This is for a diff utility I'm writing in C++.
I have a list of n character-sets {"a", "abc", "abcde", "bcd", "de"} (taken from an alphabet of k=5 different letters). I need a way to observe that the entire list can be constructed by disjunctions of the character-sets {"a", "bc", "d", "e"}. That is, "b" and "c" are linearly dependent, and every other pair of letters is independent.
In the bit-twiddling version, the character-sets above are represented as {10000, 11100, 11111, 01110, 00011}, and I need a way to observe that they can all be constructed by ORing together bitstrings from the smaller set {10000, 01100, 00010, 00001}.
In other words, I believe I'm looking for a "discrete basis" of a set of n different bit-vectors in {0,1}k. This paper claims the general problem is NP-complete... but luckily I'm only looking for a solution to small cases (k < 32).
I can think of really stupid algorithms for generating the basis. For example: For each of the k2 pairs of letters, try to demonstrate (by an O(n) search) that they're dependent. But I really feel like there's an efficient bit-twiddling algorithm that I just haven't stumbled upon yet. Does anyone know it?
EDIT: I ended up not really needing a solution to this problem after all. But I'd still like to know if there is a simple bit-twiddling solution.