A Spanning Tree (ST) of a connected undirected weighted graph G is a subgraph of G that is a tree and connects (spans) all vertices of G. A graph G can have multiple STs, each with different total weight (the sum of edge weights in the ST).

A Min(imum) Spanning Tree (MST) of G is an ST of G that has the smallest total weight among the various STs.

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

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The MST problem is a standard graph (and also optimization) problem defined as follows: Given a connected undirected weighted graph G = (V, E), select a subset of edges of G such that the graph is still connected but with minimum total weight. The output is either the actual MST of G (there can be several possible MSTs of G) or usually just the minimum total weight itself (unique).

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.

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Imagine that you work for a government who wants to link all rural villages in the country with roads.
(that is spanning tree).

The cost to build a road to connect two villages depends on the terrain, distance, etc.
(that is a complete undirected weighted graph).

You want to minimize the total building cost. How are you going to build the roads?
(that is minimum spanning tree).

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.

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The MST problem has polynomial solutions.

In this visualization, we will learn two of them: Kruskal's algorithm and Prim's algorithm. Both are classified as Greedy Algorithms.

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View the visualisation of MST algorithm on the left.

Originally, all vertices and edges in the input graph are colored with the standard black color on white background.

At the end of the MST algorithm, MST edges (and all vertices) will be colored orange and Non-MST edges will be colored grey.

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There are two different sources for specifying an input graph:

1. Draw Graph: You can draw any connected undirected weighted graph as the input graph.
2. Example Graphs: You can select from the list of example connected undirected weighted graphs to get you started.
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Kruskal's algorithm: An O(E log V) greedy MST algorithm that grows a forest of minimum spanning trees and eventually combine them into one MST.

Kruskal's requires a good sorting algorithm to sort edges of the input graph by increasing weight and another data structure called Union-Find Disjoint Sets (UFDS) to help in checking/preventing cycle.

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Kruskal's algorithm first sort the set of edges E in non-decreasing weight (there can be edges with the same weight), and if ties, by increasing smaller vertex number of the edge, and if still ties, by increasing larger vertex number of the edge.

Discussion: Is this the only possible sort criteria?

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Then, Kruskal's algorithm will perform a loop through these sorted edges (that already have non-decreasing weight property) and greedily taking the next edge e if it does not create any cycle w.r.t edges that have been taken earlier.

Without further ado, let's try Kruskal on the default example graph (that has three edges with the same weight). Go through this animated example first before continuing.

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To see on why the Greedy Strategy of Kruskal's algorithm works, we define a loop invariant: Every edge e that is added into tree T by Kruskal's algorithm is part of the MST.

At the start of Kruskal's main loop, T = {} is always part of MST by definition.

Kruskal's has a special cycle check in its main loop (using UFDS data structure) and only add an edge e into T if it will never form a cycle w.r.t the previously selected edges.

At the end of the main loop, Kruskal's can only select V-1 edges from a connected undirected weighted graph G without having any cycle. This implies that Kruskal's produces a Spanning Tree.

On the default example, notice that after taking the first 2 edges: 0-1 and 0-3, in that order, Kruskal's cannot take edge 1-3 as it will cause a cycle 0-1-3-0. Kruskal's then take edge 0-2 but it cannot take edge 2-3 as it will cause cycle 0-2-3-0.

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We have seen in the previous slide that Kruskal's algorithm will produce a tree T that is a Spanning Tree (ST) when it stops. But is it the minimum ST, i.e. the MST?

To prove this, we need to recall that before running Kruskal's main loop, we have already sort the edges in non-decreasing weight, i.e. the latter edges will have equal or larger weight than the earlier edges.

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At the start of every loop, T is always part of MST.

If Kruskal's only add a legal edge e (that will not cause cycle w.r.t the edges that have been taken earlier) with min cost, then we can be sure that w(T U e) ≤ w(T U any other unprocessed edge e' that does not form cycle) (by virtue that Kruskal's has sorted the edges, so w(e) ≤ w(e').

Therefore, at the end of the loop, the Spanning Tree T must have minimal overall weight w(T), so T is the final MST.

On the default example, notice that after taking the first 2 edges: 0-1 and 0-3, in that order, and ignoring edge 1-3 as it will cause a cycle 0-1-3-0. We can safely take the next smallest legal edge 0-2 (with weight 2) as taking any other legal edge (e.g. edge 2-3 with larger weight 3) will either create another MST with equal weight (not in this example) or another ST that is not minimum (which is this example).

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There are two parts of Kruskal's algorithm: Sorting and the Kruskal's main loop.

The sorting of edges is easy. We just store the graph using Edge List data structure and sort E edges using any O(E log E) = O(E log V) sorting algorithm (or just use C++/Java sorting library routine) by increasing weight, smaller vertex number, higher vertex number. This O(E log V) is the bottleneck part of Kruskal's algorithm as the second part is actually lighter, see below.

Kruskal's main loop can be easily implemented using Union-Find Disjoint Sets data structure. We use IsSameSet(u, v) to test if taking edge e with endpoints u and v will cause a cycle (same connected component) or not. If IsSameSet(u, v) returns false, we greedily take this next smallest and legal edge e and call UnionSet(u, v) to prevent future cycles involving this edge. This part runs in O(E) as we assume UFDS IsSameSet(u, v) and UnionSet(u, v) operations run in O(1) for a relatively small graph.

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Prim's algorithm: Another O(E log V) greedy MST algorithm that grows a Minimum Spanning Tree from a starting source vertex until it spans the entire graph.

Prim's requires a Priority Queue data structure (usually implemented using Binary Heap) to dynamically order the currently considered edges based on increasing weight, an Adjacency List data structure for fast neighbor enumeration of a vertex, and a Boolean array to help in checking cycle.

Another name of Prim's algorithm is Jarnik-Prim's algorithm.

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Prim's algorithm starts from a designated source vertex s and enqueues all edges incident to s into a Priority Queue (PQ) according to increasing weight, and if ties, by increasing vertex number (of the neighboring vertex number). Then it will repeatedly do the following greedy steps: If the vertex v of the front-most edge pair information e: (w, v) in the PQ has not been visited, it means that we can greedily extends the tree T to include vertex v and enqueue edges connected to v into the PQ, otherwise we discard edge e.

Without further ado, let's try Prim(1) on the default example graph (that has three edges with the same weight). That's it, we start Prim's algorithm from source vertex s = 1. Go through this animated example first before continuing.

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Prim's algorithm is a Greedy Algorithm because at each step of its main loop, it always try to select the next valid edge e with minimal weight (that is greedy!).

The convince us that Prim's algorithm is correct, let's go through the following simple proof: Let T be the spanning tree of graph G generated by Prim's algorithm and T* be the spanning tree of G that is known to have minimal cost, i.e. T* is the MST.

If T == T*, that's it, Prim's algorithm produces exactly the same MST as T*, we are done.

But if T != T*...

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Assume that on the default example, T = {0-1, 0-3, 0-2} but T* = {0-1, 1-3, 0-2} instead.

Let ek = (u, v) be the first edge chosen by Prim's Algorithm at the k-th iteration that is not in T* (on the default example, k = 2, e2 = (0, 3), note that (0, 3) is not in T*).

Let P be the path from u to v in T*, and let e* be an edge in P such that one endpoint is in the tree generated at the (k−1)-th iteration of Prim's algorithm and the other is not (on the default example, P = 0-1-3 and e* = (1, 3), note that vertex 1 is inside T at first iteration k = 1).

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If the weight of e* is less than the weight of ek, then Prim's algorithm would have chosen e* on its k-th iteration as that is how Prim's algorithm works.

So, it is certain that w(e*) ≥ w(ek).
(on the example graph, e* = (1, 3) has weight 1 and ek = (0, 3) also has weight 1).

When weight e* is = weight ek, the choice between the e* or ek is actually arbitrary. And whether the weight of e* is ≥ weight of ek, e* can always be substituted with ek while preserving minimal total weight of T*. (on the example graph, when we replace e* = (1, 3) with ek = (0, 3), we manage to transform T* into T).

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But if T != T*... (continued)

We can repeat the substitution process outlined earlier repeatedly until T* = T and thereby we have shown that the spanning tree generated by any instance of Prim's algorithm (from any source vertex s) is an MST as whatever the optimal MST is, it can be transformed to the output of Prim's algorithm.

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We can easily implement Prim's algorithm with two well-known data structures:

1. A Priority Queue PQ (Binary Heap or just use C++ STL priority_queue/Java PriorityQueue), and
2. A Boolean array of size V (to decide if a vertex has been taken or not, i.e. in the same connected component as source vertex s or not).

With these, we can run Prim's Algorithm in O(E log V) because we process each edge once and each time, we call Insert((w, v)) and (w, v) = ExtractMax() from a PQ in O(log E) = O(log V2) = O(2 log V) = O(log V). As there are E edges, Prim's Algorithm runs in O(E log V).

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Quiz: Having seen both Kruskal's and Prim's Algorithms, which one is the better MST algorithm?

It Depends
Prim's Algorithm
Kruskal's Algorithm

Discussion: Why?

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You have reached the end of the basic stuffs of this Min(imum) Spanning Tree graph problem and its two classic algorithms: Kruskal's and Prim's (there are others, like Boruvka's, but not discussed in this visualization). We encourage you to explore further in the Exploration Mode.

However, the harder MST problems can be (much) more challenging that its basic version.

Once you have (roughly) mastered this MST topic, we encourage you to study more on harder graph problems where MST is used as a component, e.g. approximation algorithm for NP-hard (Metric No-Repeat) TSP and Steiner Tree (soon) problems.

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For a few more challenging questions about this MST problem and/or Kruskal's/Prim's Algorithms, please practice on MST training module (no login is required, but short and of medium difficulty setting only).

However, for registered users, you should login and then go to the Main Training Page to officially clear this module (and its pre-requisites) and such achievement will be recorded in your user account.

Pro-tip: To attempt MST Online Quiz in easy or medium difficulty setting without having to clear the pre-requisites first, you have to log out first (from your profile page).

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This MST problem can be much more challenging than this basic form. Therefore we encourage you to try the following two ACM ICPC contest problems about MST: UVa 01234 - RACING and Kattis - arcticnetwork.

You are allowed to use/modify our implementation code for Kruskal's/Prim's Algorithms:
kruskal.cpp/prim.cpp
kruskal.java/prim.java
kruskal.py/prim.py
kruskal.ml/prim.ml

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As the action is being carried out, each step will be described in the status panel.

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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).

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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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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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グラフを描く

グラフの例え

Kruskal's Algorithm

Prim's Algorithm(s)

CP 4.10

CP 4.14

K5

Rail

Tessellation

s =

#### VIsuAlgoについて

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.

#### チーム

プロジェクトリーダー＆アドバイザー（2011年7月～現在)
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).

#### 利用規約

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.