Planning on the cube complex

Once again, any of the planning methods described in Chapters 5 and 6 can be adapted here, but the methods are slightly complicated by the fact that $X$ is a complex. To use sampling-based methods, a dense sequence should be generated over $X$. For example, if random sampling is used, then an $m$-cube can be chosen at random, followed by a random point in the cube. The local planning method (LPM) must take into account the connectivity of the cube complex, which requires recognizing when branches occur in the topological graph. Combinatorial methods must also take into account this connectivity. For example, a sweeping technique can be applied to produce a vertical cell decomposition, but the sweep-line (or sweep-plane) must sweep across the various $m$-cells of the complex.