# Kruskal’s Minimum Spanning Tree Algorithm | Greedy Algo-2

What is Minimum Spanning Tree?
Given a connected and undirected graph, a spanning tree of that graph is a subgraph that is a tree and connects all the vertices together. A single graph can have many different spanning trees. A minimum spanning tree (MST) or minimum weight spanning tree for a weighted, connected and undirected graph is a spanning tree with weight less than or equal to the weight of every other spanning tree. The weight of a spanning tree is the sum of weights given to each edge of the spanning tree.

How many edges does a minimum spanning tree has?
A minimum spanning tree has (V – 1) edges where V is the number of vertices in the given graph.

What are the applications of Minimum Spanning Tree?
See this for applications of MST.

Below are the steps for finding MST using Kruskal’s algorithm

1. Sort all the edges in non-decreasing order of their weight.
2. Pick the smallest edge. Check if it forms a cycle with the spanning tree formed so far. If cycle is not formed, include this edge. Else, discard it.
3. Repeat step#2 until there are (V-1) edges in the spanning tree.

The step#2 uses Union-Find algorithm to detect cycle. So we recommend to read following post as a prerequisite.
Union-Find Algorithm | Set 1 (Detect Cycle in a Graph)
Union-Find Algorithm | Set 2 (Union By Rank and Path Compression)

The algorithm is a Greedy Algorithm. The Greedy Choice is to pick the smallest weight edge that does not cause a cycle in the MST constructed so far. Let us understand it with an example: Consider the below input graph. The graph contains 9 vertices and 14 edges. So, the minimum spanning tree formed will be having (9 – 1) = 8 edges.

```After sorting:
Weight   Src    Dest
1         7      6
2         8      2
2         6      5
4         0      1
4         2      5
6         8      6
7         2      3
7         7      8
8         0      7
8         1      2
9         3      4
10        5      4
11        1      7
14        3      5```

Now pick all edges one by one from sorted list of edges
1. Pick edge 7-6: No cycle is formed, include it. 2. Pick edge 8-2: No cycle is formed, include it. 3. Pick edge 6-5: No cycle is formed, include it. 4. Pick edge 0-1: No cycle is formed, include it. 5. Pick edge 2-5: No cycle is formed, include it. 6. Pick edge 8-6: Since including this edge results in cycle, discard it.

7. Pick edge 2-3: No cycle is formed, include it. 8. Pick edge 7-8: Since including this edge results in cycle, discard it.

9. Pick edge 0-7: No cycle is formed, include it. 10. Pick edge 1-2: Since including this edge results in cycle, discard it.

11. Pick edge 3-4: No cycle is formed, include it. Since the number of edges included equals (V – 1), the algorithm stops here.

## C/C++

 `// C++ program for Kruskal's algorithm to find Minimum Spanning Tree ` `// of a given connected, undirected and weighted graph ` `#include ` `#include ` `#include ` ` `  `// a structure to represent a weighted edge in graph ` `struct` `Edge ` `{ ` `    ``int` `src, dest, weight; ` `}; ` ` `  `// a structure to represent a connected, undirected ` `// and weighted graph ` `struct` `Graph ` `{ ` `    ``// V-> Number of vertices, E-> Number of edges ` `    ``int` `V, E; ` ` `  `    ``// graph is represented as an array of edges.  ` `    ``// Since the graph is undirected, the edge ` `    ``// from src to dest is also edge from dest ` `    ``// to src. Both are counted as 1 edge here. ` `    ``struct` `Edge* edge; ` `}; ` ` `  `// Creates a graph with V vertices and E edges ` `struct` `Graph* createGraph(``int` `V, ``int` `E) ` `{ ` `    ``struct` `Graph* graph = ``new` `Graph; ` `    ``graph->V = V; ` `    ``graph->E = E; ` ` `  `    ``graph->edge = ``new` `Edge[E]; ` ` `  `    ``return` `graph; ` `} ` ` `  `// A structure to represent a subset for union-find ` `struct` `subset ` `{ ` `    ``int` `parent; ` `    ``int` `rank; ` `}; ` ` `  `// A utility function to find set of an element i ` `// (uses path compression technique) ` `int` `find(``struct` `subset subsets[], ``int` `i) ` `{ ` `    ``// find root and make root as parent of i  ` `    ``// (path compression) ` `    ``if` `(subsets[i].parent != i) ` `        ``subsets[i].parent = find(subsets, subsets[i].parent); ` ` `  `    ``return` `subsets[i].parent; ` `} ` ` `  `// A function that does union of two sets of x and y ` `// (uses union by rank) ` `void` `Union(``struct` `subset subsets[], ``int` `x, ``int` `y) ` `{ ` `    ``int` `xroot = find(subsets, x); ` `    ``int` `yroot = find(subsets, y); ` ` `  `    ``// Attach smaller rank tree under root of high  ` `    ``// rank tree (Union by Rank) ` `    ``if` `(subsets[xroot].rank < subsets[yroot].rank) ` `        ``subsets[xroot].parent = yroot; ` `    ``else` `if` `(subsets[xroot].rank > subsets[yroot].rank) ` `        ``subsets[yroot].parent = xroot; ` ` `  `    ``// If ranks are same, then make one as root and  ` `    ``// increment its rank by one ` `    ``else` `    ``{ ` `        ``subsets[yroot].parent = xroot; ` `        ``subsets[xroot].rank++; ` `    ``} ` `} ` ` `  `// Compare two edges according to their weights. ` `// Used in qsort() for sorting an array of edges ` `int` `myComp(``const` `void``* a, ``const` `void``* b) ` `{ ` `    ``struct` `Edge* a1 = (``struct` `Edge*)a; ` `    ``struct` `Edge* b1 = (``struct` `Edge*)b; ` `    ``return` `a1->weight > b1->weight; ` `} ` ` `  `// The main function to construct MST using Kruskal's algorithm ` `void` `KruskalMST(``struct` `Graph* graph) ` `{ ` `    ``int` `V = graph->V; ` `    ``struct` `Edge result[V];  ``// Tnis will store the resultant MST ` `    ``int` `e = 0;  ``// An index variable, used for result[] ` `    ``int` `i = 0;  ``// An index variable, used for sorted edges ` ` `  `    ``// Step 1:  Sort all the edges in non-decreasing  ` `    ``// order of their weight. If we are not allowed to  ` `    ``// change the given graph, we can create a copy of ` `    ``// array of edges ` `    ``qsort``(graph->edge, graph->E, ``sizeof``(graph->edge), myComp); ` ` `  `    ``// Allocate memory for creating V ssubsets ` `    ``struct` `subset *subsets = ` `        ``(``struct` `subset*) ``malloc``( V * ``sizeof``(``struct` `subset) ); ` ` `  `    ``// Create V subsets with single elements ` `    ``for` `(``int` `v = 0; v < V; ++v) ` `    ``{ ` `        ``subsets[v].parent = v; ` `        ``subsets[v].rank = 0; ` `    ``} ` ` `  `    ``// Number of edges to be taken is equal to V-1 ` `    ``while` `(e < V - 1) ` `    ``{ ` `        ``// Step 2: Pick the smallest edge. And increment  ` `        ``// the index for next iteration ` `        ``struct` `Edge next_edge = graph->edge[i++]; ` ` `  `        ``int` `x = find(subsets, next_edge.src); ` `        ``int` `y = find(subsets, next_edge.dest); ` ` `  `        ``// If including this edge does't cause cycle, ` `        ``// include it in result and increment the index  ` `        ``// of result for next edge ` `        ``if` `(x != y) ` `        ``{ ` `            ``result[e++] = next_edge; ` `            ``Union(subsets, x, y); ` `        ``} ` `        ``// Else discard the next_edge ` `    ``} ` ` `  `    ``// print the contents of result[] to display the ` `    ``// built MST ` `    ``printf``(````"Following are the edges in the constructed MST "````); ` `    ``for` `(i = 0; i < e; ++i) ` `        ``printf``(````"%d -- %d == %d "````, result[i].src, result[i].dest, ` `                                                 ``result[i].weight); ` `    ``return``; ` `} ` ` `  `// Driver program to test above functions ` `int` `main() ` `{ ` `    ``/* Let us create following weighted graph ` `             ``10 ` `        ``0--------1 ` `        ``|       | ` `       ``6|   5   |15 ` `        ``|      | ` `        ``2--------3 ` `            ``4       */` `    ``int` `V = 4;  ``// Number of vertices in graph ` `    ``int` `E = 5;  ``// Number of edges in graph ` `    ``struct` `Graph* graph = createGraph(V, E); ` ` `  ` `  `    ``// add edge 0-1 ` `    ``graph->edge.src = 0; ` `    ``graph->edge.dest = 1; ` `    ``graph->edge.weight = 10; ` ` `  `    ``// add edge 0-2 ` `    ``graph->edge.src = 0; ` `    ``graph->edge.dest = 2; ` `    ``graph->edge.weight = 6; ` ` `  `    ``// add edge 0-3 ` `    ``graph->edge.src = 0; ` `    ``graph->edge.dest = 3; ` `    ``graph->edge.weight = 5; ` ` `  `    ``// add edge 1-3 ` `    ``graph->edge.src = 1; ` `    ``graph->edge.dest = 3; ` `    ``graph->edge.weight = 15; ` ` `  `    ``// add edge 2-3 ` `    ``graph->edge.src = 2; ` `    ``graph->edge.dest = 3; ` `    ``graph->edge.weight = 4; ` ` `  `    ``KruskalMST(graph); ` ` `  `    ``return` `0; ` `} `

## Java

 `// Java program for Kruskal's algorithm to find Minimum ` `// Spanning Tree of a given connected, undirected and  ` `// weighted graph ` `import` `java.util.*; ` `import` `java.lang.*; ` `import` `java.io.*; ` ` `  `class` `Graph ` `{ ` `    ``// A class to represent a graph edge ` `    ``class` `Edge ``implements` `Comparable ` `    ``{ ` `        ``int` `src, dest, weight; ` ` `  `        ``// Comparator function used for sorting edges  ` `        ``// based on their weight ` `        ``public` `int` `compareTo(Edge compareEdge) ` `        ``{ ` `            ``return` `this``.weight-compareEdge.weight; ` `        ``} ` `    ``}; ` ` `  `    ``// A class to represent a subset for union-find ` `    ``class` `subset ` `    ``{ ` `        ``int` `parent, rank; ` `    ``}; ` ` `  `    ``int` `V, E;    ``// V-> no. of vertices & E->no.of edges ` `    ``Edge edge[]; ``// collection of all edges ` ` `  `    ``// Creates a graph with V vertices and E edges ` `    ``Graph(``int` `v, ``int` `e) ` `    ``{ ` `        ``V = v; ` `        ``E = e; ` `        ``edge = ``new` `Edge[E]; ` `        ``for` `(``int` `i=``0``; i subsets[yroot].rank) ` `            ``subsets[yroot].parent = xroot; ` ` `  `        ``// If ranks are same, then make one as root and increment ` `        ``// its rank by one ` `        ``else` `        ``{ ` `            ``subsets[yroot].parent = xroot; ` `            ``subsets[xroot].rank++; ` `        ``} ` `    ``} ` ` `  `    ``// The main function to construct MST using Kruskal's algorithm ` `    ``void` `KruskalMST() ` `    ``{ ` `        ``Edge result[] = ``new` `Edge[V];  ``// Tnis will store the resultant MST ` `        ``int` `e = ``0``;  ``// An index variable, used for result[] ` `        ``int` `i = ``0``;  ``// An index variable, used for sorted edges ` `        ``for` `(i=``0``; i

## Python

 `# Python program for Kruskal's algorithm to find ` `# Minimum Spanning Tree of a given connected,  ` `# undirected and weighted graph ` ` `  `from` `collections ``import` `defaultdict ` ` `  `#Class to represent a graph ` `class` `Graph: ` ` `  `    ``def` `__init__(``self``,vertices): ` `        ``self``.V``=` `vertices ``#No. of vertices ` `        ``self``.graph ``=` `[] ``# default dictionary  ` `                                ``# to store graph ` `         `  `  `  `    ``# function to add an edge to graph ` `    ``def` `addEdge(``self``,u,v,w): ` `        ``self``.graph.append([u,v,w]) ` ` `  `    ``# A utility function to find set of an element i ` `    ``# (uses path compression technique) ` `    ``def` `find(``self``, parent, i): ` `        ``if` `parent[i] ``=``=` `i: ` `            ``return` `i ` `        ``return` `self``.find(parent, parent[i]) ` ` `  `    ``# A function that does union of two sets of x and y ` `    ``# (uses union by rank) ` `    ``def` `union(``self``, parent, rank, x, y): ` `        ``xroot ``=` `self``.find(parent, x) ` `        ``yroot ``=` `self``.find(parent, y) ` ` `  `        ``# Attach smaller rank tree under root of  ` `        ``# high rank tree (Union by Rank) ` `        ``if` `rank[xroot] < rank[yroot]: ` `            ``parent[xroot] ``=` `yroot ` `        ``elif` `rank[xroot] > rank[yroot]: ` `            ``parent[yroot] ``=` `xroot ` ` `  `        ``# If ranks are same, then make one as root  ` `        ``# and increment its rank by one ` `        ``else` `: ` `            ``parent[yroot] ``=` `xroot ` `            ``rank[xroot] ``+``=` `1` ` `  `    ``# The main function to construct MST using Kruskal's  ` `        ``# algorithm ` `    ``def` `KruskalMST(``self``): ` ` `  `        ``result ``=``[] ``#This will store the resultant MST ` ` `  `        ``i ``=` `0` `# An index variable, used for sorted edges ` `        ``e ``=` `0` `# An index variable, used for result[] ` ` `  `            ``# Step 1:  Sort all the edges in non-decreasing  ` `                ``# order of their ` `                ``# weight.  If we are not allowed to change the  ` `                ``# given graph, we can create a copy of graph ` `        ``self``.graph ``=`  `sorted``(``self``.graph,key``=``lambda` `item: item[``2``]) ` ` `  `        ``parent ``=` `[] ; rank ``=` `[] ` ` `  `        ``# Create V subsets with single elements ` `        ``for` `node ``in` `range``(``self``.V): ` `            ``parent.append(node) ` `            ``rank.append(``0``) ` `     `  `        ``# Number of edges to be taken is equal to V-1 ` `        ``while` `e < ``self``.V ``-``1` `: ` ` `  `            ``# Step 2: Pick the smallest edge and increment  ` `                    ``# the index for next iteration ` `            ``u,v,w ``=`  `self``.graph[i] ` `            ``i ``=` `i ``+` `1` `            ``x ``=` `self``.find(parent, u) ` `            ``y ``=` `self``.find(parent ,v) ` ` `  `            ``# If including this edge does't cause cycle,  ` `                        ``# include it in result and increment the index ` `                        ``# of result for next edge ` `            ``if` `x !``=` `y: ` `                ``e ``=` `e ``+` `1`      `                ``result.append([u,v,w]) ` `                ``self``.union(parent, rank, x, y)             ` `            ``# Else discard the edge ` ` `  `        ``# print the contents of result[] to display the built MST ` `        ``print` `"Following are the edges in the constructed MST"` `        ``for` `u,v,weight  ``in` `result: ` `            ``#print str(u) + " -- " + str(v) + " == " + str(weight) ` `            ``print` `(``"%d -- %d == %d"` `%` `(u,v,weight)) ` ` `  `# Driver code ` `g ``=` `Graph(``4``) ` `g.addEdge(``0``, ``1``, ``10``) ` `g.addEdge(``0``, ``2``, ``6``) ` `g.addEdge(``0``, ``3``, ``5``) ` `g.addEdge(``1``, ``3``, ``15``) ` `g.addEdge(``2``, ``3``, ``4``) ` ` `  `g.KruskalMST() ` ` `  `#This code is contributed by Neelam Yadav `

```Following are the edges in the constructed MST
2 -- 3 == 4
0 -- 3 == 5
0 -- 1 == 10```

Time Complexity: O(ElogE) or O(ElogV). Sorting of edges takes O(ELogE) time. After sorting, we iterate through all edges and apply find-union algorithm. The find and union operations can take atmost O(LogV) time. So overall complexity is O(ELogE + ELogV) time. The value of E can be atmost O(V2), so O(LogV) are O(LogE) same. Therefore, overall time complexity is O(ElogE) or O(ElogV)