Zone Theorem and Polyhedral Decompositions

We use the zone theorem for a better analysis of an algorithm for worst-case optimal convex decomposition of nOD·convex simple polyhedra. This new analysis reveals that the simple algorithm using repeated cutting and splitting through notches which guarantees a worst-case optimal convex decomposition is quite efficient. We establish an Denr + sr:!) time and an D(nr) space bound for the algorithm where the input polyhedron has n 'edges' of which r 'are reilex and :r= ·min(n"T). For most cases r« n in practice. Thus, the second term in the time complexity is dominated by the first one in most oftbe cases in practice. However, our goal, here, is not to present an asymptotically better algorithm for the problem but to establish the fact that a simple algorithm for the problem is actually quite efficient in practice.