Time Complexity, Space Complexity, Asymptotic Notations - The Coding Shala

Home >> Algorithms >> Time and space complexity

Time Complexity, Space Complexity, and Asymptotic Notations

Time complexity describes the time taken by an algorithm and space complexity describes the memory used by an algorithm. Asymptotic Notations are languages that allow us to calculate time complexity and space complexity. Big O is most commonly used for time complexity or analysis of algorithms.

Time Complexity

In Computer Science, the time complexity of an algorithm quantifies the amount of time taken by an algorithm to run as a function of the length of the string representing the input. The time complexity of an algorithm is commonly expressed using big O notation, which excludes coefficients and lowers order terms.[From Wikipedia] In a simple way we can say how much time our code takes to execute. 

So, How to calculate the running time of an algorithm?
We can have three cases to analyze an algorithm:
  1. Worst Case
  2. Average Case
  3. Best Case
Generally, we calculate the worst-case to show the time complexity of an algorithm. In the worst-case analysis, we calculate the upper bound on the running time of an algorithm. Here, upper bound means our code won't take more time than we have described for it.

Asymptotic Notations

Asymptotic notations are mathematical tools to represent the time complexity of the algorithm. The following 3 asymptotic notation is mostly used to represent the time complexity of the algorithm.

Theta Notation(Θ):

The theta notation bounds a function from above and below, so it defines exact asymptotic behavior. For a given function g(n), we denote Θ(g(n)) is following set of functions.


Θ(g(n)) = {  f(n): there exist positive constants c1, c2 and n0 such that 

                   0 <= c1*g(n) <= f(n) <= c2*g(n)

                   for all n >= n0 }



The above definition means, if f(n) is theta of g(n), then the value f(n) is always between c1*g(n) and c2*g(n) for large values of n (n >= n0). The definition of theta also requires that f(n) must be non-negative for values of n greater than n0.

Big O Notation(O):


The Big O notation defines an upper bound of an algorithm, it bounds a function only from above.



O(g(n)) = {   f(n): there exist positive constants c and 
                    n0 such that 0 <= f(n) <= cg(n) for 
                    all n >= n0  }

Omega Notation(Ω):


Omega(Ω) notation provides an asymptotic lower bound. 

Ω(g(n)) = {   f(n): there exist positive constants c and
                    n0 such that 0 <= cg(n) <= f(n) for
                    all n >= n0 }
Asymptotic Notations - The Coding Shala
Let's see how to use these asymptotic notations.

Some rules to the analysis of loops

The following are some rules good to be remembered while we work with loops:
  • The time complexity of a function is considered as O(1) if it doesn't contain a loop.
  • The time complexity of a loop is considered as O(n) if the loop variables are incremented/decremented by a constant amount.
  • The time complexity of nested loops is equal to the number of times the innermost statement is executed. 
  • Time Complexity of a loop is considered as O(Logn) if the loop variables are divided/multiplied by a constant amount.
  • Time Complexity of a loop is considered as O(LogLogn) if the loop variables are reduced/increased exponentially by a constant amount.
  • When there are consecutive loops, we calculate time complexity as the sum of time complexities of individual loops.

Space Complexity

Space Complexity of an algorithm is the total space taken by the algorithm with respect to the input size. Space complexity includes both Auxiliary space(extra space or temporary space used by an algorithm) and space used by input.

Time and Space Complexity Example

Example 1:
What is the time, space complexity of the following code?

int a = 0, b = 0; 
for (i = 0; i < N; i++) {
a = a + rand(); 
}
for (j = 0; j < M; j++) {
b = b + rand();


}
Assume that rand() is O(1) time, O(1) space function.

Solution: O(max(N,M)) or we can say O(N+M) time, O(1) space.


Other Posts You May Like
Please leave a comment below if you like this post or found some error, it will help me to improve my content.
By

Comments

Post a Comment

Popular Posts from this Blog

Java Program to Convert Binary to Decimal - The Coding Shala

Home >> Java Programs >> Convert Binary to Decimal  Hey there, welcome back to another post. In this post, we will learn how to convert Binary to Decimal in Java. Java Program to Convert Binary to Decimal You have given a binary number, convert it to a decimal number. Example:  Input: 101 Output: 5 Input: 111 Output: 7 Binary to Decimal Conversion in Java Approach We will the below formula to convert binary to decimal: decimal number = (2^0) * (rightmost digit) + (2^1) * (second rightmost digit) + ...  for example:  binary number = 110 decimal number = 2^0 * 0 + 2^1 * 1 + 2^2 * 1 = 1 * 0 + 2 * 1 + 4 * 1 = 0 + 2 + 4 = 6 Java Program:  import java.util.Scanner; /** * https://www.thecodingshala.com/ */ public class Main { public static void printDecimal( int num) { int decimal = 0; int twos = 1; // 2^0 = 1 initial value while (num > 0) { int temp = num
Read more

N-th Tribonacci Number Solution - The Coding Shala

Home >> Programming >> N-th Tribonacci Number  Hey there, welcome back to another post. In this post, we will learn how to solve the N-th Tribonacci Number problem and will implement its solution in Java. N-th Tribonacci Number Problem Statement The Tribonacci sequence Tn is defined as follows: T0 = 0, T1 = 1, T2 = 1, and Tn+3 = Tn + Tn+1 + Tn+2 for n >= 0. Given n, return the value of Tn.  Example 1:  Input: n = 4 Output: 4 Explanation: T_3 = 0 + 1 + 1 = 2 T_4 = 1 + 1 + 2 = 4 N-th Tribonacci Number Java Solution using Bottom-Up DP This problem is similar to the Fibonacci series. We can solve it using the Bottom-Up approach of dynamic programming. Time Complexity: O(n) Space Complexity: O(n) The space complexity can be reduced to O(1) by using variables to store the previous three values instead of using an array. Java Program:  class Solution { public int tribonacci( int n) { if (n == 0) return 0; if (n == 1 || n == 2
Read more

Shell Script to find sum, product and average of given numbers - The Coding Shala

Home >> Scripting >> Sum, product and average Shell Script to find the sum, product, and the average of given numbers Read four integer numbers from the user and find the sum, product, and average of these four numbers? Code:  #! / bin / bash echo "enter four integers" read a b c d sum =$(echo "$a + $b + $c + $d" | bc - l) average=$(echo "$sum / 4" | bc - l) product=$(echo "$a * $b * $c * $d" | bc - l) echo "sum = $sum" echo "Average = $average" echo "Product = $product" Other Posts You May Like Shell script to print sum of the digits of a number Shell Script to print number in descending order Shell script to find out biggest number from three numbers Shell script to find out factorial of a number Shell Script to create a simple calculator Please leave a comment below if you like this post or found some error, it will help me to improve my content.
Read more

Shell Script to Create a Simple Calculator - The Coding Shala

Home >> Scripting >> simple calculator Shell Script to Create a Simple Calculator Write a shell script to create a simple calculator using a switch-case statement? Code:  #! / bin / bash echo "simple calculator" sum = 0 i = "y" echo "enter first number" read n1 echo "enter second number" read n2 while [ $ i = "y" ] do echo "1.Addition" echo "2.Subtraction" echo "3.Multiplication" echo "4.Division" echo "Enter choice" read ch case $ch in 1 ) sum =$(echo " $n1 + $n2" | bc - l) echo "Addition is =" $ sum ;; 2 ) sum =$(echo "$n1 - $n2" | bc - l) echo "Sub is =" $ sum ;; 3 ) sum =$(echo "$n1 * $n2" | bc - l) echo "Mul is =" $ sum ;; 4 ) sum =$(echo "$n1 / $n2" | bc - l) echo "div is =" $ sum ;; * )echo "invalid choice" esac echo "Do y
Read more

Java Program to Convert Decimal to Binary - The Coding Shala

Home >> Java Programs >> Convert Decimal to Binary  Hey there, welcome back to another post. In this post, we will learn how to convert decimal to binary in Java. Java Program to Convert Decimal to Binary You have given a decimal number. Write a Java program to convert decimal to binary. Example 1:  Input: 7 Output: 111 Input: 10 Output: 1010 Decimal to Binary Conversion using Array Approach To convert decimal to binary we can follow the below steps: Store the remainder in the array when we divide the number by 2. Divide the number by 2. Follow the above two steps until the number is above 0. Print the array in reverse order. For example:  4 % 2 = 0 => 4/2 = 2 2 % 2 = 0 => 2/2 = 1 1 % 2 = 1 => 1/2 = 0 Binary of 4 is: 100 Java Program:  import java.util.ArrayList; import java.util.List; import java.util.Scanner; /** * https://www.thecodingshala.com/ */ public class Main { public static void printBinary( int num) {
Read more