A note on counting independent terms in asymptotic expressions of computational complexity

The field of computational complexity is concerned both with the intrinsic hardness of computational problems and with the efficiency of algorithms to solve them. Given such a problem, normally one designs an algorithm to solve it and sets about establishing bounds on the algorithm’s performance as functions of the relevant input-size descriptors, particularly upper bounds expressed via the big-oh notation. In some cases, however, especially those arising in the fields of experimental algorithmics and optimization, one may have to resort to performance data on a given set of inputs in order to figure out the algorithm’s big-oh profile. In this note, we are concerned with the question of how many candidate expressions may have to be taken into account in such cases. We show that, even if we only considered upper bounds given by polynomials, the number of possibilities could be arbitrarily large for two or more descriptors. This is unexpected, given the available body of examples on algorithmic efficiency, and underlines the importance of careful and meticulous criteria. It also serves to illustrate the many facets of the big-oh notation, as well as its counter-intuitive twists.