What is Logarithmic Time Complexity? A Complete Tutorial

Logarithmic time complexity is denoted as O(log n). It is a measure of how the runtime of an algorithm scales as the input size increases. In this comprehensive tutorial. In this article, we will look in-depth into the Logarithmic Complexity. We will also do various comparisons between different logarithmic complexities, when and where such logarithmic complexities are used, several examples of logarithmic complexities, and much more. So let's get started.

What is a Logarithm?

The power to which a base needs to be raised to reach a given number is called the logarithm of that number for the respective base.
For finding logarithmic two necessary factors that need to be known are base and number. 

What is Complexity Analysis?

The primary motive to use DSA is to solve a problem effectively and efficiently. How can you decide if a program written by you is efficient or not? This is measured by complexities. Complexity is of two types:

What is Space Complexity?

The space Complexity of an algorithm is the space taken by an algorithm to run the program for a given input size. The program has some space requirements necessary for its execution these include auxiliary space and input space. The important standard for comparison of algorithms is the space taken by the algorithm to run for a given input size Hence it needs to be optimized.

What is Time Complexity?

In Computer science, there are various problems and several ways to solve each of these problems using different algorithms. These algorithms may have varied approaches, some might be too complex to Implement while some may solve the problem in a lot simpler way than others. It is hard to select a suitable and efficient algorithm out of all that are available. To make the selection of the best algorithm easy, calculation of complexity and time consumption of an algorithm is important this is why time complexity analysis is important, for this asymptotic analysis of the algorithm is done. 

There are three cases denoted by three different notations of analysis:

• Big-oh(O) Notation: Denotes the upper bound of any algorithm's runtime i.e. time is taken by the algorithm in the worst case.
• Big-omega(Ω) Notation: Denotes the best runtime of an algorithm.
• Big-Theta(Θ) notation: Denotes average case time complexity.

How to measure complexities? 

Below are some major order of Complexities are:

• Constant: If the algorithm runs for the same amount of time every time irrespective of the input size. It is said to exhibit constant time complexity.
• Linear: If the algorithm runtime is linearly proportional to the input size then the algorithm is said to exhibit linear time complexity.
• Exponential: If the algorithm runtime depends on the input value raised to an exponent then it is said to exhibit exponential time complexity.
• Logarithmic: When the algorithm runtime increases very slowly compared to an increase in input size i.e. logarithm of input size then the algorithm is said to exhibit logarithmic time complexity.

What is a Logarithm?

The power to which a base needs to be raised to reach a given number is called the logarithm of that number for the respective base.
For finding logarithmic two necessary factors that need to be known are base and number. 

Examples:

logarithm of 8 for base 2 = log₂(8) = 3, 
Explanation: 2³ = 8 Since 2 needs to be raised to a power of 3 to give 8, Thus logarithm of 8 base 2 is 3.

logarithm of 81 for base 9 = log₉(81) = 2,
Explanation: 9² = 81 Since 9 needs to be raised to a power of 2 to give 81, Thus logarithm of 81 base 9 is 2.

Note: An exponential function is the exact opposite of a logarithmic function. When a value is being multiplied repeatedly it is said to grow exponentially whereas when the value is being divided repeatedly it is said to grow logarithmically.

Different Types of Logarithmic Complexities

Now that we know what is a logarithm, let's deep dive into different types of logarithmic complexities that exists, such as:

Simple Log Complexity (Log_a b)

Simple logarithmic complexity refers to log of b to the base a. As mentioned, it refers to the time complexity in terms of base a. In design and analysis of algorithms, we generally use 2 as the base for log time complexities. The below graph shows how the simple log complexity behaves.

 There are several standard algorithms that have logarithmic time complexity:

• Merge sort
• Binary search
• Heap sort

Double Logarithm (log log N)

Double logarithm is the power to which a base must be raised to reach a value 'x' such that when the base is raised to a power 'x' it reaches a value equal to given number.

Example:

logarithm (logarithm (256)) for base 2 = log₂(log₂(256)) = log₂(8) = 3. 

Explanation: 2⁸ = 256, Since 2 needs to be raised to a power of 8 to give 256, Thus logarithm of 256 base 2 is 8. Now 2 needs to be raised to a power of 3 to give 8 so log₂(8) = 3.

N logarithm N (N * log N)

N*logN complexity refers to product of N and log of N to the base 2. N * log N time complexity is generally seen in sorting algorithms like Quick sort, Merge Sort, Heap sort. Here N is the size of data structure (array) to be sorted and log N is the average number of comparisons needed to place a value at its right place in the sorted array. 

logarithm² N (log² N)

log² N complexity refers to square of log of N to the base 2. 

N² logarithm N (N² * log N)

N²*log N complexity refers to product of square of N and log of N to the base 2. This Order of time complexity can be seen in case where an N * N * N 3D matrix needs to be sorted along the rows. The complexity of sorting each row would be N log N and for N rows it will be N * (N * log N). Thus the complexity will be N² log N,

N³ logarithm N (N³ log N)

N³*log N complexity refers to product of cube of N and log of N to the base 2. This Order of time complexity can be seen in cases where an N * N matrix needs to be sorted along the rows. The complexity of sorting each row would be N log N and for N rows it will be N * (N * log N) and for N width it will be N * N * (N log N). Thus the complexity will be N³ log N,

logarithm √N (log √N)

log √N complexity refers to log of square root of N to the base 2.

Examples To Demonstrate Logarithmic Time Complexity

Example 1: log_a b

Task: We have a number N which has an initial value of 16 and the task is to reduce the given number to 1 by repeated division of 2. 
Approach:

• Initialize a variable number_of_operation with a value 0 .
• Run a for loop from N till 1.
  • In each iteration reduce the value of N to half.
  • Increment the number_of_operation variable by one.
• Return the number_of_operation variable.

Implementation:

// C++ code for reducing a number to its logarithm
#include <bits/stdc++.h>
using namespace std;

int main()
{

    int N = 16;
    int number_of_operations = 0;

    cout << "Logarithmic reduction of N: ";
    for (int i = N; i > 1; i = i / 2) {
        cout << i << " ";
        number_of_operations++;
    }
    cout << '\n'
         << "Algorithm Runtime for reducing N to 1: "
         << number_of_operations;
}

/*package whatever //do not write package name here */

import java.io.*;

class GFG {
    public static void main (String[] args) {
        int N = 16;
        int number_of_operations = 0;

        System.out.print("Logarithmic reduction of N: ");
        for (int i = N; i > 1; i = i / 2) {
            System.out.print(i + " ");
            number_of_operations++;
        }
        System.out.println();
          System.out.print("Algorithm Runtime for reducing N to 1: " + number_of_operations);
    }
}

 # python3 code for the above approach

# Driver Code
if __name__ == "__main__": 
   
    N = 16
    number_of_operations = 0
    print("Logarithmic reduction of N: ", end = "")
    i = N
    while(i>1) :
        print( i , end = " ")
        number_of_operations += 1
        i = i // 2
    
    print()
    print("Algorithm Runtime for reducing N to 1:", number_of_operations)
    
     # This code is contributed by sanjoy_62.

// C# implementation of above approach
using System;
using System.Collections.Generic;

class GFG {

// Driver Code
public static void Main()
{
    int N = 16;
        int number_of_operations = 0;
  
        Console.Write("Logarithmic reduction of N: ");
        for (int i = N; i > 1; i = i / 2) {
            Console.Write(i + " ");
            number_of_operations++;
        }
        Console.WriteLine();
        Console.WriteLine("Algorithm Runtime for reducing N to 1: " + number_of_operations);
}
}

let number_of_operations = 0;
let n= 16;
for(let i=n; i>1; i=i/2) {
   console.log(i);
   number_of_operations++;
}

console.log(number_of_operations);

Logarithmic reduction of N: 16 8 4 2 
Algorithm Runtime for reducing N to 1: 4

Explanation:

It is clear from the above algorithm that in each iteration the value is divided by a factor of 2 starting from 16 till it reaches 1, it takes 4 operations. 

As the input value gets reduced by a factor of 2, In mathematical terms the number of operations required in this case is log₂(N), i.e. log₂(16) = 4.So, in terms of time complexity, the above algorithm takes logarithmic runtime to complete i.e.  log₂(N). 

Example 2: Binary search algorithm (log N)

Linearly Searching a value in an array of size N can be very hectic, even when the array is sorted but using binary search this can be done in a lot easier way and in lesser time as the algorithm reduces the search space by half in each operation thus gives a complexity of log₂(N), Here base is 2 because process repeatedly reduces to half. 

Consider an array Arr[] = {2, 4, 6, 8, 10, 12, 14, 16, 18}, If it is required to find the index of 8 then the algorithm will work as following:

// C++ program for finding the index of 8

#include <iostream>
using namespace std;

int find_position(int val, int Arr[], int n, int& steps)
{
    int l = 0, r = n - 1;

    while (l <= r) {
        steps++;
        int m = l + (r - l) / 2;
        if (Arr[m] == val)
            return m;
        else if (Arr[m] < val)
            l = m + 1;
        else
            r = m - 1;
    }
    return -1;
}

// Driver Code
int main()
{

    int Arr[8] = { 2, 4, 6, 8, 10, 12, 14, 16 };
    int steps = 0;

    // Function Call
    int idx = find_position(8, Arr, 8, steps);
    cout << "8 was present on index: "<<idx << endl;

    // Since the worst case runtime of Binary search is
    // log(N) so the count of steps must be less than log(N)
    cout << "Algorithm Runtime: " << steps << endl;

    return 0;
}

// Java program for finding the index of 8
import java.io.*;
class GFG {

  static int steps = 0;
  static int find_position(int val, int Arr[], int n)
  {
    int l = 0, r = n - 1;

    while (l <= r) {
      steps++;
      int m = l + (r - l) / 2;
      if (Arr[m] == val)
        return m;
      else if (Arr[m] < val)
        l = m + 1;
      else
        r = m - 1;
    }
    return -1;
  }

  // Driver Code
  public static void main (String[] args)
  {

    int Arr[] = { 2, 4, 6, 8, 10, 12, 14, 16 };
    steps = 0;

    // Function Call
    int idx = find_position(8, Arr, 8);
    System.out.println("8 was present on index: "+idx);

    // Since the worst case runtime of Binary search is
    // log(N) so the count of steps must be less than log(N)
    System.out.println("Algorithm Runtime: " + steps);

  }
}

// This code is contributed by Aman Kumar

# Python program for finding the index of 8

def find_position(val,Arr,n):
    global steps
    l=0
    r=n-1
    
    while(l<=r):
        steps+=1
        m=l+(r-l)//2
        if(Arr[m] == val):
            return m
        elif(Arr[m] < val):
            l=m+1
        else:
            l=m-1
            
    return -1
    
# Driver code
Arr=[2, 4, 6, 8, 10, 12, 14, 16]
steps=0

#  Function Call
idx = find_position(8, Arr, 8)

print("8 was present on index: {0}".format(idx))

#  Since the worst case runtime of Binary search is
# log(N) so the count of steps must be less than log(N)
print("Algorithm Runtime: {0}".format(steps))

# This code is contributed by Pushpesh Raj.

using System;

namespace GFG {
    class Program {

        static int steps = 0;

        static int FindPosition(int val, int[] arr, int n) {
            int l = 0, r = n - 1;

            while (l <= r) {
                steps++;
                int m = l + (r - l) / 2;
                if (arr[m] == val) {
                    return m;
                }
                else if (arr[m] < val) {
                    l = m + 1;
                }
                else {
                    r = m - 1;
                }
            }
            return -1;
        }

        static void Main(string[] args) {
            int[] arr = { 2, 4, 6, 8, 10, 12, 14, 16 };
            steps = 0;

            int idx = FindPosition(8, arr, 8);
            Console.WriteLine("8 was present on index: " + idx);

            Console.WriteLine("Algorithm runtime: " + steps);
        }
    }
}
//This code is contributed by Edula Vinay Kumar Reddy

// JavaScript program for finding the index of 8
let steps = 0;

function find_position(val, Arr, n) {
    let l = 0;
    let r = n - 1;

    while (l <= r) {
        steps += 1;
        let m = Math.floor(l + (r - l) / 2);
        if (Arr[m] === val) {
            return m;
        } else if (Arr[m] < val) {
            l = m + 1;
        } else {
            r = m - 1;
        }
    }
    return -1;
}

// Driver code
let Arr = [2, 4, 6, 8, 10, 12, 14, 16];

// Function Call
let idx = find_position(8, Arr, 8);

console.log(`8 was present on index: ${idx}`);

// Since the worst case runtime of Binary search is
// log(N) so the count of steps must be less than log(N)
console.log(`Algorithm Runtime: ${steps}`);

8 was present on index: 3
Algorithm Runtime: 1

Explanation:

Binary search works on Divide and conquer approach, In above example In worst case 3 comparisons will be needed to find any value in array. Also the value of log (N) where N is input size i.e. 8 for above example will be 3. Hence the algorithm can be said to exhibit logarithmic time complexity.

Example 3: Binary search algorithm (log log N)

An example where the time complexity of algorithm is Double logarithmic along with a length factor N is when prime numbers from 1 to N need to be found. 

#include <bits/stdc++.h>
using namespace std;
const long long MAX_SIZE = 1000001;

// isPrime[] : isPrime[i] is true if number is prime
// prime[] : stores all prime number less than N
// SPF[] that store smallest prime factor of number
// [for Exp : smallest prime factor of '8' and '16'
// is '2' so we put SPF[8] = 2 , SPF[16] = 2 ]
vector<long long> isprime(MAX_SIZE, true);
vector<long long> prime;
vector<long long> SPF(MAX_SIZE);

// Function generate all prime number less than N in O(n)
void manipulated_seive(int N)
{
    // 0 and 1 are not prime
    isprime[0] = isprime[1] = false;

    // Fill rest of the entries
    for (long long int i = 2; i < N; i++) {
        // If isPrime[i] == True then i is
        // prime number
        if (isprime[i]) {
            // put i into prime[] vector
            prime.push_back(i);

            // A prime number is its own smallest
            // prime factor
            SPF[i] = i;
        }

        // Remove all multiples of i*prime[j] which are
        // not prime by making isPrime[i*prime[j]] = false
        // and put smallest prime factor of i*Prime[j] as
        // prime[j] [ for exp :let i = 5 , j = 0 , prime[j]
        // = 2 [ i*prime[j] = 10 ] so smallest prime factor
        // of '10' is '2' that is prime[j] ] this loop run
        // only one time for number which are not prime
        for (long long int j = 0;
             j < (int)prime.size() && i * prime[j] < N
             && prime[j] <= SPF[i];
             j++) {
            isprime[i * prime[j]] = false;

            // put smallest prime factor of i*prime[j]
            SPF[i * prime[j]] = prime[j];
        }
    }
}

// Driver program to test above function
int main()
{
    int N = 13; // Must be less than MAX_SIZE

    manipulated_seive(N);

    // Print all prime number less than N
    for (int i = 0; i < prime.size() && prime[i] <= N; i++)
        cout << prime[i] << " ";

    return 0;
}

import java.util.*;

public class Main {
  static final int MAX_SIZE = 1000001;

  // isprime[] : isprime[i] is true if number is prime
  // prime[] : stores all prime numbers less than N
  // SPF[] that store smallest prime factor of number
  // [for Exp : smallest prime factor of '8' and '16'
  // is '2' so we put SPF[8] = 2 , SPF[16] = 2 ]
  static boolean[] isprime = new boolean[MAX_SIZE];
  static List<Integer> prime = new ArrayList<Integer>();
  static int[] SPF = new int[MAX_SIZE];

  // Function generate all prime numbers less than N in
  // O(n)
  static void manipulated_seive(int N)
  {
    Arrays.fill(isprime, true);

    // 0 and 1 are not prime
    isprime[0] = isprime[1] = false;

    // Fill rest of the entries
    for (int i = 2; i < N; i++) {
      // If isprime[i] is true then i is prime number
      if (isprime[i]) {
        // put i into prime[] list
        prime.add(i);

        // A prime number is its own smallest prime
        // factor
        SPF[i] = i;
      }

      // Remove all multiples of i*prime[j] which are
      // not prime by making isprime[i*prime[j]] =
      // false and put the smallest prime factor of
      // i*Prime[j] as prime[j] [for example: let i =
      // 5, j = 0, prime[j] = 2 [i*prime[j] = 10] so
      // the smallest prime factor of '10' is '2' that
      // is prime[j]] this loop runs only one time for
      // numbers which are not prime
      for (int j = 0;
           j < prime.size() && i * prime.get(j) < N
           && prime.get(j) <= SPF[i];
           j++) {
        isprime[i * prime.get(j)] = false;

        // put the smallest prime factor of
        // i*prime[j]
        SPF[i * prime.get(j)] = prime.get(j);
      }
    }
  }

  // Driver program to test above function
  public static void main(String[] args)
  {
    int N = 13; // Must be less than MAX_SIZE

    manipulated_seive(N);

    // Print all prime numbers less than N
    for (int i = 0;
         i < prime.size() && prime.get(i) <= N; i++) {
      System.out.print(prime.get(i) + " ");
    }
  }
}

// This code is contributed by divyansh2212

import math

MAX_SIZE = 1000001

# isprime[]: isprime[i] is True if number is prime
# prime[]: stores all prime numbers less than N
# SPF[] that store smallest prime factor of number
# [for Exp: smallest prime factor of '8' and '16'
# is '2' so we put SPF[8] = 2, SPF[16] = 2]
isprime = [True] * MAX_SIZE
prime = []
SPF = [0] * MAX_SIZE

# Function generate all prime numbers less than N in O(n)
def manipulated_seive(N):
    global isprime, prime, SPF

    # 0 and 1 are not prime
    isprime[0] = isprime[1] = False

    # Fill rest of the entries
    for i in range(2, N):
        # If isprime[i] is True then i is prime number
        if isprime[i]:
            # put i into prime[] list
            prime.append(i)

            # A prime number is its own smallest prime factor
            SPF[i] = i

        # Remove all multiples of i*prime[j] which are
        # not prime by making isprime[i*prime[j]] = False
        # and put the smallest prime factor of i*Prime[j] as
        # prime[j] [for example: let i = 5, j = 0, prime[j]
        # = 2 [i*prime[j] = 10] so the smallest prime factor
        # of '10' is '2' that is prime[j]] this loop runs
        # only one time for numbers which are not prime
        j = 0
        while j < len(prime) and i * prime[j] < N and prime[j] <= SPF[i]:
            isprime[i * prime[j]] = False

            # put the smallest prime factor of i*prime[j]
            SPF[i * prime[j]] = prime[j]

            j += 1

# Driver program to test above function
if __name__ == "__main__":
    N = 13 # Must be less than MAX_SIZE

    manipulated_seive(N)

    # Print all prime numbers less than N
    for i in range(len(prime)):
        if prime[i] <= N:
            print(prime[i], end=" ")
        else:
            break

using System;
using System.Collections.Generic;

class MainClass
{
  static readonly int MAX_SIZE = 1000001;

  // isprime[] : isprime[i] is true if number is prime
  // prime[] : stores all prime numbers less than N
  // SPF[] that store smallest prime factor of number
  // [for Exp : smallest prime factor of '8' and '16'
  // is '2' so we put SPF[8] = 2 , SPF[16] = 2 ]
  static bool[] isprime = new bool[MAX_SIZE];
  static List<int> prime = new List<int>();
  static int[] SPF = new int[MAX_SIZE];

  // Function generate all prime numbers less than N in
  // O(n)
  static void manipulated_seive(int N)
  {
    Array.Fill(isprime, true);

    // 0 and 1 are not prime
    isprime[0] = isprime[1] = false;

    // Fill rest of the entries
    for (int i = 2; i < N; i++)
    {
      // If isprime[i] is true then i is prime number
      if (isprime[i])
      {
        // put i into prime[] list
        prime.Add(i);

        // A prime number is its own smallest prime
        // factor
        SPF[i] = i;
      }

      // Remove all multiples of i*prime[j] which are
      // not prime by making isprime[i*prime[j]] =
      // false and put the smallest prime factor of
      // i*Prime[j] as prime[j] [for example: let i =
      // 5, j = 0, prime[j] = 2 [i*prime[j] = 10] so
      // the smallest prime factor of '10' is '2' that
      // is prime[j]] this loop runs only one time for
      // numbers which are not prime
      for (int j = 0;
           j < prime.Count && i * prime[j] < N
           && prime[j] <= SPF[i];
           j++)
      {
        isprime[i * prime[j]] = false;

        // put the smallest prime factor of
        // i*prime[j]
        SPF[i * prime[j]] = prime[j];
      }
    }
  }

  // Driver program to test above function
  public static void Main(string[] args)
  {
    int N = 13; // Must be less than MAX_SIZE

    manipulated_seive(N);

    // Print all prime numbers less than N
    for (int i = 0;
         i < prime.Count && prime[i] <= N; i++)
    {
      Console.Write(prime[i] + " ");
    }
  }
}

const MAX_SIZE = 1000001;

// isprime[]: isprime[i] is true if number is prime
// prime[]: stores all prime numbers less than N
// SPF[] that store smallest prime factor of number
// [for Exp: smallest prime factor of '8' and '16'
// is '2' so we put SPF[8] = 2, SPF[16] = 2]
let isprime = Array(MAX_SIZE).fill(true);
let prime = [];
let SPF = Array(MAX_SIZE).fill(0);

// Function generate all prime numbers less than N in O(n)
function manipulated_seive(N) {
    // 0 and 1 are not prime
    isprime[0] = isprime[1] = false;

    // Fill rest of the entries
    for (let i = 2; i < N; i++) {
        // If isprime[i] is true then i is prime number
        if (isprime[i]) {
            // put i into prime[] list
            prime.push(i);

            // A prime number is its own smallest prime factor
            SPF[i] = i;
        }

        // Remove all multiples of i*prime[j] which are
        // not prime by making isprime[i*prime[j]] = false
        // and put the smallest prime factor of i*Prime[j] as
        // prime[j] [for example: let i = 5, j = 0, prime[j]
        // = 2 [i*prime[j] = 10] so the smallest prime factor
        // of '10' is '2' that is prime[j]] this loop runs
        // only one time for numbers which are not prime
        let j = 0;
        while (j < prime.length && i * prime[j] < N && prime[j] <= SPF[i]) {
            isprime[i * prime[j]] = false;

            // put the smallest prime factor of i*prime[j]
            SPF[i * prime[j]] = prime[j];

            j++;
        }
    }
}

// Driver program to test above function
const N = 13; // Must be less than MAX_SIZE

manipulated_seive(N);

// Print all prime numbers less than N
console.log(prime.join(' '));

// Contributed by adityasha4x71

2 3 5 7 11 

In above example the complexity of finding prime numbers in a range of 0 to N is O(N * log (log (N))). 

Practice Problems for Logarithmic Time Complexity

Comparison of various Logarithmic Time Complexities

Below is a graph to show the comparison between different logarithmic time complexities that have been discussed above:

Conclusion

From the above discussion, we conclude that the analysis of an algorithm is very important for choosing an appropriate algorithm and the Logarithm order of complexities is one of the most optimal order of time complexities.