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

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 bit manipulation. Remarks: By default, we show e-Lecture Mode for first time (or non logged-in) visitor.

Please login if you are a repeated visitor or register for an (optional) free account first. 1. Binary Indexed (Fenwick) Tree 1-1. Cumulative Frequency Table 1-2. Range Sum Query: rsq(i, j) 1-3. Dynamic Cumulative Frequency Table 2. Modes and the First/Default Mode 3. Point Update Range Query (PU RQ) 3-1. The Visualization - Part 1 3-2. The Visualization - Part 2 3-3. The Visualization - Part 3 3-4. Range Query: rsq(j) 3-5. Range Query: rsq(i, j) 3-6. Point Update: update(i, v) 4. Second Mode 5. Range Update Point Query (RU PQ) 5-1. The RU PQ Visualization 6. Third Mode 7. Range Update Range Query (RU RQ) 7-1. The RU RQ Visualization 8. Extras 8-1. Implementation 99. Status Panel 99-1. Codetrace Panel 99-2. Media Control 99-3. Return to 'Exploration Mode' X Esc Next PgDn

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/Symbol Frequency f Cumulative Frequency cf 0 - - (index 0 is ignored) 1 0 0 2 1 1 3 0 1 4 1 2 5 2 4 == cf[4]+f[5] 6 3 7 == cf[5]+f[6] 7 2 9 8 1 10 9 1 11 10 == n 0 11 == m Pro-tip: Since you are not logged-in , you may be a first time visitor who are not aware of the following keyboard shortcuts to navigate this e-Lecture mode: [PageDown] to advance to the next slide, [PageUp] to go back to the previous slide, [Esc] to toggle between this e-Lecture mode and exploration mode. X Esc Prev PgUp Next PgDn

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/Symbol Frequency f Cumulative Frequency cf 0 - - (index 0 is ignored) 1 0 0 2 1 1 3 0 1 4 1 2 == rsq(1, 4) 5 2 4 6 3 7 7 2 9 8 1 10 9 1 11 == rsq(1, 9) 10 == n 0 11 == m Another pro-tip: We designed this visualization and this e-Lecture mode to look good on 1366x768 resolution or larger (typical modern laptop resolution in 2017). 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. X Esc Prev PgUp Next PgDn

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/Symbol Frequency f Cumulative Frequency cf 0 - - (index 0 is ignored) 1 0 0 2 1 1 3 0 1 4 1 2 5 2 4 6 3 7 7 2 → 5 9 → 12 8 1 10 → 13 9 1 → 0 11 → 13 10 == n 0 11 → 13 == m

A pure array based data structure will need O(n) per update operation. Can we do better? X Esc Prev PgUp Next PgDn

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. X Esc Prev PgUp Next PgDn

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. X Esc Prev PgUp Next PgDn

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. X Esc Prev PgUp Next PgDn

The values inside the vertices at the bottom are the values of the data (the frequency array f). X Esc Prev PgUp Next PgDn

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 . X Esc Prev PgUp Next PgDn

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? X Esc Prev PgUp Next PgDn

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? X Esc Prev PgUp Next PgDn

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? X Esc Prev PgUp Next PgDn

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. X Esc Prev PgUp Next PgDn

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. X Esc Prev PgUp Next PgDn

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. X Esc Prev PgUp Next PgDn

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). X Esc Prev PgUp Next PgDn

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. X Esc Prev PgUp Next PgDn

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. X Esc Prev PgUp Next PgDn

We have a few more extra stuffs involving this data structure. X Esc Prev PgUp Next PgDn

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++/Java/Python/OCaml implementations of this Fenwick Tree data structure in Object-Oriented Programming (OOP) fashion:

fenwicktree_ds.cpp

fenwicktree_ds.java

fenwicktree_ds.py

fenwicktree_ds.ml



Again, you are free to customize this custom library implementation to suit your needs. X Esc Prev PgUp Next PgDn

As the action is being carried out, each step will be described in the status panel. X Esc Prev PgUp Next PgDn

e-Lecture: The content of this slide is hidden and only available for legitimate CS lecturer worldwide. Drop an email to visualgo.info at gmail dot com if you want to activate this CS lecturer-only feature and you are really a CS lecturer (show your University staff profile). X Esc Prev PgUp Next PgDn

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

Spacebar: play/pause/replay Left/right arrows: step backward/step forward

-/+: decrease/increase speed

X Esc Prev PgUp Next PgDn step backward/step forwarddecrease/increase speed

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. X Esc Prev PgUp

Create RSQ / Query

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).