Counting independent terms in big-oh notation

TL;DR

This paper explores the complexity of determining the number of independent terms in big-oh notation, revealing that even for polynomial bounds with multiple variables, the possibilities can be arbitrarily large, highlighting the notation's complexity.

Contribution

It demonstrates the inherent difficulty in identifying big-oh bounds from performance data, especially showing the potential for an unbounded number of possibilities even within polynomial constraints.

Findings

Number of polynomial big-oh bounds can be arbitrarily large for multiple variables.

Determining big-oh profiles from data is computationally complex.

Highlights counter-intuitive aspects of big-oh notation.

Abstract

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 its performance as functions of the algorithm's variables, particularly upper bounds expressed via the big-oh notation. But if we were given some inscrutable code and were asked to figure out its big-oh profile from performance data on a given set of inputs, how hard would we have to grapple with the various possibilities before zooming in on a reasonably small set of candidates? Here we show that, even if we restricted our search to upper bounds given by polynomials, the number of possibilities could be arbitrarily large for two or more variables. This is unexpected, given the available body of examples on algorithmic efficiency, and serves to illustrate the many facets of the big-oh notation, as well as its counter-intuitive twists.