Asymptotic Analysis of Time Complexity in Algorithms

Welcome! In my previous post, I wrote about the use of different notations for Asymptotic (Algorithmic) Analysis. Today, let’s discuss how the analysis is done.

Why is it called Asymptotic Analysis?

In every case, for a given function, f(n), we define a  function g(n) which approximates f(n) at higher values of n. Thus, g(n) is a curve, approximating f(n) at higher values of n. Such a curve is called an Asymptotic Curve. For this reason, the Algorithm Analysis is called Asymptotic Analysis.

Rates of Growth

The Rate at which the running time increases as a function of input is called rate of growth. In case of a total time of execution
C0n2 + C1 + (C2 + C3)n, since the smaller elements and constants are insignificant for large values of n, the actual rate of grown is considered to be n2.
This, rate of growth: O(n2)

Here is a chart with commonly used Rates of Growth:


Performing Asymptotic Analysis

So, with the theory out of our way, how do we actually perform the Analysis?

There are a few general rules I read and have compiled to the best of my abilities. Read on to know more!

1. Consecutive Statements

The running time of each consecutive statement is addded.

    // Constant time
    a = A + 1
    // Executes n times
    for(i = 0; i < n; i++){
        b = b + 2 // Constant time
    }

Total time of execution: constant c * n + c = cn = O(n)

2. Loops

The running time of loops is the running of all the statements inside the loop, multiplied by the number of iterations in the loop.

    // executing n times
    for(i = 0; i < n; i++){
        m = m + i; // constant time
    }

Total time of execution: a constant c * n = cn = O(n)

3. Nested Loops

The running time of nested loops (in my observation) is generally the highest. Analysis is done from inside out. Total running time is the product of the sizes of all the loops.

    // outer loop executing n times
    for(i = 0; i < n; i++){
        // inner loop executing n times
        for(j = 0; j < n; j++ ){
            m = m + i; // constant time
        }
    }

Total time of execution: a constant c * n * n = cn2 = O(n2)

4. Logarithmic Complexity

An Algorithm has logarithmic complexity, if it takes a constant time to cut the problem size by a fraction (usually 1/2). As an example, let us consider the following,

    for(i = 1; i <= n; i *= 2){
        print i
    }

The value of i is doubled after each step. so, in k times execution, after kth step, 2k = n, and at (k + 1)th step, we come out of the loop.

log(2k) = log(n)
k * log(2) = log(n)
// Assuming base – 2
k = log(n)

5. If – else

Worst-case running time is calculated as:
Time(test) + Time of if block or else block, whichever is greater

    // test: constant (C0)
    if(length() == 0){
        return false; // if : constant (C1)
    } else {
        // else : ((C2) + (C3)) * n
        for(n = 0; n < length(); n++){ // for loop : n
            // nested test : (C2)
            if ( list[n] == list[n + 1]){
                return false; // if : (C3)
            }
        }
    } 

Total time of execution: C0 + C1 + (C2 + C3) * n = O(n)

We generally use these guidelines to calculate the Time or Space complexity of an Algorithm.

So, that’s it for now. Stay tuned for more awesome Algorithms!

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