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# Big O Calculator + Online Solver With Free Steps

**Big-O Calculator** is an online tool that helps you compute the complexity domination of two algorithms. It conveys the rate of growth or decline of a function.

The **Big-O calculator** only considers the dominating term of the function when computing Big-O for a specific function g(n). The term that gets bigger quickly is the dominating term.

For instance, $n^2$ grows faster than n, $ g(n) = 2n^2 + 10n + 13 $ would have a large $ O(n^2) $ complexity. This is somewhat similar to the expedient method of determining limits for **fractional polynomials**, in which you are ultimately just concerned with the dominating term for the **numerators** and **denominators**.

## What Is a Big-O Calculator?

**Big-O CalculatorÂ ****is an online calculator that helps to evaluate the performance of an algorithm. **

As the input increases, it calculates how long it takes to execute the **function** or how effectively the function is scaled. Efficiency is measured in terms of both **temporal complexity** and **spatial complexity**.

The length of the function’s execution in terms of its processing cycles is measured by its **time complexity**. The degree of **space complexity** is related to how much memory the function uses.

The algorithm’s upper bound, **Big-O**, is occasionally used to denote how well it handles the worst scenario. Finding our stuff on the first attempt is the best-case situation, which doesn’t provide us with anything valuable.

## How To Use a Big O Calculator?

You can use the** Big-O Calculator** by following the given detailed guidelines, and the calculator will surely provide you with the desired results. You can therefore follow the given instructions to get the Big-O for the given function.

### Step 1

Enter the dominated function **f(n)** in the provided entry box.

### Step 2

Enter the dominating function **g(n)** in the provided entry box.

### Step 3

Finally, simply click the “**Submit**” button, and the whole step-by-step solution for the Big O domination will be displayed.

As we have discussed before, the** dominating function g(n)** only dominates if the calculated result is zero. As the calculator follows the given notation:

\[\lim_{n\to\infty} \frac{f(n)}{g(n)} = 0 \]

## How Does Big-O Calculator Work?

The** Big O CalculatorÂ **works by calculating the big-O notation for the given functions. It specifically uses the letter **O** since a function’s growth rate is also known as the **function’s order**. A function described in the big O notation usually only provides an upper constraint on the **function’s development rate**.

There must be positive constants c and k such that $ 0 \leq f(n) \leq cg(n) $ for every $ n \geq k $, according to the expression f(n) = O(g(n)). For the function f, the values of **c** and **k** must be constant and independent of n.

The **calculator** eliminates uncertainty by using the worst-case scenario; the algorithm will never do worse than we anticipate.

### Best Case and Worst Case Scenario

We only take into account the worst-case scenario when calculating Big O. However, it can also be crucial to take into account average cases and best-case scenarios.

The** ideal scenario**, for instance, would be if the value was the array’s first item while looking for it in an unsorted array. This would lead to O(1). In contrast, the worst-case scenario would be O(n) if the value sought after was the array’s final item or was not present.

**Best case:** Locate the item in the first place of an array.

**Worst case:** Locate the item in the last place of an array.

### Why Use Big O?

**Big-O** is used because it helps to quickly analyze how fast the function runs depending upon its input. There may be a variety of options for any given issue. However, if you use seconds to estimate execution time, you are subject to variations brought on by physical phenomena.

The amount of storage on the processor required to execute the solution, the CPU speed, and any other algorithms running simultaneously on the system are all examples of this.

To measure the efficiency of an algorithm **Big O calculator** is used. Each algorithm has unique **time** and **space complexity.** The ideal response will typically be a combination of the two.

For instance, if we want a rapid response and aren’t concerned about space constraints, an appropriate alternative could be an approach with reduced time complexity but higher space complexity such as **Merge Sort**.

### Common Big O Functions

Following are a few of the most popular Big O functions:

#### Constant Function

The Big-O notation for the constant function is:

**Constant Function = O(1)Â **

#### Logarithmic Function

The notation used for logarithmic function is given as:

**Log Function = O(log(n))**

#### Linear Function

Linear functions are denoted as:

**Linear Function = O(n)Â **

#### Quadratic Function

The Big-O notation for the quadratic function is:

**Quadratic Function = $O(n^2)$Â **

#### Cubic Function

The Big-0 notation for the cubic function is given as:

**Cubic Function = $O(n^3))$**

#### Exponential Function

The Big-O notation is given as:

**Exponential Function = $O(2^n)$**

With this knowledge, you can easily use the** Big-O calculator** to solve the time and space complexity of the functions.

## Solved Examples

Let’s explore some examples to better understand the working of the **Big-O calculator**.

**Example 1**

Prove that:

\[ 4^2 = O(8^n) \]

**Solution**

\[ f(n) = 4^n \]

\[ g(n) = 8^n \]

For all n$\leq$ k, we have:

\[ 4^n \leq C.8^n \]

Assuming k =2, the equation 1 is given as:

\[ 4^n \leq C.8^n \]

\[ \frac{4^n}{8^n} \leq C. \frac{8^n}{ 8^n}; for\ all\ n \geq 2 \]

\[ \frac{1}{2} ^n \leq C.(1) ; for\ all\ n\geq 2 \]

If we have n=2, then C becomes:

\[ C= \frac{1}{2}^2 =\frac{1}{4} \]

Substituting the value of C in equation 1 gives:

\[ 4^n \leq \frac{1}{4} .8^n ; for\ all\ n\geq 2 \]

\[ 4^n \leq \frac{1}{4} .(2^n. 4^n) ; for\ all\ n\geq 2 \]

\[ 1 \leq \frac{2^n}{4} ; for\ all\ n\geq 2 \]

\[ 1 \leq \frac{2^n}{2^2}; for\ all\ n\geq 2\]

\[ 1 \leq 2^{(n-2)}\]

From the above, we can say that $4^n$ belongs to $O(8^n)$.

**Example 2**

Prove that $f(n) \in O(n^3)$, where $f(n) = 3n^3 + 2n + 7$.

**Solution**

Let $ n \leq 1 $,

The function is given as:

\[ f(n) = 3n^3 + 2n + 7 \]

\[ f(n) = 3n^3 + 2n + 7 \leq 3n^3 + 2n^3 + 7n^3 \]

\[ f(n) = 12n^3 \]

From above we can say thatÂ $ f(n) \in O(n^3) $

Consequently for all positive n $ f(n) = 3n^3 + 2n + 7 \geq n^3 $.

**Example 3**

Prove thatÂ $ f(n) \in O(n^3) $, where $ f(n) = n^3 + 20n + 1 $ is $ O(n^3) $

**Solution**

The function f(n) belongs to $ O(n^3) $ if and only if $ f(n) \leq c.n^3 $ for some $ n \geq n_{0} $.

By using the above condition:

\[ n^3 + 20n + 1 \leq c.n^3 \]

\[ 1 + \frac{20}{n^2}Â + \frac{1}{n^3} \leq c \]

Therefore $ n \geq 1 $ and $ c \geq 22 $,

From this we can say thatÂ $ f(n) \in O(n^3) $.