Non-Recursive Algorithm

Analyzing the time efficiency of non recursive algorithms. Let us start with a very simple example that demonstrates all the principal steps typically taken in analyzing such algorithms.

EXAMPLE 1: Consider the problem of finding the value of the largest element in a list of n numbers. For simplicity, we assume that the list is implemented as an array. The following is pseudocode of a standard algorithm for solving the problem.

ALGORITHM MaxElement(A[0..n − 1])
//Determines the value of the largest element in a given array
//Input: An array A[0..n − 1] of real numbers
//Output: The value of the largest element in A
maxval ← A[0]
for i ← 1 to n − 1 do
if A[i] > maxval
maxval ← A[i]
return maxval

The obvious measure of an input’s size here is the number of elements in the array, i.e., n. The operations that are going to be executed most often are in the algorithm’s for loop. There are two operations in the loop’s body: the comparison A[i] > maxval and the assignment maxval ← A[i]. Which of these two operations should we consider basic? Since the comparison is executed on each repetition of the loop and the assignment is not, we should consider the comparison to be the algorithm’s basic operation. Note that the number of comparisons will be the same for all arrays of size n; therefore, in terms of this metric, there is no need to distinguish among the worst, average, and best cases here.

Let us denote C(n) the number of times this comparison is executed and try to find a formula expressing it as a function of size n. The algorithm makes one comparison on each execution of the loop, which is repeated for each value of the loop’s variable i within the bounds 1 and n − 1, inclusive. Therefore, we get the following sum for C(n):

This is an easy sum to compute because it is nothing other than 1 repeated n − 1 times. Thus,

Here is a general plan to follow in analyzing non recursive algorithm.

General Plan for Analyzing the Time Efficiency of Non-recursive Algorithms

1. Decide on a parameter (or parameters) indicating an input’s size.
2. Identify the algorithm’s basic operation. (As a rule, it is located in the innermost loop.)
3. Check whether the number of times the basic operation is executed depends only on the size of an input. If it also depends on some additional property, the worst-case, average-case, and, if necessary, best-case efficiencies have to be investigated separately.
4. Set up a sum expressing the number of times the algorithm’s basic operation is executed.
5. Using standard formulas and rules of sum manipulation, either find a closed form formula for the count or, at the very least, establish its order of growth.