# Simplified Voronoi Diagrams

The Voronoi diagram has proved to be a useful tool in a variety of contexts in computational geometry. Our interest here is in using the diagram to simplify the planning of collision-free paths for a robot among obstacles, the so-called generalized movers' problem. The Voronoi diagram, as usually defined, is a strong deformation retract of free space so that free space can be continuously deformed onto the diagram. In particular, any path in free space can be continuously deformed onto the diagram. This means that the diagram is complete for path planning, i.e., searching the original space for paths can be reduced to a search on the diagram. Reducing the dimension of the set to be searched usually reduces the time complexity of the search. Secondly, the diagram leads to robust paths, i.e., paths that are maximally clear of obstacles.