# indicate constant time complexity in terms of big-o notation

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## Big O notation: definition and examples · YourBasic

Big O notation is a convenient way to describe how fast a function is growing. It is often used in computer science when estimating time complexity.

## Big O notation: definition and examples

yourbasic.org

Big O notation is a convenient way to describe how fast a function is growing.

Definition Constant time Linear time Quadratic time Sloppy notation Ω and Θ notation Key takeaways

## Definition

When we compute the time complexity T() of an algorithm we rarely get an exact result, just an estimate. That’s fine, in computer science we are typically only interested in how fast T() is growing as a function of the input size .

For example, if an algorithm increments each number in a list of length , we might say: “This algorithm runs in () time and performs (1) work for each element”.

Here is the formal mathematical definition of Big O.

Let T() and f() be two positive functions. We write **T() ∊ (f())**, and say that T() has order of f(), if there are positive constants M and n₀ such that T() ≤ M·f() for all ≥ n₀.

This graph shows a situation where all of the conditions in the definition are met.

In essence:

T() ∊ (f()) means that T() doesn't grow faster than f().

## Constant time

Let’s start with the simplest possible example: **T() ∊ (1)**.

According to the definition this means that there are constants M and n₀ such that T() ≤ M when ≥ n₀. In other words, T() ∊ (1) means that T() is smaller than some fixed constant, whose value isn’t stated, for all large enough values of .

An algorithm with T() ∊ (1) is said to have **constant time complexity**.

## Linear time

In the Time complexity article, we looked at an algorithm with complexity T() = -1. Using Big O notation this can be written as **T() ∊ ()**. (If we choose M = 1 and n₀ = 1, then T() = - 1 ≤ 1· when ≥ 1.)

An algorithm with T() ∊ () is said to have **linear time complexity**.

## Quadratic time

The second algorithm in the Time complexity article had time complexity T() = 2/2 - /2. With Big O notation, this becomes **T() ∊ (2)**, and we say that the algorithm has **quadratic time complexity**.

## Sloppy notation

The notation T() ∊ (f()) can be used even when f() grows **much faster** than T(). For example, we may write T() = - 1 ∊ (2). This is indeed true, but not very useful.

## Ω and Θ notation

**Big Omega**is used to give a

**lower bound**for the growth of a function. It’s defined in the same way as Big O, but with the inequality sign turned around:

Let T() and f() be two positive functions. We write **T() ∊ Ω(f())**, and say that T() is big omega of f(), if there are positive constants m and n₀ such that T() ≥ m(f()) for all ≥ n₀.

**Big Theta**is used to indicate that a function is bounded both from above and below.

**T() ∊ Θ(f())**if T() is both (f()) and Ω(f()).

### Example

T() = 33 + 2 + 7 ∊ Θ(3)

If ≥ 1, then T() = 33 + 2 + 7 ≤ 33 + 23 + 73 = 123. Hence T() ∊ (3).

On the other hand, T() = 33 + 2 + 7 > 3 for all positive . Therefore T() ∊ Ω(3).

And consequently T() ∊ Θ(3).

## Key takeaways

When analyzing algorithms you often come across the following time complexities.

Complexity Θ(1) Good news Θ(log ) Θ() Θ( log )

Θ(k), where k ≥ 2 Bad news

Θ(k), where k ≥ 2 Θ(!)

### ( log ) is really good

The first four complexities indicate an excellent algorithm. An algorithm with worst-case time complexity W() ∊ ( log ) scales very well, since logarithms grow very slowly.

log2 1,000 ≈ 10 log2 1,000,000 ≈ 20

log2 1,000,000,000 ≈ 30

In fact, Θ( log ) time complexity is very close to linear – it takes roughly twice the time to solve a problem twice as big.

log growth rate is close to linear

### Ω(2) is pretty bad

The last three complexities typically spell trouble. Algorithms with time complexity Ω(2) are useful only for small input: shouldn’t be more than a few thousand.

10,0002 = 100,000,000

An algorithm with quadratic time complexity scales poorly – if you increase the input size by a factor 10, the time increases by a factor 100.

### Further reading

Time complexity of array/list operations [Java, Python]

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## Big O Notation Quiz

Do you know anything about the Big O Notation Algorithm? Test your knowledge with this quiz. In computer science, the Big O Notation is utilized to group algori...

## Big O Notation Quiz

22 Questions | Total Attempts: 16947

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Do you know anything about the Big O Notation Algorithm? Test your knowledge with this quiz. In computer science, the Big O Notation is utilized to group algorithms according to how their run time or space conditions change as the input size grows. In analytic number theory, the Big O Notation is often used to convey the arithmetical function. This Big O Notation quiz can be a valuable tool for practicing for an exam.

Questions and Answers

1.

What is the time complexity of the insert(index) method in ArrayList?

A. O(n) B. O(n^2) C. O(nlogn) D. O(logn) 2.

Indicate constant time complexity in terms of Big-O notation.

A. O(n) B. O(1) C. O(logn) D. O(n^2) 3.

Indicate exponential time complexity in terms of big-O notation?

A. O(n) B. O(n^2) C. O(2^n) D. O(logn) 4.

Find the slowest time.

A. O(n) B. O(n^2) C. O(n!) D. O(2^n) 5.

What is the time complexity of the ArrayList remove(index) method?

A. O(n) B. O(2n) C. O(logn) D. O(n^2) 6.

What is the time complexity of adding an item in front of a LinkedList?

A. O(logn) B. O(1) C. O(n^2) D. O(2^n) 7.

What is the time complexity of adding elements at the beginning of ArrayList?

A. O(n) B. O(n^2) C. O(2n) D. O(nlogn) 8.

Indicate logarithm polynomial time complexity.

A.

O(n^const(const=2,3…) )

B. O(n^2) C. O(2n) D. O(2^n) 9.

What is the time complexity of the insert(index) method in ArrayList?

A. O(n) B. O(2n) C. O(logn) D. O(nlogn) 10.

What is the time complexity of the recursive Binary Search algorithm?

A. O(n) B. O(2^n) C. O(logn) D. O(nlogn) 11.

What the time complexity of the linear search algorithm?

A. O(n) B. O(n^2) C. O(2^n) D. O(1) 12.

Search a binary search tree costs?

A. O(n) B. O(n^2) C. O(logn) D. O(nlogn) 13.

Element insertion to a Binary Search tree costs?

A. O(n) B. O(n^2) C. O(logn) D. O(2^n) 14.

Insert and remove items from a heap costs?

A. O(n) B. O(n^2) C. O(logn) D. O(1) 15.

The average time complexity of the Selection sort is?

A. O(n) B. O(2^n) C. O(logn) D. O(nlogn) 16.

The average time complexity of the Heap sort is?

A. O(n) B. O(2^n) C. O(logn) D. O(nlogn) 17.

The average time complexity of Quicksort is?

A. O(n) B. O(n^2) C. O(2+nlogn) D. O(nlogn) 18.

The average time complexity of Insertion sort is?

A. O(n) B. O(n^2) C. O(2^n) D. O(logn) 19.

A hash table uses hashing to transform an item's key into a table index so that iterations, retrievals, and deletions can be performed in expected ___________ time.

A. O(n) B. O(logn) C. O(1) D. O(false) 20.

Average time complexity of Merge sort is?

A. O(n) B. O(2^n) C. O(logn) D. O(nlogn) 21.

The average time complexity of Shell sort is?

A. O(n) B. O(n^2) C. O(n^1.25) D. O(n^2.25) 22.

The average time complexity of Bubble sort is?

A. O(n^2) B. O(n) C. O(logn) D. O(nlogn) Related Topics Database

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## What is Big O Notation Explained: Space and Time Complexity

Do you really understand Big O? If so, then this will refresh your understanding before an interview. If not, don’t worry — come and join us for some endeavors in computer science. If you have taken some algorithm related courses, you’ve probably heard of the term Big O notation. If

JANUARY 16, 2020 / #BIG O NOTATION

## What is Big O Notation Explained: Space and Time Complexity

Shen Huang

Do you really understand Big O? If so, then this will refresh your understanding before an interview. If not, don’t worry — come and join us for some endeavors in computer science.

If you have taken some algorithm related courses, you’ve probably heard of the term **Big O notation**. If you haven’t, we will go over it here, and then get a deeper understanding of what it really is.

Big O notation is one of the most fundamental tools for computer scientists to analyze the cost of an algorithm. It is a good practice for software engineers to understand in-depth as well.

This article is written with the assumption that you have already tackled some code. Also, some in-depth material also requires high-school math fundamentals, and therefore can be a bit less comfortable to total beginners. But if you are ready, let’s get started!

In this article, we will have an in-depth discussion about Big O notation. We will start with an example algorithm to open up our understanding. Then, we will go into the mathematics a little bit to have a formal understanding. After that we will go over some common variations of Big O notation. In the end, we will discuss some of the limitations of Big O in a practical scenario. A table of contents can be found below.

### Table of Contents

What is Big O notation, and why does it matter

Formal Definition of Big O notation

Big O, Little O, Omega & Theta

Complexity Comparison Between Typical Big Os

Time & Space Complexity

Best, Average, Worst, Expected Complexity

Why Big O doesn’t matter

In the end…

So let’s get started.

### 1. What is Big O Notation, and why does it matter

“Big O notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. It is a member of a family of notations invented by Paul Bachmann, Edmund Landau, and others, collectively called Bachmann–Landau notation or asymptotic notation.”

— Wikipedia’s definition of Big O notation

In plain words, Big O notation describes the **complexity** of your code using algebraic terms.

To understand what Big O notation is, we can take a look at a typical example, **O(n²)**, which is usually pronounced **“Big O squared”**. The letter **“n”** here represents the **input size**, and the function **“g(n) = n²”** inside the **“O()”** gives us an idea of how complex the algorithm is with respect to the input size.

A typical algorithm that has the complexity of O(n²) would be the **selection sort** algorithm. Selection sort is a sorting algorithm that iterates through the list to ensure every element at index **i** is the **ith** smallest/largest element of the list. The **CODEPEN** below gives a visual example of it.

The algorithm can be described by the following code. In order to make sure the ith element is the ith smallest element in the list, this algorithm first iterates through the list with a for loop. Then for every element it uses another for loop to find the smallest element in the remaining part of the list.

SelectionSort(List) {

for(i from 0 to List.Length) {

SmallestElement = List[i]

for(j from i to List.Length) {

if(SmallestElement > List[j]) {

SmallestElement = List[j]

} }

Swap(List[i], SmallestElement)

} }

In this scenario, we consider the variable **List** as the input, thus input size n is the **number of elements inside List**. Assume the if statement, and the value assignment bounded by the if statement, takes constant time. Then we can find the big O notation for the SelectionSort function by analyzing how many times the statements are executed.

First the inner for loop runs the statements inside n times. And then after **i** is incremented, the inner for loop runs for n-1 times… …until it runs once, then both of the for loops reach their terminating conditions.

Selection Sort Loops Illustrated

This actually ends up giving us a geometric sum, and with some high-school math we would find that the inner loop will repeat for 1+2 … + n times, which equals n(n-1)/2 times. If we multiply this out, we will end up getting n²/2-n/2.

When we calculate big O notation, we only care about the **dominant terms**, and we do not care about the coefficients. Thus we take the n² as our final big O. We write it as O(n²), which again is pronounced “Big O squared”.

Now you may be wondering, what is this **“dominant term”** all about? And why do we not care about the coefficients? Don’t worry, we will go over them one by one. It may be a little bit hard to understand at the beginning, but it will all make a lot more sense as you read through the next section.

### 2. Formal Definition of Big O notation

Once upon a time there was an Indian king who wanted to reward a wise man for his excellence. The wise man asked for nothing but some wheat that would fill up a chess board.

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