Efficient Enumeration of Enclosed Vector Spaces

Abstract

In this paper, we address several problems concerning vector spaces enclosed in a given set. Let V be a vector space over a finite field of cardinality c, and let $S \subseteq V$ be a set of vectors. A space enclosed in S is a vector subspace W of V that is also contained in S: $W \subseteq S$. We focus on enumeration problems, where the task is to list all solutions, and we first provide an algorithm to enumerate all spaces that are enclosed in S. Our algorithm is further adapted to solve two more problems: the enumeration of (inclusion-)maximal enclosed spaces, and the problem of finding an enclosed space of maximum dimension. The latter problem arises in the context of Boolean functions' regularity detection. It can also be seen as a dual version of the well-known linear span: indeed, the span is the minimum-dimension vector space that contains a given set of vectors S, and it is a fundamental concept in linear algebra. Our proposed algorithms are based on the binary partition paradigm, and have total time complexity $e^{\frac{1}{2\ln c}\ln^2 n - Θ(\log n \log \log n)}$, where $n= |\inputset|$. The first version, for enumerating all enclosed spaces, also achieves a delay (time between consecutive outputs) of O(n). Our algorithms provide a quadratic speed-up with respect to a brute-force approach, although the speed-up appears even greater in our experimental evaluation on boolean vector spaces.