Big O Notation

Definition

The big o notation is simplified analysis of an algorithm’s efficiency. It is also known as Landau’s Symbol. Big O Notation is used in Computer Science and Mathematics to describe the Asymptotic behavior of functions. Let’s study some of Big O Notation characteristics;

  1. Big O Notation enlightens about the complexity of a target algorithm for a given input considered as “N”,
  2. Big O Notation is the abstraction of the efficiency in terms of the machine independent, it shall perform the same way on every operation system and hardware.
  3. Big O Notation does not concern about how much of time that an algorithm takes but how it performs under certain situations.
  4. Big O Notation gives us the “Time” and the “Space” constraints.

In addition to the second characteristic. When we run a program, we will have performance that is how much of time or hardware resource is used, and complexity how the algorithm acts and grows. Note that the Complexity affects the performance, but the other way around is not possible.

Furthermore, there are three types of measurements that I’ll explain and demonstrate those measurements in a different chapter.

Rules

Drop the constants

If you have a function that has a running time of 5N, that is realized as O(n). Because n gets bigger and the 5 is not the consideration anymore,

Observe all inputs

Different inputs and variables have different weights on identifying the notation. If you iterate in two different arrays you get O(a*b). Study the following pseudo code;

method(int[] array1, int[] array2){

For(int a : array1){//O(a)

For(int b : array1){//O(b)

System.out.println(“match”);//O(1)

}

}

}

Drop the low order terms

Certain terms dominate the other terms so in this case drop the lower terms here is the sequence of the notations:

O(1) < O(log n) < O(n) < O(n log n) < O(n2) < O(2n) < O(n!)

Complexities

Big O Notation can be used to describe the Time Complexity and the Space Complexity of algorithms. Each of these subjects are different terms.

Time Complexity

The Time Complexity corresponds to the amount of time that an Algorithm takes to run. The Time Complexity has also Best, Average and Worst cases.

Space Complexity

The Space Complexity describes how much of a space the algorithm allocates in the memory according to the amount of given data.

Cases

In Big O Notation we have three cases Worst, Best and Average cases. When algorithms are analyzed, generally “Worst Case” is referred. It doesn’t mean that the rest of the cases are less important, but depending on the input, the Worst Case has a weight.

Notations

image courtesy www.bigocheatsheet.com

O(1)

Constant time is a basic statement that has only Constants, in a different way the values that are solid and will not change. Regardless of amount of the data, the code executes the process in the time amount, this can be a variable definition, access in an array or a print out. The simplest examples would go for this:

Int x = (9/2)*12-1;

To find out the Big O Notation of such constants;

1.int a = (9/2)*12-1;//O(1)

2.int b = 100/2; //O(1)

3.int result a+b; //O(1)

System.out.println(result); //O(1)

Total time spent: O(1) + O(1) + O(1) + O(1) = O(1)

4*O(1)

(Constants are dropped)

O(n)

linear time is known as the completion time grows for the given amount of data. A good example to it is the linear search in a loop that iterates through N amount of elements.

1.for(int i=0; i<N; i++){//O(n)

2.System.out.println(i)//O(1)

}

Total time spent: O(n)*O(1)=O(n)

1.int x = 55*3+(10-9); //O(1)

for(int i=0; i<N; i++){//O(n){
3.System.out.println(i)//O(1)

}

Total time spent: O(1) + O(n)*O(1)=O(n)

(Drop low order terms)

O(n2)

the time completion of quadratic algorithms is proportional to the square of the given amount of data. Commonly we spot these sorts of algorithms in nested loops. In addition, the more nested loops exist, the square will become cubic O(n3) or more. The good example is the Bubble Sort. Furthermore, such algorithms dramatically hamper the performance if the amount of data increases.

O(2N)

Exponential algorithms are those whose performance doubles for every given input. We can basically spot such algorithms in recursive methods of calculations. For the sake of simplicity, I’ll give an example of Fibonacci numbers. As you will see, the method will call itself twice for the given input.

public int fibonacciNumbers(int number) {

  if (number <= 1) {

    return number;

  } else {

    return fibonacciNumbers(number - 1) + fibonacciNumbers(number - 2);

  }

}

O(n!)

Factorial algorithm that calculates all permutation of a given array is considered as O(n!). The most suitable example of this is the travelling sales man problem with the brute-force search.

O(log n)

The best way to explain logarithmic algorithms is to search in chunks of data by halving them and the rest will be assumed as O(n). Since we split it we will gain some time while looking up. The good world example would be the Binary Search Tree for these types of algorithms and they are very efficient, because increasing the amount of data has a minor effect at some point, because the amount of data is halved each time as it is seen with the Binary Search Tree.

O(n log n)

Quasilinear algorithms are the ones hard to spot. The given values will be compared once. Essentially each comparison will reduce the possible final sorted data structure in half like in O(log n). Furthermore, the number of comparisons that will be performed is equal to log in the factorial of “n”. The comparisons are also equal to n log n this is how O(n log n) algorithms are formed. The mathematical representation as follows;

Sum of comparisons are equal to log n!

Divided comparisons are equal to log n + log(n-1) + … +log(1)//End of comparisons

The outcome is equal to n log n

Examples of Quasilinear complexities are Merge and Heap Sorts.

Further Reading

  1. Big O Cheat Sheet: A great web site that points out the Time and Space complexities of Data Structures and Algorithms.
  2. Stackoverflow Post: While making my research, I found an explanation of John very easy to explain Notations by using the same example.

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