Common Big-O Notations

Big-O notation is a way to measure the time and space complexity of an algorithm. It describes the upper bound of the complexity in the worst-case scenario. Let’s look into the different types of time complexities:

1. Linear Time Complexity: Big O(n) Complexity

Linear time complexity means that the running time of an algorithm grows linearly with the size of the input.

For example, consider an algorithm that traverses through an array to find a specific element:

Code Snippet

bool findElement(int arr[], int n, int key)
{
    for (int i = 0; i < n; i++) {
        if (arr[i] == key) {
            return true;
        }
    }
    return false;
}

2. Logarithmic Time Complexity: Big O(log n) Complexity

Logarithmic time complexity means that the running time of an algorithm is proportional to the logarithm of the input size.

For example, a binary search algorithm has a logarithmic time complexity:

Code Snippet

int binarySearch(int arr[], int l, int r, int x)
{
    if (r >= l) {
        int mid = l + (r - l) / 2;
        if (arr[mid] == x)
            return mid;
        if (arr[mid] > x)
            return binarySearch(arr, l, mid - 1, x);
        return binarySearch(arr, mid + 1, r, x);
    }
    return -1;
}

3. Quadratic Time Complexity: Big O(n²) Complexity

Quadratic time complexity means that the running time of an algorithm is proportional to the square of the input size.

For example, a simple bubble sort algorithm has a quadratic time complexity:

Code Snippet

void bubbleSort(int arr[], int n)
{
    for (int i = 0; i < n - 1; i++) {
        for (int j = 0; j < n - i - 1; j++) {
            if (arr[j] > arr[j + 1]) {
                swap(&arr[j], &arr[j + 1]);
            }
        }
    }
}

4. Cubic Time Complexity: Big O(n³) Complexity

Cubic time complexity means that the running time of an algorithm is proportional to the cube of the input size.

For example, a naive matrix multiplication algorithm has a cubic time complexity:

Code Snippet

void multiply(int mat1[][N], int mat2[][N], int res[][N])
{
    for (int i = 0; i < N; i++) {
        for (int j = 0; j < N; j++) {
            res[i][j] = 0;
            for (int k = 0; k < N; k++)
                res[i][j] += mat1[i][k] * mat2[k][j];
        }
    }
}

5. Polynomial Time Complexity: Big O(n^k) Complexity

Polynomial time complexity refers to the time complexity of an algorithm that can be expressed as a polynomial function of the input size n. In Big O notation, an algorithm is said to have polynomial time complexity if its time complexity is O(n^k), where k is a constant and represents the degree of the polynomial.

Algorithms with polynomial time complexity are generally considered efficient, as the running time grows at a reasonable rate as the input size increases. Common examples of algorithms with polynomial time complexity include linear time complexity O(n), quadratic time complexity O(n²), and cubic time complexity O(n³).

6. Exponential Time Complexity: Big O(2ⁿ) Complexity

Exponential time complexity means that the running time of an algorithm doubles with each addition to the input data set.

For example, the problem of generating all subsets of a set is of exponential time complexity:

Code Snippet

void generateSubsets(int arr[], int n)
{
    for (int i = 0; i < (1 << n); i++) {
        for (int j = 0; j < n; j++) {
            if (i & (1 << j)) {
                cout << arr[j] << " ";
            }
        }
        cout << endl;
    }
}

Factorial Time Complexity: Big O(n!) Complexity

Factorial time complexity means that the running time of an algorithm grows factorially with the size of the input. This is often seen in algorithms that generate all permutations of a set of data.

Here’s an example of a factorial time complexity algorithm, which generates all permutations of an array:

Code Snippet

void permute(int* a, int l, int r)
{
    if (l == r) {
        for (int i = 0; i <= r; i++) {
            cout << a[i] << " ";
        }
        cout << endl;
    }
    else {
        for (int i = l; i <= r; i++) {
            swap(a[l], a[i]);
            permute(a, l + 1, r);
            swap(a[l], a[i]); // backtrack
        }
    }
}

If we plot the most common Big O notation examples, we would have graph like this:

Big O Notation Tutorial – A Guide to Big O Analysis

Big O notation is a powerful tool used in computer science to describe the time complexity or space complexity of algorithms. It provides a standardized way to compare the efficiency of different algorithms in terms of their worst-case performance. Understanding Big O notation is essential for analyzing and designing efficient algorithms.

In this tutorial, we will cover the basics of Big O notation, its significance, and how to analyze the complexity of algorithms using Big O.

Table of Content

What is Big-O Notation?
Definition of Big-O Notation:
Why is Big O Notation Important?
Properties of Big O Notation
Common Big-O Notations
How to Determine Big O Notation?
Mathematical Examples of Runtime Analysis
Algorithmic Examples of Runtime Analysis
Algorithm Classes with Number of Operations and Execution Time
Comparison of Big O Notation, Big Ω (Omega) Notation, and Big θ (Theta) Notation
Frequently Asked Questions about Big O Notation

Common Big-O Notations

1. Linear Time Complexity: Big O(n) Complexity

2. Logarithmic Time Complexity: Big O(log n) Complexity

3. Quadratic Time Complexity: Big O(n2) Complexity

4. Cubic Time Complexity: Big O(n3) Complexity

5. Polynomial Time Complexity: Big O(nk) Complexity

6. Exponential Time Complexity: Big O(2n) Complexity

Factorial Time Complexity: Big O(n!) Complexity

Big O Notation Tutorial – A Guide to Big O Analysis

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3. Quadratic Time Complexity: Big O(n²) Complexity

4. Cubic Time Complexity: Big O(n³) Complexity

5. Polynomial Time Complexity: Big O(n^k) Complexity

6. Exponential Time Complexity: Big O(2ⁿ) Complexity