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A Binary Indexed (Fenwick) Tree is a data structure that provides efficient methods for implementing dynamic cumulative frequency tables.


This Fenwick Tree data structure uses many bit manipulation techniques. In this visualization, we will refer to this data structure using the term Fenwick Tree as the abbreviation 'BIT' of Binary Indexed Tree is usually associated with the usual bit manipulation.


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Suppose that we have a multiset of integers s = {2,4,5,6,5,6,8,6,7,9,7} (not necessarily sorted). There are m = 11 elements in s. Also suppose that the largest integer that we will ever use is n = 10 and we never use integer 0. For example, these integers represent student (integer) scores from [1..10]. Notice that m is independent of n.


We can create a frequency table f from s with a trivial O(m) time loop. We can then create cumulative frequency table cf from frequency table f in O(n) time using technique similar to DP 1D prefix sum.


Index/Score/SymbolFrequency fCumulative Frequency cf
0-- (index 0 is ignored)
100
211
301
412
524 == cf[4]+f[5]
637 == cf[5]+f[6]
729
8110
9111
10 == n011 == m

Pro-tip 1: Since you are not logged-in, you may be a first time visitor (or not an NUS student) who are not aware of the following keyboard shortcuts to navigate this e-Lecture mode: [PageDown]/[PageUp] to go to the next/previous slide, respectively, (and if the drop-down box is highlighted, you can also use [→ or ↓/← or ↑] to do the same),and [Esc] to toggle between this e-Lecture mode and exploration mode.

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With such cumulative frequency table cf, we can perform Range Sum Query: rsq(i, j) to return the sum of frequencies between index i and j (inclusive), in efficient O(1) time, again using the DP 1D prefix sum (i.e., the inclusion-exclusion principle). For example, rsq(5, 9) = rsq(1, 9) - rsq(1, 4) = 11-2 = 9.


Index/Score/SymbolFrequency fCumulative Frequency cf
0-- (index 0 is ignored)
100
211
301
412 == rsq(1, 4)
524
637
729
8110
9111 == rsq(1, 9)
10 == n011 == m

Pro-tip 2: We designed this visualization and this e-Lecture mode to look good on 1366x768 resolution or larger (typical modern laptop resolution in 2021). We recommend using Google Chrome to access VisuAlgo. Go to full screen mode (F11) to enjoy this setup. However, you can use zoom-in (Ctrl +) or zoom-out (Ctrl -) to calibrate this.

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A dynamic data structure need to support (frequent) updates in between queries. For example, we may update (add) the frequency of score 7 from 2 → 5 and update (subtract) the frequency of score 9 from 1 → 0, thereby updating the table into:


Index/Score/SymbolFrequency fCumulative Frequency cf
0-- (index 0 is ignored)
100
211
301
412
524
637
72 → 59 → 12
8110 → 13
91 → 011 → 13
10 == n011 → 13 == m

A pure array based data structure will need O(n) per update operation. Can we do better?


Pro-tip 3: Other than using the typical media UI at the bottom of the page, you can also control the animation playback using keyboard shortcuts (in Exploration Mode): Spacebar to play/pause/replay the animation, / to step the animation backwards/forwards, respectively, and -/+ to decrease/increase the animation speed, respectively.

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Introducing: Fenwick Tree data structure.


There are three mode of usages of Fenwick Tree in this visualization.


The first mode is the default Fenwick Tree that can handle both Point Update (PU) and Range Query (RQ) in O(log n) where n is the largest index/key in the data structure. Remember that the actual number of keys in the data structure is denoted by another variable m. We abbreviate this default type as PU RQ that simply stands for Point Update Range Query.


This clever arrangement of integer keys idea is the one that originally appears in Peter M. Fenwick's 1994 paper.

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You can click the 'Create' menu to create a frequency array f where f[i] denotes the frequency of appearance of key i in our original array of keys s.


IMPORTANT: This frequency array f is not the original array of keys s. For example, if you enter {0,1,0,1,2,3,2,1,1,0}, it means that you are creating 0 one, 1 two, 0 three, 1 four, ..., 0 ten (1-based indexing). The largest index/integer key is n = 10 in this example as in the earlier slides.


If you have the original array s of m elements, e.g., {2,4,5,6,5,6,8,6,7,9,7} from earlier slides (s does not need to be necessarily sorted), you can do an O(m) pass to convert s into frequency table f of n indices/integer keys. (We will provide this alternative input method in the near future).


You can click the 'Randomize' button to generate random frequencies.

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Although conceptually this data structure is a tree, it will be implemented as an integer array called ft that ranges from index 1 to index n (we sacrifice index 0 of our ft array). The values inside the vertices of the Fenwick Tree shown above are the values stored in the 1-based Fenwick Tree ft array.

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The values inside the vertices at the bottom are the values of the data (the frequency array f).

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The value stored in index i in array ft, i.e., ft[i] is the cumulative frequency of keys in range [i-LSOne(i)+1 .. i]. Visually, this range is shown by the edges of the Fenwick Tree. For details of LSOne(i) operation, see our bitmask visualization page.

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The function rsq(j) returns the cumulative frequencies from the first index 1 (ignoring index 0) to index j.


This value is the sum of sub-frequencies stored in array ft with indices related to j via this formula j' = j-LSOne(j). This relationship forms a Fenwick Tree, specifically, the 'interrogation tree' of Fenwick Tree.


We apply this formula iteratively until j is 0. (We will add that dummy vertex 0 later).


Discussion: Do you understand what does this function compute?


This function runs is O(log n), regardless of m. Discussion: Why?

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rsq(i, j) returns the cumulative frequencies from index i to j, inclusive.


If i = 1, the previous slide is sufficient.
If i > 1, we simply need to return: rsq(j)–rsq(i–1).


Discussion: Do you understand the reason?


This function also runs in O(log n), regardless of m. Discussion: Why?

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To update the frequency of a key (an index) i by v (v is either positive or negative; |v| does not necessarily be one), we use update(i, v).


Indices that are related to i via i' = i+LSOne(i) will be updated by v when i < ft.size() (Note that ft.size() is N+1 (as we ignore index 0). These relationships form a variant of Fenwick Tree structure called the 'updating tree'.


Discussion: Do you understand this operation and on why we avoided index 0?


This function also runs in O(log n), regardless of m. Discussion: Why?

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The second mode of Fenwick Tree is the one that can handle Range Update (RU) but only able to handle Point Query (PQ) in O(log n).


We abbreviate this type as RU PQ.

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Create the data and try running the Range Update or Point Query algorithms on it. Creating the data for this type means inserting several intervals. For example, if you enter [2,4],[3,5], it means that we are updating range 2 to 4 by +1 and then updating range 3 to 5 by +1, thus we have the following frequency table: 0,1,2,2,1 that means 0 one, 1 two, 2 threes, 2 fours, 1 five.

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The vertices at the top shows the values stored in the Fenwick Tree (the ft array).


The vertices at the bottom shows the values of the data (the frequency table f).


Notice the clever modification of Fenwick Tree used in this RU PQ type: We increase the start of the range by +1 but decrease one index after the end of the range by -1 to achieve this result.

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The third mode of Fenwick Tree is the one that can handle both Range Update (RU) and Range Query (RQ) in O(log n), making this type on par with Segment Tree with Lazy Update that can also do RU RQ in O(log n).

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Create the data and try running the Range Update or Range Query algorithms on it.


Creating the data is inserting several intervals, similar as RU PQ version. But this time, you can also do Range Query efficiently.

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In Range Update Range Query Fenwick Tree, we need to have two Fenwick Trees. The vertices at the top shows the values of the first Fenwick Tree (BIT1[] array), the vertices at the middle shows the values of the second Fenwick Tree (BIT2[] array), while the vertices at the bottom shows the values of the data (the frequency table). The first Fenwick Tree behaves the same as in RU PQ version. The second Fenwick Tree is used to do clever offset to allow Range Query again.

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We have a few more extra stuffs involving this data structure.

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Unfortunately, this data structure is not yet available in C++ STL, Java API, Python or OCaml Standard Library as of 2020. Therefore, we have to write our own implementation.


Please look at the following C++/Python/Java/OCaml implementations of this Fenwick Tree data structure in Object-Oriented Programming (OOP) fashion:
fenwicktree_ds.cpp | py | java | ml


Again, you are free to customize this custom library implementation to suit your needs.


You have reached the last slide. 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

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 his friend Dr Suhendry Effendy) and beyond. Today, a few 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/equivalent, CS2040/equivalent, CS3230, CS3233, and CS4234), as advocators of online learning, we hope that curious minds around the world will find these visualizations 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. However, we are currently experimenting with a mobile (lite) version of VisuAlgo to be ready by April 2022.

VisuAlgo is an ongoing project and more complex visualizations 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 12 visualization modules so that every visualization module in VisuAlgo have online quiz component.

We have translated VisuAlgo pages into three main languages: English, Chinese, and Indonesian. Currently, we have also written public notes about VisuAlgo in various languages:

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, Senior 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

Final Year Project/UROP students 5 (Aug 2021-Dec 2022)
Liu Guangyuan, Manas Vegi, Sha Long, Vuong Hoang Long

Final Year Project/UROP students 6 (Aug 2022-Apr 2023)
Lim Dewen Aloysius, Ting Xiao

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/Instagram/TikTok posts, course webpages, blog reviews, emails, 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 (https://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 real examinations in NUS.

List of Publications

This work has been presented briefly at the CLI Workshop at the 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) and this link for the short update in 2015 (to link VisuAlgo name with the previous project).

This work is done mostly by my past students. 

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.

Privacy Policy

Version 1.1 (Updated Fri, 14 Jan 2022).

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Since Wed, 22 Dec 2021, only National University of Singapore (NUS) staffs/students and approved CS lecturers outside of NUS who have written a request to Steven can login to VisuAlgo, anyone else in the world will have to use VisuAlgo as an anonymous user that is not really trackable other than what are tracked by Google Analytics.

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For other CS lecturers worldwide who have written to Steven, a VisuAlgo account (your (non-NUS) email address, you can use any display name, and encrypted password) is needed to distinguish your online credential versus the rest of the world. Your account will be tracked similarly as a normal NUS student account above but it will have CS lecturer specific features, namely the ability to see the hidden slides that contain (interesting) answers to the questions presented in the preceding slides before the hidden slides. You can also access Hard setting of the VisuAlgo Online Quizzes. You can freely use the material to enhance your data structures and algorithm classes. Note that there can be other CS lecturer specific features in the future.

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