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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.


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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?

Prim's Algorithm
Kruskal's Algorithm
It Depends
Diskussion: Warum?
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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.


Try them to consolidate and improve your understanding about this graph problem.


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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Alle Schritte werden in der Status Anzeige erklärt während sie passieren
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Kontrolliere die Animation mit Hilfe deiner Tastatur! Die Tasten sind:

Leertaste: start/stop/wiederholen
Pfeiltaste rechts/links: ein Schritt vor oder zurück
-/+: senke/erhöhe die Geschwindigkeit
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Kehre zum 'Exploration Mode' zurück und beginne zu Erforschen
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Graph zeichnen

Beispiel Graphen

Kruskal's Algorithm

Prim's Algorithm(s)

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Nutzungsbedingungen

Über

VisuAlgo wurde konzeptioniert 2011 von Dr Steven Halim als ein Tool um seinen Studenten zu helfen Datenstrukturen und Algorithmen besser zu verstehen, indem sie die Grundlagen alleine und in ihrem eigenen Tempo lernen können.
VisuAlgo enthält viele fortgeschrittene Algorithmen die auch in Dr Steven Halim's Buch ('Competitive Programming', co-author ist sein Bruder Dr Felix Halim) und mehr. Heute, können die Visualisierungen/Animationen vieler fortgeschrittener Algorithmen nur auf VisoAlgo gefunden werden.
Obwohl die Visualisierungen speziell für die verschiedenen Datenstruktur und Algorithmik Kurse der National University of Singapore (NUS) gemacht sind, freuen wir uns, als Befürworter des Online Lernens, wenn auch andere neugierige Geister unsere Visualisierungen nützlich finden.
VisuAlgo ist nicht designed um gut auf kleinen Touchscreens (z,B, Smartphones) zu funktionieren, da die Darstellung komplexer Algorithmen viele Pixel benötigt und click-and-drag Aktionen zur Interaktion. Die minimale Bildschirmauflösung für ein akzeptables Benutz Erlebnis ist 1024x768 und nur die Startseite ist einigermaßen mobilfähig.
VisuAlgo ist ein laufendes Projekt und weitere komplexe Visualisierungen werden weiterhin entwickelt.
Die aufregendste Entwicklung ist der automatisierte Fragen Generator und Überprüfer (das Online Quiz System), dass Studenten erlaubt deren Wissen über grundlegende Datenstrukturen und Algorithmen zu testen. Die Fragen werden mit der Hilfe einiger Regeln zufällig generiert und die Antworten der Studenten werden automatisch von unserem Bewertungs Server bewertet. Das Online Quiz System, wenn es von mehr Informatik Tutoren übernommen wird, sollte eigentlich grundlegende Datenstrucktur- und Algorithmikfragen in Klausuren an vielen Universitäten ersetzten. Indem man ein wenig (allerdings nicht null) Gewicht darauf legt, dass das Online Quiz bestanden wird, kann ein Informatik Tutor (stark) das Können seiner Studenten was solche grundlegenden Fragen betrifft erhöhen, da die Studenten eine nahezu unendlich Anzahl ein Trainingsfragen beantworten können bevor sie das Online Quiz machen. Der Training Modus enthält aktuell Fragen für 12 Visualisierungsmodule. Die letzten 8 werden bald folgen, sodass es für alle Visualisierungsmodule ein Online Quiz gibt.
Eine weitere aktive Abteilung ist das Internationalisierungs Sub-Projekt von VisuAlgo. Wir wollen eine Datenbank für alle Informatik Begriffe aus alle englischen Texte im VisuAlgo System anlegen. Das ist eine große Aufgabe und benötigt Crowdsourcing. Sobald das System funktionstüchtig ist, werden wir VisuAlgo Besucher dazu einladen. Besonders wenn sie keine englischen Muttersprachler sind. Aktuel, haben wir auch verschiedene Notizen in verschiedenen Sprachen über VisuAlgo:
zh, id, kr, vn, th.

Mannschaft

Projektleiter & Berater (Juli 2011 bis heute)
Dr Steven Halim, Senior Lecturer, School of Computing (SoC), National University of Singapore (NUS)
Dr Felix Halim, Software Engineer, Google (Mountain View)

Studentische Hilfskräfte 1 (Jul 2011-Apr 2012)
Koh Zi Chun, Victor Loh Bo Huai

Abschlussprojekt/UROP Studenten 1 (Jul 2012-Dec 2013)
Phan Thi Quynh Trang, Peter Phandi, Albert Millardo Tjindradinata, Nguyen Hoang Duy

Abschlussprojekt/UROP Studenten 2 (Jun 2013-Apr 2014)
Rose Marie Tan Zhao Yun, Ivan Reinaldo

Studentische Hilfskräfte 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

Abschlussprojekt/UROP Studenten 3 (Jun 2014-Apr 2015)
Erin Teo Yi Ling, Wang Zi

Abschlussprojekt/UROP Studenten 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.

Danksagungen
Dieses Projekt wird durch den großzügigen Teaching Enhancement Grant des NUS Centre for Development of Teaching and Learning (CDTL) ermöglicht.

Nutzungsbedingungen

VisuAlgo ist kostenlos für die Informatik-Community dieses Planeten (natürlich auch von Leute nicht von der Erde). Wenn dir VisuAlgo gefällt, ist die einzige Bezahlung um die wir bitten, das du anderen Informatik Studenten und Tutoren von dieser Seite erzählst. =) über Facebook, Twitter, Kurs Internet Seit, Blog Eintrag, Email usw.

Bist du ein Datenstruktur oder Algorithmik Student/Tutor, darfst du diese Webseite für deine Kurse nutzen. Solltest du Screenshots (Videos) von dieser Seite machen, darfst du diese woanders verwenden, solange du die URL dieser Seite (http://visualgo.net) als Referenz angibst. Es ist allerdings NICHT erlaubt VisuAlgo (client-Side) Dateien herunter zu laden und diese auf deiner eigenen Website zu hosten, da das ein  Plagiat wäre. Es ist auch NICHT erlaubt eine Anspaltung dieser Website zu machen und Varianten von VisuAlgo zu erstellen. Eine private Nutzung einer offline Kopie (client-side) von VisuAlgo ist erlaubt.

Beachte allerdings das VisuAlgo's Online Quiz System von Natur aus eine schwere Server-seitige Komponente hat und es gibt keinen einfachen Weg die Server-seitige Scripts und Datenbanken lokal zu speichern. Aktuell kann die allgemeinen Öffentlichkeit nur den 'Trainings Modus' nutzen um an das Online Quiz System zu kommen. Der 'Test-Modus' ist eine kontrollierterte Umgebung in der zufällig generierte Fragen und automatische Überprüfung für eine echte Prüfung in NUS genutzt werden. Andere interessierte Informatik Tutoren sollten Steven kontaktieren, wenn sie auch diesen 'Test-Modus' ausprobieren wollen.

Liste der Publikationen

Diese Arbeit wurde kurz beim CLI Workshop beim ACM ICPC Weltfinale 2012 (Polen, Warschau) und bei der IOI Konferenz bei IOI 2012 (Italien, Sirmione-Montichiari). Du kannst du diesen Link klicken um unser 2012 Paper über dieses System zu lesen (Es hieß 2012 noch nicht VisuAlgo).
Diese Arbeit wurde wurde hauptsächlich von ehemaligen Studenten gemacht. Die letzten Ergebnisse sind hier: Erin, Wang Zi, Rose, Ivan.

Bug Reports oder Anfragen zu neuen Features

VisuAgo ist kein fertiges Projekt. Dr Steven Halim arbeitet aktiv daran VisuAlgo zu verbessern. Wenn du beim benutzten von VisuAlgo in einer Visualisierung/Online Quiz einen Bug findest oder ein neues Feature möchtest, kontaktiere bitte Dr Steven Halim. Sein Kontakt ist die Verkettung seines Namens und at gmail dot com.