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Sorting algorithms simply reorder elements (integers, numbers, strings, etc) of an array (a list) in a certain order (increasing, decreasing, lexicographical, etc). There are many different sorting algorithms, and each has its own advantages and limitations. In this visualization, we assume that we will sort integers in increasing order.



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Here you can create a new list, which you can either define yourself, or let the computer generate a random list.
To sort the elements, click Sort then Go.
Some sorting algorithms will have certain options; you may toggle the options as you wish before clicking Go.

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View the visualisation/animation of the chosen sorting algorithm here.

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Listed here are the common sorting algorithms. Select the respective algorithm names to switch among the different sorting algorithms.


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Given an array of N items 
  1. Compare a pair of adjacent items (a, b)
  2. Swap the pair if the items are out of order (in this case a > b)
  3. Repeat until we reach the end of array (the last pair)
  4. The largest item will be at the last position N
  5. Reduce N by 1 and go to Step 1
Try Bubble Sort

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Comparison and swap require time that is bounded by a constant c.

There are two nested loops in Bubble Sort:
 - Outer loop runs for exactly N iterations
 - Inner loops:
    when i=0, (N−1) iterations (of comparisons and possibly swaps)
    when i=1, (N−2) iterations
    …
    when i=(N−1), 0 iterations

Total number of iterations = 0+1+…+(N−1) = N*(N−1)/2
Total time = c*N*(N−1)/2 = O(N^2)
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Bubble Sort is inefficient with a O(N^2) time
complexity
However, it has an interesting property.
    Given the following array, how many times will the
inner loop swap a pair of item?

[3, 6, 11, 25, 39]

Idea
If we go through the inner loop with no swapping
    => the array is sorted
    => can stop early!

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Given an array of N items:
  1. Find the smallest item X, in the range of [0…N−1]
  2. Swap X with the 0-th item
  3. Increase the lower-bound by 1 and go back to Step 1
Try Selection Sort

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void selectionSort(int a[], int N) {
    for (int i = 0; i < N-1; i++) {                                // O(N)
        find the minimum element from (i+1)-th to the last element // O(N)
        swap it with the i-th element                              // O(1)
    }
}
Total: O(N^2)
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Quiz: How many (real) swaps are required to sort [29, 10, 14, 37, 13] by Selection Sort?

3
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Similar to how most people arrange a hand of
poker cards
  1. Start with one card in your hand
  2. Pick the next card and insert it into its proper sorted order
  3. Repeat previous step for all cards
Cards         Try Insertion Sort
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void insertionSort(int a[], int N) {
    for (int i = 1; i < N; i++) {
        next <- a[i] // next is the item to be inserted
        shift next to proper position
    }
}
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  1. Outer-loop executes (N−1) times
  2. Number of times inner-loop is executed depends on
    the input
  3. Therefore, the best-case time is O(N)
  4. And the worst-case time is O(N^2)
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Quiz: What is the complexity of Insertion Sort?

O(N^2)
O(N log N)
O(N)
O(1)
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  1. Merge each pair of elements into sets of 2
  2. Merge each pair of sets of 2 into sets of 4
  3. Repeat previous step for sets of 4 … 
  4. Final step: merge 2 sets of N/2 elements to obtain a
    fully sorted set
Try Merge Sort

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Quick Sort is a divide-and-conquer algorithm
  1. Divide step
    • Choose an item p (known as (known as pivot) and partition the items of a[i...j] into two parts
      • Items that are smaller than p
      • Items that are greater than or equal to p
    • Recursively sort the two parts
  2. Conquer step
    • Do nothing!
Try Quick Sort

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Same as Quick Sort except for randomly selecting the pivot.

Try Randomized Quick Sort

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Try Counting Sort
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- Treats each data to be sorted as a character string
- It is not using comparison, i.e. no comparison between the data is needed
- In each iteration:
  1. Organize the data into groups according to the next character in each data
  2. The groups are then “concatenated” for next iteration
Try Radix Sort

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It's QUIZ time!
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Quiz: Which of these algorithms run in O(N log N)?

QuickSort (Deterministic)
Bubble Sort
Insertion Sort
Merge Sort
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Quiz: Which of these algorithms has worst case time complexity of Θ(N^2) for sorting N integers?

Radix Sort
Bubble Sort
Merge Sort
Selection Sort
Insertion Sort
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We don't provide source code for these basic sorting algorithms. Usually, sorting is just a small part in problem solving process and nowadays, most of programming languages have their own sorting functions so we don't really have to re-code them. In C++, you can use STL algorithm::sort. In Java, you can use Collections.sort. If the comparison function is problem-specific, we may need to supply additional comparison function to those built-in sorting routines.
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Do you think sorting is just as simple as calling built-in sort routine? Try these two online judge problems to find out more: Kattis - sortofsorting or
Kattis - sidewayssorting

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Note: Please Sign up/Login before attempting the training!

Test your understanding here!
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As the action is being carried out, each step will be described in the status panel.

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You can also follow the pseudocode highlights to trace the algorithm.

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Control the animation with the player controls! Keyboard shortcuts are:

Spacebar: play/pause/replay
Left/right arrows: step backward/step forward
-/+: decrease/increase speed
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Return to 'Exploration Mode' to start exploring!


Note that if you notice any bug in this visualization or if you want to request for a new visualization feature, do not hesitate to drop an email to the project leader: Dr Steven Halim via his email address: stevenhalim at gmail dot com.

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About Team Terms of use

About

VisuAlgo was conceptualised in 2011 by Dr Steven Halim as a tool to help his students better understand data structures and algorithms, by allowing them to learn the basics on their own and at their own pace.

VisuAlgo contains many advanced algorithms that are discussed in Dr Steven Halim's book ('Competitive Programming', co-authored with his brother Dr Felix Halim) and beyond. Today, some of these advanced algorithms visualization/animation can only be found in VisuAlgo.

Though specifically designed for National University of Singapore (NUS) students taking various data structure and algorithm classes (e.g. CS1010, CS1020, CS2010, CS2020, CS3230, and CS3230), as advocators of online learning, we hope that curious minds around the world will find these visualisations useful too.

VisuAlgo is not designed to work well on small touch screens (e.g. smartphones) from the outset due to the need to cater for many complex algorithm visualizations that require lots of pixels and click-and-drag gestures for interaction. The minimum screen resolution for a respectable user experience is 1024x768 and only the landing page is relatively mobile-friendly.

VisuAlgo is an ongoing project and more complex visualisations are still being developed.

The most exciting development is the automated question generator and verifier (the online quiz system) that allows students to test their knowledge of basic data structures and algorithms. The questions are randomly generated via some rules and students' answers are instantly and automatically graded upon submission to our grading server. This online quiz system, when it is adopted by more CS instructors worldwide, should technically eliminate manual basic data structure and algorithm questions from typical Computer Science examinations in many Universities. By setting a small (but non-zero) weightage on passing the online quiz, a CS instructor can (significantly) increase his/her students mastery on these basic questions as the students have virtually infinite number of training questions that can be verified instantly before they take the online quiz. The training mode currently contains questions for 12 visualization modules. We will soon add the remaining 8 visualization modules so that every visualization module in VisuAlgo have online quiz component.

Another active branch of development is the internationalization sub-project of VisuAlgo. We want to prepare a database of CS terminologies for all English text that ever appear in VisuAlgo system. This is a big task and requires crowdsourcing. Once the system is ready, we will invite VisuAlgo visitors to contribute, especially if you are not a native English speaker. Currently, we have also written public notes about VisuAlgo in various languages: zh, id, kr, vn, th.

Team

Project Leader & Advisor (Jul 2011-present)
Dr Steven Halim, Senior Lecturer, School of Computing (SoC), National University of Singapore (NUS)
Dr Felix Halim, Software Engineer, Google (Mountain View)

Undergraduate Student Researchers 1 (Jul 2011-Apr 2012)
Koh Zi Chun, Victor Loh Bo Huai

Final Year Project/UROP students 1 (Jul 2012-Dec 2013)
Phan Thi Quynh Trang, Peter Phandi, Albert Millardo Tjindradinata, Nguyen Hoang Duy

Final Year Project/UROP students 2 (Jun 2013-Apr 2014)
Rose Marie Tan Zhao Yun, Ivan Reinaldo

Undergraduate Student Researchers 2 (May 2014-Jul 2014)
Jonathan Irvin Gunawan, Nathan Azaria, Ian Leow Tze Wei, Nguyen Viet Dung, Nguyen Khac Tung, Steven Kester Yuwono, Cao Shengze, Mohan Jishnu

Final Year Project/UROP students 3 (Jun 2014-Apr 2015)
Erin Teo Yi Ling, Wang Zi

Final Year Project/UROP students 4 (Jun 2016-Dec 2017)
Truong Ngoc Khanh, John Kevin Tjahjadi, Gabriella Michelle, Muhammad Rais Fathin Mudzakir

List of translators who have contributed ≥100 translations can be found at statistics page.

Acknowledgements
This project is made possible by the generous Teaching Enhancement Grant from NUS Centre for Development of Teaching and Learning (CDTL).

Terms of use

VisuAlgo is free of charge for Computer Science community on earth. If you like VisuAlgo, the only payment that we ask of you is for you to tell the existence of VisuAlgo to other Computer Science students/instructors that you know =) via Facebook, Twitter, course webpage, blog review, email, etc.

If you are a data structure and algorithm student/instructor, you are allowed to use this website directly for your classes. If you take screen shots (videos) from this website, you can use the screen shots (videos) elsewhere as long as you cite the URL of this website (http://visualgo.net) and/or list of publications below as reference. However, you are NOT allowed to download VisuAlgo (client-side) files and host it on your own website as it is plagiarism. As of now, we do NOT allow other people to fork this project and create variants of VisuAlgo. Using the offline copy of (client-side) VisuAlgo for your personal usage is fine.

Note that VisuAlgo's online quiz component is by nature has heavy server-side component and there is no easy way to save the server-side scripts and databases locally. Currently, the general public can only use the 'training mode' to access these online quiz system. Currently the 'test mode' is a more controlled environment for using these randomly generated questions and automatic verification for a real examination in NUS. Other interested CS instructor should contact Steven if you want to try such 'test mode'.

List of Publications

This work has been presented briefly at the CLI Workshop at the ACM ICPC World Finals 2012 (Poland, Warsaw) and at the IOI Conference at IOI 2012 (Sirmione-Montichiari, Italy). You can click this link to read our 2012 paper about this system (it was not yet called VisuAlgo back in 2012).

This work is done mostly by my past students. The most recent final reports are here: Erin, Wang Zi, Rose, Ivan.

Bug Reports or Request for New Features

VisuAlgo is not a finished project. Dr Steven Halim is still actively improving VisuAlgo. If you are using VisuAlgo and spot a bug in any of our visualization page/online quiz tool or if you want to request for new features, please contact Dr Steven Halim. His contact is the concatenation of his name and add gmail dot com.