A New Algorithm for Enumeration of Cells of Hyperplane Arrangements and a Comparison with Avis and Fukuda's Reverse Search

Abstract

We design a new algorithm, called Incremental Enumeration (IncEnu), for the enumeration of full-dimensional cells of hyperplane arrangements (or dually, for the enumeration of vertices of generator-presented zonotopes). The algorithm is based on an incremental construction of the graph of cells of the arrangement. IncEnu is compared to Avis and Fukuda's Reverse Search (RS), including its later improvements by Sleumer and others. The basic versions of IncEnu and RS are not directly comparable, since they solve different numbers of linear programs (LPs) of different sizes. We therefore reformulate our algorithm as a version that permits comparison with RS in terms of the number of LPs solved. The result is that both IncEnu and RS have “the same” complexity-theoretic properties (compactness, output-polynomiality, worst-case bounds, tightness of bounds). In spite of the fact that IncEnu and RS have the same asymptotic bounds, it is proved that IncEnu is faster than RS by a nontrivial additive term. Our computational experiments show that for most test cases IncEnu is significantly faster than RS in practice. Based on the results obtained, we conjecture that IncEnu is $\mathcal{O}(d)$ times faster for nondegenerate arrangements, where $d$ denotes the dimension of the arrangement.