robot path planners

Abstract: We consider graph search models, in which an invisible, omniscient and fast intruder can hide anywhere in the graph, and the goal is to design a \emph{search strategy} composed of a sequence of movements of \emph{searchers} that guarantees the capture of the intruder regardless of its actions. Our point of departure is the connected search model in which the cleared subgraph is always connected during the search process. In this paper, we initiate the study of the subgraph searching problems which arise from the following question: what happens if the number of searchers available is not sufficient to clear a given graph $G$? We propose an approach in which one wants to clear in a connected way as large subgraph as possible with the given $k$ searchers. In this context we study two problems: one problem, called the \emph{subgraph search}, is to find the maximum-sized subgraph whose search number is at most $k$, and the other one, called the \emph{protected subgraph search}, is to perform a partial search on $G$ leading to as many cleared edges as possible. Informally speaking, the difference lies in the fact that in the latter a recontamination may come from an edge never to be cleared and hence the partial search strategy needs to avoid such recontaminations. Having proved that both problems are computationally hard in general, we give efficient algorithms for $n$-vertex trees: an $\bigo(\min\{kn, n \log n\})$ time algorithm for the subgraph search and an $\bigo(\min\{2^{6k}kn, n^{8.571} \log n\})$ time algorithm for the protected subgraph search.

Abstract: This work presents an algorithm, together with its correctness proof, to determine the minimum perimeter polygon (MPP) for digital objects given as regular complexes in the triangular plane tiling. Such objects are edge-adjacency-connected sets of triangle tiles that have no end tiles, and the point set union of all their tiles forms a simple polygon. Nevertheless, the boundary paths of the objects are not assumed to be simple. Then the MPP is a weakly simple polygon that coincides with the relative convex hull (i.e., geodesic hull) of a set $A$ with respect to a simple polygon $B$, where $A\subset B$, but $A$ is not necessarily a polygon, in fact it is generally not connected. Our MPP algorithm relies on constructing and iteratively constraining cones of visibility through forthcoming boundary tiles, it uses the structure of the canonical boundary path, the MPP frontier is the shortest polygonal curve following this path. We also propose a boundary tracing algorithm to obtain such paths from the objects.