On the Computational Complexity of Finding the Maxima of a Set of Vectors

Let U1,U2,...,Udbe totally ordered sets and let V be a set of n d-dimensional vectors in U1× U2× ... × Ud. A partial ordering is defined on V in a natural way. We consider the problem of finding all maximal elements of V with respect to the partial ordering. The computational complexity of the problem is defined to be the maximum number of required comparisons of two components and is denoted by Cd(n). It is trivial that C1(n) = n-1 and Cd(n) ≤ 0 (n2) for d ≥ 2. previous results are Cd(n) ≤ 0 (n log2n) for d = 2,3. in this paper, we show 1. Cd(n) ≤ 0 (n(log2n)d-2) for d ≥ 4. 2. Cd(n) ≥ ⌈log2n! ⌉ for d ≥ 2.