A Discrete Topological Complexity of Discrete Motion Planning

Abstract: In this paper we generalize the discrete r-homotopy to the discrete (s, r)-homotopy. Then by this notion, we introduce the discrete motion planning for robots which can move discreetly. Moreover, in this case the number of motion planning, called discrete topological complexity, required for these robots is reduced. Then we prove some properties of discrete topological complexity; For instance, we show that a discrete motion planning in a metric space X exists if and only if X is a discrete contractible space. Also, we prove that the discrete topological complexity depends only on the strictly discrete homotopy type of spaces.