Some results on the asymptotic behavior of functions on subsets of the natural numbers

When analyzing the computational complexity of divide and conquer algorithms, the complexity function is usually specified by means of a recurrence relation where the argument is restricted to a subset S of the natural numbers, N. This system is then used to characterize the asymptotic behavior of the algorithm for a corresponding restricted set of inputs. A careful and sometimes complicated argument is then carried out to make assertions about the asymptotic behavior of the algorithm for all inputs (Aho, Hopcroft and Ullman [1974]). In certain cases, the argument can be greatly simplified if the complexity function and the set S have special properties. In this paper, we develop these properties for some important classes of asymptotic behavior.