On some properties of the Voronoi diagram for planning multiple paths in free space ?

Most of the emphasis in path planning, a topic of much interest in several domains, has been on finding the optimal path or at most k optimal paths, but in domains such as adversarial planning, one of the agents might deliberately take less optimal paths to confuse the opponent, and by the same token an agent, for inferring opponent’s intent, has to consider all possible paths that the opponent might take. We study the problem of computing all representative paths with different properties, such as all representative paths with at most L loops, in free space among polygonal regions using a framework of Voronoi diagram. We prove that given any path, a homotopic path can always be obtained from the Voronoi diagram of the regions. We also show that all representative paths with a given property might not be always obtainable from the Voronoi diagram even after searching through the entire graph. Thus, the Voronoi diagram, though widely used, is inadequate to represent certain important properties of free space. Further, we show how our findings can be applied for entity re-identification, a problem of much importance in the military domain.