Stable placements for spider robots

We study the problem of computing the set of admissible and stable placements of spider robots, a simple case of legged robots. The environment consists of a set of <italic>n</italic> points in the plane representing authorized footholds. We show that the space of admissible and stable placements of such robots has size &thgr;(<italic>n</italic><supscrpt>2</supscrpt>) and can be constructed in <italic>O</italic>(<italic>n</italic><supscrpt>2</supscrpt> log <italic>n</italic>) time and <italic>O</italic>(<italic>n</italic><supscrpt>2</supscrpt>) space. We give also efficient algorithms for several related problems.