do nó de destino. Ou seja, a conjugação entre algoritmos de descoberta de caminhos e campos de influência tem como resultado algoritmos de planeamento de movimento que resolvem em simultâneo os problemas dos algoritmos de descoberta de caminhos e de campos de influência. Como será demonstrado ao longo desta tese, é possível combinar campos de influência e o algoritmo A*, o algoritmo de Dijkstra, e o algoritmo da pesquisa pelo melhor primeiro, de modo a obter algoritmos híbridos que são adaptativos. Além disso, esses algoritmos podem gerar caminhos suaves que se assemelham aos que são efetuados por seres humanos, embora a suavidade de caminhos não seja o foco principal desta tese. No entanto, nem sempre é possível realizar essa conjugação entre os campos de influência e os algoritmos de descoberta de caminhos; um exemplo é o algoritmo de pesquisa na franja (“fringe search”, do inglês), bem como o novo algoritmo de pesquisa proposto nesta tese, que se designa por algoritmo de pesquisa pelo melhor vizinho primeiro (“best neighbor first search”, do inglês)

Simulation and Comparison of Efficency in Pathfinding algorithms in Games

There are several routes to go from point A to point B in many computer games and computer player have to choose the best route. To do this, the pathfinding algorithms is used. Currently, several algorithms have been proposed for routing in games so that the general challenges of them is high consumption of memory and a long Execution time. Due to these problems, the development and introduction of new algorithms will be continued. At the first part of this article, in addition to basic and important used algorithms, the new algorithm BIDDFS is introduced.In the second part, these algorithms in the various modes, are simulated on 2D-Grid, and compared based on their efficency (memory consumption and execution time) , Simulated algorithms include: Dijkstra, Iddfs, Biddfs, Bfs (Breadth), Greedy Best First Search, Ida*, A*, Jump point search, HPA*

Influence map-based pathfinding algorithms in video games

Path search algorithms, i.e., pathfinding algorithms, are used to solve shortest path problems by intelligent agents, ranging from computer games and applications to robotics. Pathfinding is a particular kind of search, in which the objective is to find a path between two nodes. A node is a point in space where an intelligent agent can travel. Moving agents in physical or virtual worlds is a key part of the simulation of intelligent behavior. If a game agent is not able to navigate through its surrounding environment without avoiding obstacles, it does not seem intelligent. Hence the reason why pathfinding is among the core tasks of AI in computer games. Pathfinding algorithms work well with single agents navigating through an environment. In realtime strategy (RTS) games, potential fields (PF) are used for multi-agent navigation in large and dynamic game environments. On the contrary, influence maps are not used in pathfinding. Influence maps are a spatial reasoning technique that helps bots and players to take decisions about the course of the game. Influence map represent game information, e.g., events and faction power distribution, and is ultimately used to provide game agents knowledge to take strategic or tactical decisions. Strategic decisions are based on achieving an overall goal, e.g., capture an enemy location and win the game. Tactical decisions are based on small and precise actions, e.g., where to install a turret, where to hide from the enemy. This dissertation work focuses on a novel path search method, that combines the state-of-theart pathfinding algorithms with influence maps in order to achieve better time performance and less memory space performance as well as more smooth paths in pathfinding.Algoritmos de pathfinding são usados por agentes inteligentes para resolver o problema do caminho mais curto, desde a àrea jogos de computador até à robótica. Pathfinding é um tipo particular de algoritmos de pesquisa, em que o objectivo é encontrar o caminho mais curto entre dois nós. Um nó é um ponto no espaço onde um agente inteligente consegue navegar. Agentes móveis em mundos físicos e virtuais são uma componente chave para a simulação de comportamento inteligente. Se um agente não for capaz de navegar no ambiente que o rodeia sem colidir com obstáculos, não aparenta ser inteligente. Consequentemente, pathfinding faz parte das tarefas fundamentais de inteligencia artificial em vídeo jogos. Algoritmos de pathfinding funcionam bem com agentes únicos a navegar por um ambiente. Em jogos de estratégia em tempo real (RTS), potential fields (PF) são utilizados para a navegação multi-agente em ambientes amplos e dinâmicos. Pelo contrário, os influence maps não são usados no pathfinding. Influence maps são uma técnica de raciocínio espacial que ajudam agentes inteligentes e jogadores a tomar decisões sobre o decorrer do jogo. Influence maps representam informação de jogo, por exemplo, eventos e distribuição de poder, que são usados para fornecer conhecimento aos agentes na tomada de decisões estratégicas ou táticas. As decisões estratégicas são baseadas em atingir uma meta global, por exemplo, a captura de uma zona do inimigo e ganhar o jogo. Decisões táticas são baseadas em acções pequenas e precisas, por exemplo, em que local instalar uma torre de defesa, ou onde se esconder do inimigo. Esta dissertação foca-se numa nova técnica que consiste em combinar algoritmos de pathfinding com influence maps, afim de alcançar melhores performances a nível de tempo de pesquisa e consumo de memória, assim como obter caminhos visualmente mais suaves