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Given an undirected weighted graph G = (V, E) and an integer s that partitions the set of vertices V = [0, 1, ..., |V|-1] into a set of required vertices R = [0, 1, ..., s-1] and a set of Steiner vertices S = [s, s+1, ..., |V|-1], the General Steiner Tree problem is a problem of finding a subset S' ⊆ S of the Steiner vertices and a spanning tree T = (R ⋃ S', E) of minimum weight. The weight of the tree T is simply the sum of its edge weights.

This General Steiner Tree problem is a generalization of the more well-known Minimum Spanning Tree problem (MST).

Unlike MST, which has a polynomial solution (e.g., O(E log V) Kruskal's/Prim's algorithm), this general Steiner-Tree problem is an NP-hard combinatorial optimization problem.

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There are a few other variants of this general Steiner-Tree problem, e.g., the Euclidean-Steiner-Tree and the Metric-Steiner-Tree problems.

In the Euclidean-Steiner-Tree problem, there are V distinct points (set R) on an Euclidean (2-dimensional) plane, the job is to find a(n additional, possibly empty) set of Steiner points S (can be anywhere in the 2-D plane) and a spanning tree T = (R ⋃ S, E) such that the weight of T is minimized. The weight of any two points is simply the Euclidean distance of those two points.

In the Metric-Steiner-Tree problem, it is like the Euclidean-Steiner-Tree, but this time the additional Steiner points are given as a set of S upfront. The weight of any two points must satisfy metric space properties.

Euclidean-Steiner Tree, Metric-Steiner-Tree, and the general Steiner-Tree that is visualized in this website, are all NP-hard optimization problems.

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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View the visualisation of the selected Steiner-Tree algorithms here.

Originally, all vertices and edges in the input graph are colored with the standard black outline. As the visualization goes on, the color light blue will be used to denote the set of required vertices R and the color orange will be used to show Steiner vertices that are currently used.

At the end of the selected Steiner-Tree algorithm, we show the best spanning tree = the Steiner tree T.

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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There are two different ways to specify an input graph:

1. Edit Graph: You can edit the currently displayed undirected weighted graph into any other undirected weighted graph.
2. Example Graphs: You can select from the list of example undirected weighted graphs to get you started.

One day, we will provide an easy way to relabel vertices in any currently loaded graph so that the set of required vertices R is always numbered with [0, 1, ..., s-1] (thus the rest are the potential Steiner vertices S).

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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There are 3 special cases of the general Steiner-Tree problem with polynomial solutions:

1. s = 2, that implies that the only required vertices R = [0, 1], we can simply run O((V+E) log V) Dijkstra's algorithm to find the Shortest Path Spanning Tree that connects source vertex 0 with destination vertex 1 (or vice versa). This spanning tree will also be the required Steiner Tree.
2. s = |V|, that implies that all of the |V| vertices, i.e., R = [0, 1, ..., |V|-1] are all required, we can simply run O(E log V) Kruskal's or Prim's algorithm to find the full Minimum Spanning Tree (MST) of the entire graph. This MST will also be the required Steiner Tree.
3. If G is a tree: Details TBA.
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But when s = [3, 4, ..., |V|-2], we have no choice but to run exponential algorithms as these cases fall into general cases of the NP-hard general Steiner-Tree problem.

One possible way is to try all possible subsets of the |V|-s vertices that can be part of the optimal Steiner Tree. Each time, we combine vertices in R = [0, 1, ..., s-1] with the currently chosen subset of potential Steiner vertices, run an O(E) Kruskal's algorithm (without re-sorting the edge list by weight anymore) for each of the 2|V|-s possible subsets, and report the best Steiner Tree. This is an exponential algorithm and this visualization page shows this algorithm.

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There is a more optimized exponential algorithm called the Dreyfus-Wagner Dynamic Programming (DP) algorithm that avoids recomputation of sub-problems.

Unfortunately this part has not been digitized/visualized yet and is in the pipeline.

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Please see CS4234 lecture note for the details.

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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#### Tentang

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.

#### Tim

Pemimpin & Penasihat Proyek (Jul 2011-sekarang)
Associate Professor Steven Halim, School of Computing (SoC), National University of Singapore (NUS)
Dr Felix Halim, Senior Software Engineer, Google (Mountain View)

Murid-Murid S1 Peniliti 1
CDTL TEG 1: Jul 2011-Apr 2012: Koh Zi Chun, Victor Loh Bo Huai

Murid-Murid Proyek Tahun Terakhir/UROP 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

Murid-Murid S1 Peniliti 2
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

Murid-Murid Proyek Tahun Terakhir/UROP 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

Murid-Murid S1 Peniliti 3
Optiver: Aug 2023-Oct 2023: Bui Hong Duc, Oleh Naver, Tay Ngan Lin

Murid-Murid Proyek Tahun Terakhir/UROP 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.

Ucapan Terima Kasih
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.

#### Syarat Guna

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.

#### Kebijakan Privasi

Version 1.2 (Updated Fri, 18 Aug 2023).

Since Fri, 18 Aug 2023, we no longer use Google Analytics. Thus, all cookies that we use now are solely for the operations of this website. The annoying cookie-consent popup is now turned off even for first-time visitors.

Since Fri, 07 Jun 2023, thanks to a generous donation by Optiver, anyone in the world can self-create a VisuAlgo account to store a few customization settings (e.g., layout mode, default language, playback speed, etc).