How can I find T (1) when I measure the complexity of an algorithm

Question

Question 01: How can I find T (1) when I measure the complexity of an algorithm?

For example I have this algorithm

Int Max1 (int *X, int N)
{
  int a ;
  if (N==1) return X[0] ;
  a = Max1 (X, N‐1);
  if (a > X[N‐1]) return a;
  else return X[N‐1];
}

How can I find T(1)?

Question 2 :

T(n)= T(n-1) + 1 ==> O(n)

what is the meaning of the "1" in this equation

cordially

thks Sirco , it's my first question in stackoverflow i don't knew the rules — Ado Riadh, Jan 03 '13 at 15:44
By the way, this question might be more appropriate on Theoretical Computer Science: http://cstheory.stackexchange.com/ — Kyle Strand, Jan 03 '13 at 18:18

AbstractChaos · Answer 1 · 2013-01-03T16:00:17.010

Max1(X,N-1) Is the actual algorithm the rest is a few checks which would be O(1) as regardless of input the time taken will be the same.

The Max1 function I can only assume is finding array highest number in array this would be O(n) as it will increase in time in a linear fashion to the number of input n.

Also as far as I can tell 1 stands for 1 in most algorithms only letters have variable meanings, if you mean how they got

T(n-1) + 1 to O(n), it is due to the fact you ignore coefficients and lower order terms so the 1 is both cases is ignored to make O(n)

score 1 · Answer 2 · answered Jan 03 '13 at 17:20

Answer 1. You are looking for a complexity. You must decide what case complexity you want: best, worst, or average. Depending on what you pick, you find T(1) in different ways:

Best: Think of the easiest input of length 1 that your algorithm could get. If you're searching for an element in a list, the best case is that the element is the first thing in the list, and you can have T(1) = 1.
Worst: Think of the hardest input of length 1 that your algorithm could get. Maybe your linear search algorithm executes 1 instruction for most inputs of length 1, but for the list [77], you take 100 steps (this example is a bit contrived, but it's entirely possible for an algorithm to take more or less steps depending on properties of the input unrelated to the input's "size"). In this case, your T(1) = 100.
Average: Think of all the inputs of length 1 that your algorithm could get. Assign probabilities to these inputs. Then, calculate the average T(1) of all possibilities to get the average-case T(1).

In your case, for inputs of length 1, you always return, so your T(n) = O(1) (the actual number depends on how you count instructions).

Answer 2. The "1" in this context indicates a precise number of instructions, in some system of instruction counting. It is distinguished from O(1) in that O(1) could mean any number (or numbers) that do not depend on (change according to, trend with, etc.) the input. Your equation says "The time it takes to evaluate the function on an input of size n is equal to the time it takes to evaluate the function on an input of size n - 1, plus exactly one additional instruction".

score 1 · Answer 3 · answered Jan 03 '13 at 18:17

T(n) is what's called a "function of n," which is to say, n is a "variable" (meaning that you can substitute in different values in its place), and each particular (valid) value of n will determine the corresponding value of T(n). Thus T(1) simply means "the value of T(n) when n is 1."

So the question is, what is the running-time of the algorithm for an input value of 1?

How can I find T (1) when I measure the complexity of an algorithm

3 Answers3