7    VisuAlgo    Binary Heap
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Create(A) - O(N log N)

Create(A) - O(N)

Insert(v)

ExtractMax()

HeapSort()

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Sorted Example

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Sorted Example

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About Team Terms of use
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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-Apr 2017)
Truong Ngoc Khanh, John Kevin Tjahjadi, Gabriella Michelle

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.

A Binary (Max) Heap is a complete binary tree that maintains the Max Heap property.


Binary Heap is one possible data structure to model an efficient Priority Queue.


To focus the discussion scope, we design this visualization to show a Binary Max Heap that contains distinct integers only. However, as this visualization only accept integers, it is easy to convert a Binary Max Heap into a Binary Min Heap by re-creating a Binary Max Heap with the negation of every integer in the original Binary Max Heap.

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You can view the visualisation of Binary Max Heap here!


You should see a complete binary tree and all vertices except the root satisfy the Max Heap property (A[parent(i)] > A[i] — remember that we disallow duplicate integers).


Important fact to memorize at this point: If we have a Binary Heap of N items, since we will store it as a complete binary tree, its height will not be taller than O(log N)! This fact is important in the analysis of all Binary Heap-related operations.

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A complete binary tree can be stored efficiently as a compact array (1-based, do you know why?). VisuAlgo displays the index of each vertex as a red label below each vertex. Read those indices in sorted order from 1 to N, then you will see the vertices of the complete binary tree from top to down, left to right.


This way, we can implement basic binary tree traversal operations with simple index manipulations (with help of bit shift manipulation):

  1. parent(i) = i>>1, index i divided by 2 (integer division),
  2. left(i) = i<<1, index i multiplied by 2,
  3. right(i) = (i<<1)+1, index i multiplied by 2 and added by 1.
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In this visualization, you can perform five (5) common/standard Binary Max Heap operations (there are others, but currently we do not show all possible operations of a Binary Max Heap for pedagogical reason in a certain NUS module).


Insert(v): Insertion of a new item v into a Binary Max Heap can only be done at the last index plus 1 to maintain the compact array = complete binary tree property. However, the Max Heap property may still be violated. This operation then fixes Max Heap property from the insertion point upwards (if necessary) and stop when there is no longer Max Heap property violation. [We recommend that you stop this e-Lecture mode now and try the animation of insertions of random integers].


Discussion: Do you know why swapping with parent when there is a Max Heap property violation during insertion is always a correct strategy?


Analysis: The worst case is when we insert a new item v that is greater than the value of the current root. Such insertion causes Insert(v) to fix Max Heap property from a leaf up to the root and therefore runs in O(log N) as a complete binary tree can never be taller than O(log N).

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ExtractMax(): The reporting and then the deletion of the maximum item (the root) of a Binary Max Heap requires an existing item to replace the root, do you know why? That item must be the last index for the same reason: To maintain the compact array = complete binary tree property. Because we promote a leaf vertex to the root vertex of a Binary Max Heap, it will very likely violates the Max Heap property. This operation then fixes Binary Max Heap property from the root downwards by comparing the current value with the its child/the larger of its two children (if necessary) [We recommend that you stop this e-Lecture mode now and try the animation on several ExtractMax() operations].


Do you know why if a vertex has two children, we have to check (and possibly swap) that vertex with the larger of its two children during the downwards fix of Max Heap property?


Analysis: This operation also runs in O(log N) as in the worst case, the last vertex (a leaf) happens to have the smallest value. When it is promoted to the root, it will eventually trickle back to one of the leaf via a path that has up to O(log N) edges.

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Create(A) - O(N log N): Simply insert all N items of the (comma separated) input array into an initially empty Binary Max Heap one by one. [We recommend that you stop this e-Lecture mode now and try the animation of Create(A), the O(N log N version).]


Analysis: This operation is clearly O(N log N) as we call O(log N) Insert(v) operation N times. Try the 'Sorted example' for the extreme case of this operation. If we insert values in increasing order into an initially empty Binary Max Heap, then every insertion triggers a path from the insertion point (a new leaf) upwards to the root — a clear O(N log N) worst case behavior.


PS: We also provide the 'Random' option to generate a random Binary Max Heap.

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Create(A) - O(N): This faster version of Create operation was invented by Robert W. Floyd in 1964. It takes advantage of the fact that a compact array = complete binary tree and all leaves (half of the vertices) are Binary Max Heap by default, do you know why? This operation then fixes Binary Max Heap property (if necessary) only from the last internal vertex back to the root.


Analysis: A loose analysis gives another O(N/2 log N) = O(N log N) complexity but it is actually just O(2*N) = O(N). Try the same 'Sorted example' and see that on the extreme case, this operation is far superior than the O(N log N) version. [We recommend that you stop this e-Lecture mode now and try opening two copies of VisuAlgo on two browser windows. Execute different CreateHeap versions on the worst case 'Sorted example' to see the somewhat dramatic differences of the two.]

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HeapSort(): John William Joseph Williams invented HeapSort() algorithm in 1964, together with this Binary Heap data structure. HeapSort() operation (assuming the Binary Max Heap has been created in O(N)) is very easy. Simply call the O(log N) ExtractMax() operation N times. [We recommend that you stop this e-Lecture mode now and try this HeapSort animation.]


Analysis: HeapSort() runs in O(N log N) — an optimal comparison-based sorting algorithm.

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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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