Given an undirected, connected and weighted graph, find Minimum Spanning Tree (MST) of the graph using Prim’s algorithm.

Input : Adjacency List representation of above graph Output : Edges in MST 0 - 1 1 - 2 2 - 3 3 - 4 2 - 5 5 - 6 6 - 7 2 - 8Note :There are two possible MSTs, the other MST includes edge 0-7 in place of 1-2.

We have discussed below Prim’s MST implementations.

- Prim’s Algorithm for Adjacency Matrix Representation (In C/C++ with time complexity O(v
^{2}) - Prim’s Algorithm for Adjacency List Representation (In C with Time Complexity O(ELogV))

The second implementation is time complexity wise better, but is really complex as we have implemented our own priority queue. STL provides priority_queue, but the provided priority queue doesn’t support decrease key operation. And in Prim’s algorithm, we need a priority queue and below operations on priority queue :

- ExtractMin : from all those vertices which have not yet been included in MST, we need to get vertex with minimum key value.
- DecreaseKey : After extracting vertex we need to update keys of its adjacent vertices, and if new key is smaller, then update that in data structure.

The algorithm discussed here can be modified so that decrease key is never required. The idea is, not to insert all vertices in priority queue, but only those which are not MST and have been visited through a vertex that has included in MST. We keep track of vertices included in MST in a separate boolean array inMST[].

1) Initialize keys of all vertices as infinite and parent of every vertex as -1. 2) Create an empty priority_queue pq. Every item of pq is a pair (weight, vertex). Weight (or key) is used used as first item of pair as first item is by default used to compare two pairs. 3) Initialize all vertices as not part of MST yet. We use boolean array inMST[] for this purpose. This array is required to make sure that an already considered vertex is not included in pq again. This is where Ptim's implementation differs from Dijkstra. In Dijkstr's algorithm, we didn't need this array as distances always increase. We require this array here because key value of a processed vertex may decrease if not checked. 4) Insert source vertex into pq and make its key as 0. 5) While either pq doesn't become empty a) Extract minimum key vertex from pq. Let the extracted vertex be u. b) Include u in MST using inMST[u] = true. c) Loop through all adjacent of u and do following for every vertex v. // If weight of edge (u,v) is smaller than // key of v and v is not already in MST If inMST[v] = false && key[v] > weight(u, v) (i) Update key of v, i.e., do key[v] = weight(u, v) (ii) Insert v into the pq (iv) parent[v] = u 6) Print MST edges using parent array.

Below is C++ implementation of above idea.

`// STL implementation of Prim's algorithm for MST ` `#include<bits/stdc++.h> ` `using` `namespace` `std; ` `# define INF 0x3f3f3f3f ` ` ` `// iPair ==> Integer Pair ` `typedef` `pair<` `int` `, ` `int` `> iPair; ` ` ` `// This class represents a directed graph using ` `// adjacency list representation ` `class` `Graph ` `{ ` ` ` `int` `V; ` `// No. of vertices ` ` ` ` ` `// In a weighted graph, we need to store vertex ` ` ` `// and weight pair for every edge ` ` ` `list< pair<` `int` `, ` `int` `> > *adj; ` ` ` `public` `: ` ` ` `Graph(` `int` `V); ` `// Constructor ` ` ` ` ` `// function to add an edge to graph ` ` ` `void` `addEdge(` `int` `u, ` `int` `v, ` `int` `w); ` ` ` ` ` `// Print MST using Prim's algorithm ` ` ` `void` `primMST(); ` `}; ` ` ` `// Allocates memory for adjacency list ` `Graph::Graph(` `int` `V) ` `{ ` ` ` `this` `->V = V; ` ` ` `adj = ` `new` `list<iPair> [V]; ` `} ` ` ` `void` `Graph::addEdge(` `int` `u, ` `int` `v, ` `int` `w) ` `{ ` ` ` `adj[u].push_back(make_pair(v, w)); ` ` ` `adj[v].push_back(make_pair(u, w)); ` `} ` ` ` `// Prints shortest paths from src to all other vertices ` `void` `Graph::primMST() ` `{ ` ` ` `// Create a priority queue to store vertices that ` ` ` `// are being preinMST. This is weird syntax in C++. ` ` ` `// Refer below link for details of this syntax ` ` ` `priority_queue< iPair, vector <iPair> , greater<iPair> > pq; ` ` ` ` ` `int` `src = 0; ` `// Taking vertex 0 as source ` ` ` ` ` `// Create a vector for keys and initialize all ` ` ` `// keys as infinite (INF) ` ` ` `vector<` `int` `> key(V, INF); ` ` ` ` ` `// To store parent array which in turn store MST ` ` ` `vector<` `int` `> parent(V, -1); ` ` ` ` ` `// To keep track of vertices included in MST ` ` ` `vector<` `bool` `> inMST(V, ` `false` `); ` ` ` ` ` `// Insert source itself in priority queue and initialize ` ` ` `// its key as 0. ` ` ` `pq.push(make_pair(0, src)); ` ` ` `key[src] = 0; ` ` ` ` ` `/* Looping till priority queue becomes empty */` ` ` `while` `(!pq.empty()) ` ` ` `{ ` ` ` `// The first vertex in pair is the minimum key ` ` ` `// vertex, extract it from priority queue. ` ` ` `// vertex label is stored in second of pair (it ` ` ` `// has to be done this way to keep the vertices ` ` ` `// sorted key (key must be first item ` ` ` `// in pair) ` ` ` `int` `u = pq.top().second; ` ` ` `pq.pop(); ` ` ` ` ` `inMST[u] = ` `true` `; ` `// Include vertex in MST ` ` ` ` ` `// 'i' is used to get all adjacent vertices of a vertex ` ` ` `list< pair<` `int` `, ` `int` `> >::iterator i; ` ` ` `for` `(i = adj[u].begin(); i != adj[u].end(); ++i) ` ` ` `{ ` ` ` `// Get vertex label and weight of current adjacent ` ` ` `// of u. ` ` ` `int` `v = (*i).first; ` ` ` `int` `weight = (*i).second; ` ` ` ` ` `// If v is not in MST and weight of (u,v) is smaller ` ` ` `// than current key of v ` ` ` `if` `(inMST[v] == ` `false` `&& key[v] > weight) ` ` ` `{ ` ` ` `// Updating key of v ` ` ` `key[v] = weight; ` ` ` `pq.push(make_pair(key[v], v)); ` ` ` `parent[v] = u; ` ` ` `} ` ` ` `} ` ` ` `} ` ` ` ` ` `// Print edges of MST using parent array ` ` ` `for` `(` `int` `i = 1; i < V; ++i) ` ` ` `printf` `(` `"%d - %d\n"` `, parent[i], i); ` `} ` ` ` `// Driver program to test methods of graph class ` `int` `main() ` `{ ` ` ` `// create the graph given in above fugure ` ` ` `int` `V = 9; ` ` ` `Graph g(V); ` ` ` ` ` `// making above shown graph ` ` ` `g.addEdge(0, 1, 4); ` ` ` `g.addEdge(0, 7, 8); ` ` ` `g.addEdge(1, 2, 8); ` ` ` `g.addEdge(1, 7, 11); ` ` ` `g.addEdge(2, 3, 7); ` ` ` `g.addEdge(2, 8, 2); ` ` ` `g.addEdge(2, 5, 4); ` ` ` `g.addEdge(3, 4, 9); ` ` ` `g.addEdge(3, 5, 14); ` ` ` `g.addEdge(4, 5, 10); ` ` ` `g.addEdge(5, 6, 2); ` ` ` `g.addEdge(6, 7, 1); ` ` ` `g.addEdge(6, 8, 6); ` ` ` `g.addEdge(7, 8, 7); ` ` ` ` ` `g.primMST(); ` ` ` ` ` `return` `0; ` `} ` |

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Time complexity : O(E Log V))

**A Quicker Implementation using array of vectors representation of a weighted graph : **

`// STL implementation of Prim's algorithm for MST ` `#include<bits/stdc++.h> ` `using` `namespace` `std; ` `# define INF 0x3f3f3f3f ` ` ` `// iPair ==> Integer Pair ` `typedef` `pair<` `int` `, ` `int` `> iPair; ` ` ` `// To add an edge ` `void` `addEdge(vector <pair<` `int` `, ` `int` `> > adj[], ` `int` `u, ` ` ` `int` `v, ` `int` `wt) ` `{ ` ` ` `adj[u].push_back(make_pair(v, wt)); ` ` ` `adj[v].push_back(make_pair(u, wt)); ` `} ` ` ` `// Prints shortest paths from src to all other vertices ` `void` `primMST(vector<pair<` `int` `,` `int` `> > adj[], ` `int` `V) ` `{ ` ` ` `// Create a priority queue to store vertices that ` ` ` `// are being preinMST. This is weird syntax in C++. ` ` ` `// Refer below link for details of this syntax ` ` ` `priority_queue< iPair, vector <iPair> , greater<iPair> > pq; ` ` ` ` ` `int` `src = 0; ` `// Taking vertex 0 as source ` ` ` ` ` `// Create a vector for keys and initialize all ` ` ` `// keys as infinite (INF) ` ` ` `vector<` `int` `> key(V, INF); ` ` ` ` ` `// To store parent array which in turn store MST ` ` ` `vector<` `int` `> parent(V, -1); ` ` ` ` ` `// To keep track of vertices included in MST ` ` ` `vector<` `bool` `> inMST(V, ` `false` `); ` ` ` ` ` `// Insert source itself in priority queue and initialize ` ` ` `// its key as 0. ` ` ` `pq.push(make_pair(0, src)); ` ` ` `key[src] = 0; ` ` ` ` ` `/* Looping till priority queue becomes empty */` ` ` `while` `(!pq.empty()) ` ` ` `{ ` ` ` `// The first vertex in pair is the minimum key ` ` ` `// vertex, extract it from priority queue. ` ` ` `// vertex label is stored in second of pair (it ` ` ` `// has to be done this way to keep the vertices ` ` ` `// sorted key (key must be first item ` ` ` `// in pair) ` ` ` `int` `u = pq.top().second; ` ` ` `pq.pop(); ` ` ` ` ` `inMST[u] = ` `true` `; ` `// Include vertex in MST ` ` ` ` ` `// Traverse all adjacent of u ` ` ` `for` `(` `auto` `x : adj[u]) ` ` ` `{ ` ` ` `// Get vertex label and weight of current adjacent ` ` ` `// of u. ` ` ` `int` `v = x.first; ` ` ` `int` `weight = x.second; ` ` ` ` ` `// If v is not in MST and weight of (u,v) is smaller ` ` ` `// than current key of v ` ` ` `if` `(inMST[v] == ` `false` `&& key[v] > weight) ` ` ` `{ ` ` ` `// Updating key of v ` ` ` `key[v] = weight; ` ` ` `pq.push(make_pair(key[v], v)); ` ` ` `parent[v] = u; ` ` ` `} ` ` ` `} ` ` ` `} ` ` ` ` ` `// Print edges of MST using parent array ` ` ` `for` `(` `int` `i = 1; i < V; ++i) ` ` ` `printf` `(` `"%d - %d\n"` `, parent[i], i); ` `} ` ` ` `// Driver program to test methods of graph class ` `int` `main() ` `{ ` ` ` `int` `V = 9; ` ` ` `vector<iPair > adj[V]; ` ` ` ` ` `// making above shown graph ` ` ` `addEdge(adj, 0, 1, 4); ` ` ` `addEdge(adj, 0, 7, 8); ` ` ` `addEdge(adj, 1, 2, 8); ` ` ` `addEdge(adj, 1, 7, 11); ` ` ` `addEdge(adj, 2, 3, 7); ` ` ` `addEdge(adj, 2, 8, 2); ` ` ` `addEdge(adj, 2, 5, 4); ` ` ` `addEdge(adj, 3, 4, 9); ` ` ` `addEdge(adj, 3, 5, 14); ` ` ` `addEdge(adj, 4, 5, 10); ` ` ` `addEdge(adj, 5, 6, 2); ` ` ` `addEdge(adj, 6, 7, 1); ` ` ` `addEdge(adj, 6, 8, 6); ` ` ` `addEdge(adj, 7, 8, 7); ` ` ` ` ` `primMST(adj, V); ` ` ` ` ` `return` `0; ` `} ` |

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Note : Like Dijkstra’s priority_queue implementation, we may have multiple entries for same vertex as we do not (and we can not) make isMST[v] = true in if condition.

`// If v is not in MST and weight of (u,v) is smaller ` ` ` `// than current key of v ` ` ` `if` `(inMST[v] == ` `false` `&& key[v] > weight) ` ` ` `{ ` ` ` `// Updating key of v ` ` ` `key[v] = weight; ` ` ` `pq.push(make_pair(key[v], v)); ` ` ` `parent[v] = u; ` `} ` |

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But as explained in Dijkstra’s algorithm, time complexity remains O(E Log V) as there will be at most O(E) vertices in priority queue and O(Log E) is same as O(Log V).

Unlike Dijkstra’s implementation, a boolean array inMST[] is mandatory here because the key values of newly inserted items can be less than the key values of extracted items. So we must not consider extracted items.

This article is contributed by **Shubham Agrawal**. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above

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