Algorithm Complexity Analysis

Profiler : A tool to measure how ast a program runs.

Complexity Analysis is a tool that allows us to explain how an algorithm behaves as the input grows later.

Example: The maximum element in an array. Given an input array A of size n.

var Max= A[0];
for(var i = 0; i < n ; ++i){
  if(A[i] >= Max){
    M=A[i];
  }
}

Instructions:

Look up A[0]
Assign to M from A[0] -> (M=A[0])
Assign to i -> (var i = 0)
Comparing i < n
For each loop; icrease i (++i) and compare (i<n) so that 2*n instruction each iteration.
Just empty for statement body instructions calculate this:

f(n) = 4 + 2n

For body instructions:

Look up A[i]
Compare A[i] with M
In if statement, look up A[i]
Assign to M from A[i] this four istructions invoke n time iteration (Worst-Case). So that instructions are 4n from for body.

Thus, Instructions calculate this algorithm f(n) = 4 + 2n + 4n = 4 + 6n

When analysing algorithm, we often consider the worst-case scenarios.

Dropping all factors (constants). E.i. For above example drop 4. f(n) = 4 + 6n -> f(n) = 6n -> f(n) = n
Keepin the largest growing term (n, n^2, n^3) Example: f(n) = 2n + 8 -> f(n) = n In math, we can't do. But in computer science, we should do. Because, we should ignore some instructions. For example, in java program, we accept JVM runtime. Another examples;

f(n) = 5n + 12 -> f(n) = n

f(n) = 1234 -> f(n) = 1

f(n) = n^2 + 2n + 8 -> f(n) = n^2

f(n) = n^3 + 3n^2 + 4n + 8 -> f(n) = n^3

huseyinsimsekk / Algorithm