A reduction is a way of transforming/converting one problem into another problem.

Suppose you have a problem A which you do not know how to solve. However, you know an algorithm to solve problem B. If you can "transform" an instance α of problem A into an instance β of problem B, you can use the known algorithm for B to solve the "transformed" instance β, and obtain the solution for α from the solution β, by "reversing the transformation". We then say that A reduces to B.

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Through reductions, we have another tool to solve problems.

Furthermore, with an efficient, polynomial-time "transformation", reductions provide the ability to compare the "hardness" of two problems A and B.

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Given two decision problems A and B, a polynomial-time reduction from A to B,
denoted A ≤p B, is a transformation from instance α of A to instance β of B such that:

1. α is a YES-instance for A if and only if β is a YES-instance for B.
(A YES-instance is an instance of the decision problem whose answer is YES/True).
2. The transformation takes polynomial time in the size of α.

With a polynomial-time reduction, we will be able to claim:
If B is "easily solvable", then so is A.
If A is "hard", then so is B.

However, if B is "hard", it does not imply that A is "hard".
Likewise, if A is "easy", it does not mean that B is "easy".

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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In this (static) visualization page (no animation), we want to show the (well-known) network of reductions from different NP-complete decision problems to another. In theory, since all these problems are NP-complete, they can all be reduced to one another, but the transformation can become very convoluted for some problems.

This network of reductions shown in this page is a combination from "Introductions to Algorithms (CLRS, 4th edition)", Karp's 21 NP-complete problems, "Computers and Intractability (Garey and Johnson, 79), etc.

The content of this visualization will be added over time as more and more NP-complete problems are discussed in NUS algorithm classes.

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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A vertex contains the (abbreviated) name of an NP-complete decision problem. Hover over it to see its full name. You can also click on the vertex to open the relevant slide that formally describes the problem.

A directed edge between vertices opens the slide which contains the proof of reduction from one problem to the other, in the direction of the arrow, as found in many computational complexity resources (books/Internet/etc). The reduction proof is generally if and only if (iff). Green arrow indicates that we have digitize the reduction proof. Black arrow indicates that we have not done so. We hope to add as much content to this page gradually over time.

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The CIRCUIT-SATISFIABILITY problem (C-SAT) is the decision problem of determining whether a given Boolean circuit - essentially a Directed Acyclic Graph - (using AND ∧, OR ∨, NOT ¬ gates) has an assignment of its n binary inputs that makes the output True (1). In other words, it asks whether the n inputs to a given Boolean circuit can be consistently set to True (1) or False (0) such that the circuit outputs True (1). If that is the case, the circuit is called satisfiable. Otherwise, the circuit is called unsatisfiable.

YES-instance (Sample certificate: False, True, True, False)

Important: L ≤p C-SAT for every language L ∈ NP.

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Definitions: Literal is a Boolean variable or its negation, e.g., xi, i.
Clause: A disjunction (OR ∨) of literals, e.g, Cj = x1 ∨ x̄2 ∨ x3.

The CONJUNCTIVE-NORMAL-FORM-SATISFIABILITY (CNF-SAT), or sometimes just called as SATISFIABILITY (SAT) problem, is the decision problem of determining whether a given formula Φ that is conjunction (AND ∧) of clauses, e.g., Φ = C1 ∧ C2 ∧ C3 ∧ C4, has a satisfying truth assignment.

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3-CNF-SAT(isfiability), usually just called as 3-SAT problem, is a CNF-SAT where each clause contains exactly 3 literals corresponding to different variables.

Example: Φ = (x̄1 ∨ x2 ∨ x3) ∧ (x1 ∨ x̄2 ∨ x3) ∧ (x̄1 ∨ x2 ∨ x4) is a YES-instance.
(Sample certificate: x1 = x2 = x4 = True, x3 = False).

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A clique is a set of vertices C of graph G = (V, E) such that ∃ an edge between every pair of distinct vertices in the set (u, v ∈ C, ∃ (u, v)). Set C is a complete subgraph of G. Then the size of the clique is the number of vertices in the set C.

The CLIQUE decision problem asks the following question on a graph G = (V, E):
Does there exist a complete subgraph in G of at least size k?

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Given a graph G = (V, E), a subset C ⊆ V is said to be a vertex cover if for each edge (u, v) ∈ E, either u ∈ C and/or v ∈ C.

Given a graph G = (V, E), the VERTEX-COVER (VC) problem asks:
Does there exist a vertex cover of size ≤ k?

Note that we have built a specialized mvc visualization page that discusses this problem and various strategies to attack this problem in a much detailed manner.

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Given an undirected unweighted graph G = (V, E),
the HAMILTONIAN-CYCLE (HC, sometimes abbreviated as HAM-CYCLE) problem asks:
Does there exist a (simple) cycle passing through all vertices exactly once?

Left/YES-instance (Sample certificate: 0-1-3-4-2-0) ------- ------- ------- Right/NO-instance

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Given an undirected complete graph with non-negative weight on edges and b ∈ R+,
the TRAVELING-SALESPERSON(/previously MAN)-PROBLEM (TSP) problem asks:
Does there exist a tour of cost at most b?

YES-instance for b = 108 but NO-instance for b = 107 as OPT = 108

Note that we have built a specialized tsp visualization page that discusses this problem and various strategies to attack this problem in a much detailed manner.

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Given a multiset S of N (usually non-negative) Integers: {A1, A2, ..., AN},
Is there exist a subset of S which sums to W?

Example: N = 5, S = {5, 1, 5, 1, 4}, and W = 7,
then it is a YES-instance with certificate indices {0, 1, 3} or values {5, 1, 1} that sums to 7.

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Given n items described by non-negative Integer pairs (w1, v1), (w2, v2), ..., (wn, vn), capacity W and threshold V, is there a subset of item with total weight at most W and total value at least V?

YES-instance: n = 5: {(12, 4), (1, 2), (4, 10), (1, 1), (2, 2)}, W = 15, and V = 15, certificate: take everything except item (12, 4) with total weight 8 and total value 15.

NO-instance: use the same instance as above but V = 16.

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Given a graph G = (V, E), a subset X ⊆ V is said to be an independent set if for each pair of vertices u, v ∈ X, then (u, v) ∉ E.

Given a graph G = (V, E), the INDEPENDENT-SET (IS) problem asks:
Does there exist an independent set of size ≥ k?

Note that we have built a specialized mvc visualization page that discusses this problem and various strategies to attack this problem in a much detailed manner (remember that a set of vertices is IS if and only if its complement is a VC).

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Given a set T of non-negative Integers, can we partition T into two sets of equal sum?

YES-instance: T = [18, 2, 8, 5, 7, 24], certificate: S1 = [18, 5, 7, 2] and S2 = [8, 24]
NO-instance: T = [1, 2]

PS: Obviously if the sum of T is odd, we cannot partition T into two sets with equal sum (immediately a NO instance). Without loss of generality, PARTITION problem is usually asked on instances when the sum of T is even.

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Given a graph G = (V, E), a dominating set D is a subset of vertices such that for all other vertices u that are not in D, there exists some vertex v in D such that (u, v) is an edge in G (i.e., u is adjacent to v).

The Dominating-Set decision problem asks if given an integer k, there exists a dominating set in G of at most size k.

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Given a set of sets S = {S1, S2, ..., Sn}, a set H is said to be a hitting set of S
if for all Si, H ∩ Si ≠ ∅ (i.e., H has a non-empty intersection with all Si).

Given the set of sets S and a non-negative integer k, the HITTING-SET (HS) problem asks:
Is there exists a hitting set of S of size ≤ k?.

Example: S = {S1, S2, S3, S4}, S1 = {2, 3}, S2 = {6, 7}, S3 = {1, 4, 5, 7}, S4 = {3, 7, 8}, k = 2, then it is a YES-instance with certificate H = {3, 7}.

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C-SAT ≤p CNF-SAT reduction is shown in CLRS Chapter 34.

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CNF-SAT ≤p 3-SAT reduction is shown in CLRS Chapter 34.

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The key idea (major hint) is that 3-CNF-SAT has k clauses (disjunction of 3 literals) but IS is on a graph G = (V, E). How about creating k triangles (with 3 vertices = 3 literals each) and do something involving a variable and its negation.

Can you work out the details of the poly-time reduction and its proof yourself?
Try first before clicking next.

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For each clause, create 3 vertices in G, one for each literal.
We connect these 3 literals of a clause into a triangle sub-graph.
We also connect a literal to each of its negation (in any other clauses).
The IS criteria will ensure that we will only pick exactly one variable per triangle and we will not pick a variable in one clause and its negation in any other clause.

Take note that this reduction runs in poly-time.

Theorem: YES-instance for 3-SAT if and only YES-instance for IS.

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The key idea (major hint) is to that VC has a graph G with V vertices and E edges but HC has n sets of Integers. How about converting the edges into sets of size two?

Can you work out the details of the poly-time reduction and its proof yourself?
Try first before clicking next.

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Take an instance (G = (V, E), k) of VC.
Then for each edge e = (u, v), e ∈ E, we create a set Se = {u, v} of HC.
Thus the transformed instance of HC is: ({Se | e ∈ E}, k).

Take note that this reduction runs in poly-time, in O(E).

It is easy to see that a YES-instance for VC if and only YES-instance for HC.

Proof omitted: The vertices in C of VC correspond to elements in H of HS and vice versa.

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The key idea (major hint) is to build a weighted complete graph G' such that the HC in G turns out to be the least cost tour in G'.

Can you work out the details of the poly-time reduction and its proof yourself?
Try first before clicking next.

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Let G = (V, E) be an instance α of HC. We build an instance β of TSP as follows: create a complete graph G' on the same V vertices, but for each pair u, v ∈ V: if u, v ∈ E, then w(u, v) = 1, else w(u, v) = 2 (or anything greater than 1).

Take note that this reduction runs in poly-time, to be precise O(n2) as at most C(n, 2) edges are added from G to G'.

Theorem: G has a Hamiltonian cycle if and only if G' has a TSP tour of cost at most n.

Proof: Split into two parts:
• (if) G' has a TSP tour of cost at most n (YES-instance of TSP) → G has a Hamiltonian cycle (YES-instance of HC)
• (only if) G has a Hamiltonian cycle (YES-instance of HC) → G' has a TSP tour of cost at most n (YES-instance of TSP)

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The content of this interesting slide (the answer of the usually intriguing discussion point from the earlier slide) is hidden and only available for legitimate CS lecturer worldwide. This mechanism is used in the various flipped classrooms in NUS.

If you are really a CS lecturer (or an IT teacher) (outside of NUS) and are interested to know the answers, please drop an email to stevenhalim at gmail dot com (show your University staff profile/relevant proof to Steven) for Steven to manually activate this CS lecturer-only feature for you.

FAQ: This feature will NOT be given to anyone else who is not a CS lecturer.

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The content of this interesting slide (the answer of the usually intriguing discussion point from the earlier slide) is hidden and only available for legitimate CS lecturer worldwide. This mechanism is used in the various flipped classrooms in NUS.

If you are really a CS lecturer (or an IT teacher) (outside of NUS) and are interested to know the answers, please drop an email to stevenhalim at gmail dot com (show your University staff profile/relevant proof to Steven) for Steven to manually activate this CS lecturer-only feature for you.

FAQ: This feature will NOT be given to anyone else who is not a CS lecturer.

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The key idea (major hint) is to that PARTITION has n Integers only but KNAPSACK has n pair of Integers. How about duplicating the Integers? Also PARTITION has a specific target value (half of total of the n Integers) that can also be used as appropriate parameter for KNAPSACK.

Can you work out the details of the poly-time reduction and its proof yourself?
Try first before clicking next.

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Given a PARTITION instance α with T = [w1, w2, ..., wn] with total sum S = ∑i=[1..n] wi, construct a KNAPSACK instance β: {(w1, w1), (w2,w2), ..., (wn,wn)}
with capacity W = S/2 and threshold V = S/2.

Take note that this reduction runs in poly-time, to be precise O(n · log (wmax)) as it just copies n weights to n (weight, weight-as-value) pairs. If we assume wmax fits in standard 32/64-bit signed Integers, then log (wmax)) is just 32/64 respectively and this reduction runs in O(n).

Theorem: YES-instance for PARTITION if and only YES-instance for KNAPSACK.

Proof: Split into two parts:
• YES-instance for PARTITION → YES-instance for KNAPSACK.
• YES-instance for KNAPSACK → YES-instance for PARTITION.

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The content of this interesting slide (the answer of the usually intriguing discussion point from the earlier slide) is hidden and only available for legitimate CS lecturer worldwide. This mechanism is used in the various flipped classrooms in NUS.

If you are really a CS lecturer (or an IT teacher) (outside of NUS) and are interested to know the answers, please drop an email to stevenhalim at gmail dot com (show your University staff profile/relevant proof to Steven) for Steven to manually activate this CS lecturer-only feature for you.

FAQ: This feature will NOT be given to anyone else who is not a CS lecturer.

🕑

The content of this interesting slide (the answer of the usually intriguing discussion point from the earlier slide) is hidden and only available for legitimate CS lecturer worldwide. This mechanism is used in the various flipped classrooms in NUS.

If you are really a CS lecturer (or an IT teacher) (outside of NUS) and are interested to know the answers, please drop an email to stevenhalim at gmail dot com (show your University staff profile/relevant proof to Steven) for Steven to manually activate this CS lecturer-only feature for you.

FAQ: This feature will NOT be given to anyone else who is not a CS lecturer.

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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Initially conceived in 2011 by Associate Professor Steven Halim, VisuAlgo aimed to facilitate a deeper understanding of data structures and algorithms for his students by providing a self-paced, interactive learning platform.

Featuring numerous advanced algorithms discussed in Dr. Steven Halim's book, 'Competitive Programming' — co-authored with Dr. Felix Halim and Dr. Suhendry Effendy — VisuAlgo remains the exclusive platform for visualizing and animating several of these complex algorithms even after a decade.

While primarily designed for National University of Singapore (NUS) students enrolled in various data structure and algorithm courses (e.g., CS1010/equivalent, CS2040/equivalent (including IT5003), CS3230, CS3233, and CS4234), VisuAlgo also serves as a valuable resource for inquisitive minds worldwide, promoting online learning.

Initially, VisuAlgo was not designed for small touch screens like smartphones, as intricate algorithm visualizations required substantial pixel space and click-and-drag interactions. For an optimal user experience, a minimum screen resolution of 1366x768 is recommended. However, since April 2022, a mobile (lite) version of VisuAlgo has been made available, making it possible to use a subset of VisuAlgo features on smartphone screens.

VisuAlgo remains a work in progress, with the ongoing development of more complex visualizations. At present, the platform features 24 visualization modules.

Equipped with a built-in question generator and answer verifier, VisuAlgo's "online quiz system" enables students to test their knowledge of basic data structures and algorithms. Questions are randomly generated based on specific rules, and students' answers are automatically graded upon submission to our grading server. As more CS instructors adopt this online quiz system worldwide, it could effectively eliminate manual basic data structure and algorithm questions from standard Computer Science exams in many universities. By assigning a small (but non-zero) weight to passing the online quiz, CS instructors can significantly enhance their students' mastery of these basic concepts, as they have access to an almost unlimited number of practice questions that can be instantly verified before taking the online quiz. Each VisuAlgo visualization module now includes its own online quiz component.

VisuAlgo has been translated into three primary languages: English, Chinese, and Indonesian. Additionally, we have authored public notes about VisuAlgo in various languages, including Indonesian, Korean, Vietnamese, and Thai:

id, kr, vn, th.

#### Team

Associate Professor Steven Halim, School of Computing (SoC), National University of Singapore (NUS)
Dr Felix Halim, Senior Software Engineer, Google (Mountain View)

CDTL TEG 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
Jun 2013-Apr 2014 Rose Marie Tan Zhao Yun, Ivan Reinaldo

CDTL TEG 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 2
Jun 2014-Apr 2015: Erin Teo Yi Ling, Wang Zi
Jun 2016-Dec 2017: Truong Ngoc Khanh, John Kevin Tjahjadi, Gabriella Michelle, Muhammad Rais Fathin Mudzakir
Aug 2021-Apr 2023: Liu Guangyuan, Manas Vegi, Sha Long, Vuong Hoang Long, Ting Xiao, Lim Dewen Aloysius

Optiver: Aug 2023-Oct 2023: Bui Hong Duc, Oleh Naver, Tay Ngan Lin

Final Year Project/UROP students 3
Aug 2023-Apr 2024: Xiong Jingya, Radian Krisno, Ng Wee Han

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

Acknowledgements
NUS CDTL gave Teaching Enhancement Grant to kickstart this project.

For Academic Year 2023/24, a generous donation from Optiver will be used to further develop VisuAlgo.

VisuAlgo is generously offered at no cost to the global Computer Science community. If you appreciate VisuAlgo, we kindly request that you spread the word about its existence to fellow Computer Science students and instructors. You can share VisuAlgo through social media platforms (e.g., Facebook, YouTube, Instagram, TikTok, Twitter, etc), course webpages, blog reviews, emails, and more.

Data Structures and Algorithms (DSA) students and instructors are welcome to use this website directly for their classes. If you capture screenshots or videos from this site, feel free to use them elsewhere, provided that you cite the URL of this website (https://visualgo.net) and/or the list of publications below as references. However, please refrain from downloading VisuAlgo's client-side files and hosting them on your website, as this constitutes plagiarism. At this time, we do not permit others to fork this project or create VisuAlgo variants. Personal use of an offline copy of the client-side VisuAlgo is acceptable.

Please note that VisuAlgo's online quiz component has a substantial server-side element, and it is not easy to save server-side scripts and databases locally. Currently, the general public can access the online quiz system only through the 'training mode.' The 'test mode' offers a more controlled environment for using randomly generated questions and automatic verification in real examinations at NUS.

List of Publications

This work has been presented 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).

Bug Reports or Request for New Features

VisuAlgo is not a finished project. Associate Professor 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 Associate Professor Steven Halim. His contact is the concatenation of his name and add gmail dot com.