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).

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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).

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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:

Draw Graph: You can draw any connected undirected weighted graph as the input graph.

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 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 Listdata 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 e_{k} = (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, e_{2} = (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 e_{k}, 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(e_{k}). (on the example graph, e* = (1, 3) has weight 1 and e_{k} = (0, 3) also has weight 1).

When weight e* is = weight e_{k}, the choice between the e* or e_{k} is actually arbitrary. And whether the weight of e* is ≥ weight of e_{k}, e* can always be substituted with e_{k} while preserving minimal total weight of T*. (on the example graph, when we replace e* = (1, 3) with e_{k} = (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:

A Priority Queue PQ (Binary Heap or just use C++ STL priority_queue/Java PriorityQueue), and

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 V^{2}) = 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?

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.

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 Algorithm that can be downloaded Here (ch4_03_kruskal_prim.cpp/java, from the companion Competitive Programming book website).

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当操作进行时，状态面板将会有每个步骤的描述。

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

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.

项目领导和顾问（2011年7月至今） Dr Steven Halim, Senior Lecturer, School of Computing (SoC), National University of Singapore (NUS) Dr Felix Halim, Software Engineer, Google (Mountain View)