Technicalities

In practice, we neglect certain technical details when we state and solve recurrences. A good example of a detail that is often glossed over is the assumption of integer arguments to functions. Normally, the running time T (n) of an algorithm is only defined when n is an integer, since for most algorithms, the size of the input is always an integer. For example, the recurrence describing the worst-case running time of MERGE-SORT is really

Boundary conditions represent another class of details that we typically ignore. Since the running time of an algorithm on a constant-sized input is a constant, the recurrences that arise from the running times of algorithms generally have T(n) = Θ(1) for sufficiently small n. Consequently, for convenience, we shall generally omit statements of the boundary conditions of recurrences and assume that T (n) is constant for small n. For example, we normally state recurrence (4.1) as

without explicitly giving values for small n. The reason is that although changing the value of T (1) changes the solution to the recurrence, the solution typically doesn't change by more than a constant factor, so the order of growth is unchanged.

When we state and solve recurrences, we often omit floors, ceilings, and boundary conditions. We forge ahead without these details and later determine whether or not they matter. They usually don't, but it is important to know when they do. Experience helps, and so do some theorems stating that these details don't affect the asymptotic bounds of many recurrences encountered in the analysis of algorithms (see Theorem 4.1). In this chapter, however, we shall address some of these details to show the fine points of recurrence solution methods.