# Prim’s MST for Adjacency List Representation | Greedy Algo-6

We recommend to read following two posts as a prerequisite of this post.

We have discussed Prim’s algorithm and its implementation for adjacency matrix representation of graphs. The time complexity for the matrix representation is O(V^2). In this post, O(ELogV) algorithm for adjacency list representation is discussed.
As discussed in the previous post, in Prim’s algorithm, two sets are maintained, one set contains list of vertices already included in MST, other set contains vertices not yet included. With adjacency list representation, all vertices of a graph can be traversed in O(V+E) time using BFS. The idea is to traverse all vertices of graph using BFS and use a Min Heap to store the vertices not yet included in MST. Min Heap is used as a priority queue to get the minimum weight edge from the cut. Min Heap is used as time complexity of operations like extracting minimum element and decreasing key value is O(LogV) in Min Heap.

Following are the detailed steps.
1) Create a Min Heap of size V where V is the number of vertices in the given graph. Every node of min heap contains vertex number and key value of the vertex.
2) Initialize Min Heap with first vertex as root (the key value assigned to first vertex is 0). The key value assigned to all other vertices is INF (infinite).
3) While Min Heap is not empty, do following
…..a) Extract the min value node from Min Heap. Let the extracted vertex be u.
…..b) For every adjacent vertex v of u, check if v is in Min Heap (not yet included in MST). If v is in Min Heap and its key value is more than weight of u-v, then update the key value of v as weight of u-v.

Let us understand the above algorithm with the following example:

Initially, key value of first vertex is 0 and INF (infinite) for all other vertices. So vertex 0 is extracted from Min Heap and key values of vertices adjacent to 0 (1 and 7) are updated. Min Heap contains all vertices except vertex 0.
The vertices in green color are the vertices included in MST.

Since key value of vertex 1 is minimum among all nodes in Min Heap, it is extracted from Min Heap and key values of vertices adjacent to 1 are updated (Key is updated if the a vertex is not in Min Heap and previous key value is greater than the weight of edge from 1 to the adjacent). Min Heap contains all vertices except vertex 0 and 1.

Since key value of vertex 7 is minimum among all nodes in Min Heap, it is extracted from Min Heap and key values of vertices adjacent to 7 are updated (Key is updated if the a vertex is not in Min Heap and previous key value is greater than the weight of edge from 7 to the adjacent). Min Heap contains all vertices except vertex 0, 1 and 7.

Since key value of vertex 6 is minimum among all nodes in Min Heap, it is extracted from Min Heap and key values of vertices adjacent to 6 are updated (Key is updated if the a vertex is not in Min Heap and previous key value is greater than the weight of edge from 6 to the adjacent). Min Heap contains all vertices except vertex 0, 1, 7 and 6.

The above steps are repeated for rest of the nodes in Min Heap till Min Heap becomes empty

## C++

 `// C / C++ program for Prim's MST for adjacency list representation of graph ` ` `  `#include ` `#include ` `#include ` ` `  `// A structure to represent a node in adjacency list ` `struct` `AdjListNode { ` `    ``int` `dest; ` `    ``int` `weight; ` `    ``struct` `AdjListNode* next; ` `}; ` ` `  `// A structure to represent an adjacency list ` `struct` `AdjList { ` `    ``struct` `AdjListNode* head; ``// pointer to head node of list ` `}; ` ` `  `// A structure to represent a graph. A graph is an array of adjacency lists. ` `// Size of array will be V (number of vertices in graph) ` `struct` `Graph { ` `    ``int` `V; ` `    ``struct` `AdjList* array; ` `}; ` ` `  `// A utility function to create a new adjacency list node ` `struct` `AdjListNode* newAdjListNode(``int` `dest, ``int` `weight) ` `{ ` `    ``struct` `AdjListNode* newNode = (``struct` `AdjListNode*)``malloc``(``sizeof``(``struct` `AdjListNode)); ` `    ``newNode->dest = dest; ` `    ``newNode->weight = weight; ` `    ``newNode->next = NULL; ` `    ``return` `newNode; ` `} ` ` `  `// A utility function that creates a graph of V vertices ` `struct` `Graph* createGraph(``int` `V) ` `{ ` `    ``struct` `Graph* graph = (``struct` `Graph*)``malloc``(``sizeof``(``struct` `Graph)); ` `    ``graph->V = V; ` ` `  `    ``// Create an array of adjacency lists.  Size of array will be V ` `    ``graph->array = (``struct` `AdjList*)``malloc``(V * ``sizeof``(``struct` `AdjList)); ` ` `  `    ``// Initialize each adjacency list as empty by making head as NULL ` `    ``for` `(``int` `i = 0; i < V; ++i) ` `        ``graph->array[i].head = NULL; ` ` `  `    ``return` `graph; ` `} ` ` `  `// Adds an edge to an undirected graph ` `void` `addEdge(``struct` `Graph* graph, ``int` `src, ``int` `dest, ``int` `weight) ` `{ ` `    ``// Add an edge from src to dest.  A new node is added to the adjacency ` `    ``// list of src.  The node is added at the begining ` `    ``struct` `AdjListNode* newNode = newAdjListNode(dest, weight); ` `    ``newNode->next = graph->array[src].head; ` `    ``graph->array[src].head = newNode; ` ` `  `    ``// Since graph is undirected, add an edge from dest to src also ` `    ``newNode = newAdjListNode(src, weight); ` `    ``newNode->next = graph->array[dest].head; ` `    ``graph->array[dest].head = newNode; ` `} ` ` `  `// Structure to represent a min heap node ` `struct` `MinHeapNode { ` `    ``int` `v; ` `    ``int` `key; ` `}; ` ` `  `// Structure to represent a min heap ` `struct` `MinHeap { ` `    ``int` `size; ``// Number of heap nodes present currently ` `    ``int` `capacity; ``// Capacity of min heap ` `    ``int``* pos; ``// This is needed for decreaseKey() ` `    ``struct` `MinHeapNode** array; ` `}; ` ` `  `// A utility function to create a new Min Heap Node ` `struct` `MinHeapNode* newMinHeapNode(``int` `v, ``int` `key) ` `{ ` `    ``struct` `MinHeapNode* minHeapNode = (``struct` `MinHeapNode*)``malloc``(``sizeof``(``struct` `MinHeapNode)); ` `    ``minHeapNode->v = v; ` `    ``minHeapNode->key = key; ` `    ``return` `minHeapNode; ` `} ` ` `  `// A utilit function to create a Min Heap ` `struct` `MinHeap* createMinHeap(``int` `capacity) ` `{ ` `    ``struct` `MinHeap* minHeap = (``struct` `MinHeap*)``malloc``(``sizeof``(``struct` `MinHeap)); ` `    ``minHeap->pos = (``int``*)``malloc``(capacity * ``sizeof``(``int``)); ` `    ``minHeap->size = 0; ` `    ``minHeap->capacity = capacity; ` `    ``minHeap->array = (``struct` `MinHeapNode**)``malloc``(capacity * ``sizeof``(``struct` `MinHeapNode*)); ` `    ``return` `minHeap; ` `} ` ` `  `// A utility function to swap two nodes of min heap. Needed for min heapify ` `void` `swapMinHeapNode(``struct` `MinHeapNode** a, ``struct` `MinHeapNode** b) ` `{ ` `    ``struct` `MinHeapNode* t = *a; ` `    ``*a = *b; ` `    ``*b = t; ` `} ` ` `  `// A standard function to heapify at given idx ` `// This function also updates position of nodes when they are swapped. ` `// Position is needed for decreaseKey() ` `void` `minHeapify(``struct` `MinHeap* minHeap, ``int` `idx) ` `{ ` `    ``int` `smallest, left, right; ` `    ``smallest = idx; ` `    ``left = 2 * idx + 1; ` `    ``right = 2 * idx + 2; ` ` `  `    ``if` `(left < minHeap->size && minHeap->array[left]->key < minHeap->array[smallest]->key) ` `        ``smallest = left; ` ` `  `    ``if` `(right < minHeap->size && minHeap->array[right]->key < minHeap->array[smallest]->key) ` `        ``smallest = right; ` ` `  `    ``if` `(smallest != idx) { ` `        ``// The nodes to be swapped in min heap ` `        ``MinHeapNode* smallestNode = minHeap->array[smallest]; ` `        ``MinHeapNode* idxNode = minHeap->array[idx]; ` ` `  `        ``// Swap positions ` `        ``minHeap->pos[smallestNode->v] = idx; ` `        ``minHeap->pos[idxNode->v] = smallest; ` ` `  `        ``// Swap nodes ` `        ``swapMinHeapNode(&minHeap->array[smallest], &minHeap->array[idx]); ` ` `  `        ``minHeapify(minHeap, smallest); ` `    ``} ` `} ` ` `  `// A utility function to check if the given minHeap is ampty or not ` `int` `isEmpty(``struct` `MinHeap* minHeap) ` `{ ` `    ``return` `minHeap->size == 0; ` `} ` ` `  `// Standard function to extract minimum node from heap ` `struct` `MinHeapNode* extractMin(``struct` `MinHeap* minHeap) ` `{ ` `    ``if` `(isEmpty(minHeap)) ` `        ``return` `NULL; ` ` `  `    ``// Store the root node ` `    ``struct` `MinHeapNode* root = minHeap->array[0]; ` ` `  `    ``// Replace root node with last node ` `    ``struct` `MinHeapNode* lastNode = minHeap->array[minHeap->size - 1]; ` `    ``minHeap->array[0] = lastNode; ` ` `  `    ``// Update position of last node ` `    ``minHeap->pos[root->v] = minHeap->size - 1; ` `    ``minHeap->pos[lastNode->v] = 0; ` ` `  `    ``// Reduce heap size and heapify root ` `    ``--minHeap->size; ` `    ``minHeapify(minHeap, 0); ` ` `  `    ``return` `root; ` `} ` ` `  `// Function to decreasy key value of a given vertex v. This function ` `// uses pos[] of min heap to get the current index of node in min heap ` `void` `decreaseKey(``struct` `MinHeap* minHeap, ``int` `v, ``int` `key) ` `{ ` `    ``// Get the index of v in  heap array ` `    ``int` `i = minHeap->pos[v]; ` ` `  `    ``// Get the node and update its key value ` `    ``minHeap->array[i]->key = key; ` ` `  `    ``// Travel up while the complete tree is not hepified. ` `    ``// This is a O(Logn) loop ` `    ``while` `(i && minHeap->array[i]->key < minHeap->array[(i - 1) / 2]->key) { ` `        ``// Swap this node with its parent ` `        ``minHeap->pos[minHeap->array[i]->v] = (i - 1) / 2; ` `        ``minHeap->pos[minHeap->array[(i - 1) / 2]->v] = i; ` `        ``swapMinHeapNode(&minHeap->array[i], &minHeap->array[(i - 1) / 2]); ` ` `  `        ``// move to parent index ` `        ``i = (i - 1) / 2; ` `    ``} ` `} ` ` `  `// A utility function to check if a given vertex ` `// 'v' is in min heap or not ` `bool` `isInMinHeap(``struct` `MinHeap* minHeap, ``int` `v) ` `{ ` `    ``if` `(minHeap->pos[v] < minHeap->size) ` `        ``return` `true``; ` `    ``return` `false``; ` `} ` ` `  `// A utility function used to print the constructed MST ` `void` `printArr(``int` `arr[], ``int` `n) ` `{ ` `    ``for` `(``int` `i = 1; i < n; ++i) ` `        ``printf``(````"%d - %d "````, arr[i], i); ` `} ` ` `  `// The main function that constructs Minimum Spanning Tree (MST) ` `// using Prim's algorithm ` `void` `PrimMST(``struct` `Graph* graph) ` `{ ` `    ``int` `V = graph->V; ``// Get the number of vertices in graph ` `    ``int` `parent[V]; ``// Array to store constructed MST ` `    ``int` `key[V]; ``// Key values used to pick minimum weight edge in cut ` ` `  `    ``// minHeap represents set E ` `    ``struct` `MinHeap* minHeap = createMinHeap(V); ` ` `  `    ``// Initialize min heap with all vertices. Key value of ` `    ``// all vertices (except 0th vertex) is initially infinite ` `    ``for` `(``int` `v = 1; v < V; ++v) { ` `        ``parent[v] = -1; ` `        ``key[v] = INT_MAX; ` `        ``minHeap->array[v] = newMinHeapNode(v, key[v]); ` `        ``minHeap->pos[v] = v; ` `    ``} ` ` `  `    ``// Make key value of 0th vertex as 0 so that it ` `    ``// is extracted first ` `    ``key[0] = 0; ` `    ``minHeap->array[0] = newMinHeapNode(0, key[0]); ` `    ``minHeap->pos[0] = 0; ` ` `  `    ``// Initially size of min heap is equal to V ` `    ``minHeap->size = V; ` ` `  `    ``// In the followin loop, min heap contains all nodes ` `    ``// not yet added to MST. ` `    ``while` `(!isEmpty(minHeap)) { ` `        ``// Extract the vertex with minimum key value ` `        ``struct` `MinHeapNode* minHeapNode = extractMin(minHeap); ` `        ``int` `u = minHeapNode->v; ``// Store the extracted vertex number ` ` `  `        ``// Traverse through all adjacent vertices of u (the extracted ` `        ``// vertex) and update their key values ` `        ``struct` `AdjListNode* pCrawl = graph->array[u].head; ` `        ``while` `(pCrawl != NULL) { ` `            ``int` `v = pCrawl->dest; ` ` `  `            ``// If v is not yet included in MST and weight of u-v is ` `            ``// less than key value of v, then update key value and ` `            ``// parent of v ` `            ``if` `(isInMinHeap(minHeap, v) && pCrawl->weight < key[v]) { ` `                ``key[v] = pCrawl->weight; ` `                ``parent[v] = u; ` `                ``decreaseKey(minHeap, v, key[v]); ` `            ``} ` `            ``pCrawl = pCrawl->next; ` `        ``} ` `    ``} ` ` `  `    ``// print edges of MST ` `    ``printArr(parent, V); ` `} ` ` `  `// Driver program to test above functions ` `int` `main() ` `{ ` `    ``// Let us create the graph given in above fugure ` `    ``int` `V = 9; ` `    ``struct` `Graph* graph = createGraph(V); ` `    ``addEdge(graph, 0, 1, 4); ` `    ``addEdge(graph, 0, 7, 8); ` `    ``addEdge(graph, 1, 2, 8); ` `    ``addEdge(graph, 1, 7, 11); ` `    ``addEdge(graph, 2, 3, 7); ` `    ``addEdge(graph, 2, 8, 2); ` `    ``addEdge(graph, 2, 5, 4); ` `    ``addEdge(graph, 3, 4, 9); ` `    ``addEdge(graph, 3, 5, 14); ` `    ``addEdge(graph, 4, 5, 10); ` `    ``addEdge(graph, 5, 6, 2); ` `    ``addEdge(graph, 6, 7, 1); ` `    ``addEdge(graph, 6, 8, 6); ` `    ``addEdge(graph, 7, 8, 7); ` ` `  `    ``PrimMST(graph); ` ` `  `    ``return` `0; ` `} `

## Java

 `// Java program for Prim's MST for ` `// adjacency list representation of graph ` `import` `java.util.LinkedList; ` `import` `java.util.PriorityQueue; ` `import` `java.util.Comparator; ` ` `  `public` `class` `prims { ` `    ``class` `node1 { ` ` `  `        ``// Stores destination vertex in adjacency list ` `        ``int` `dest; ` ` `  `        ``// Stores weight of a vertex in adjacency list ` `        ``int` `weight; ` ` `  `        ``// Constructor ` `        ``node1(``int` `a, ``int` `b) ` `        ``{ ` `            ``dest = a; ` `            ``weight = b; ` `        ``} ` `    ``} ` `    ``static` `class` `Graph { ` ` `  `        ``// Number of vertices in the graph ` `        ``int` `V; ` ` `  `        ``// List of adjacent nodes of a given vertex ` `        ``LinkedList[] adj; ` ` `  `        ``// Constructor ` `        ``Graph(``int` `e) ` `        ``{ ` `            ``V = e; ` `            ``adj = ``new` `LinkedList[V]; ` `            ``for` `(``int` `o = ``0``; o < V; o++) ` `                ``adj[o] = ``new` `LinkedList<>(); ` `        ``} ` `    ``} ` ` `  `    ``// class to represent a node in PriorityQueue ` `    ``// Stores a vertex and its corresponding ` `    ``// key value ` `    ``class` `node { ` `        ``int` `vertex; ` `        ``int` `key; ` `    ``} ` ` `  `    ``// Comparator class created for PriorityQueue ` `    ``// returns 1 if node0.key > node1.key ` `    ``// returns 0 if node0.key < node1.key and ` `    ``// returns -1 otherwise ` `    ``class` `comparator ``implements` `Comparator { ` ` `  `        ``@Override` `        ``public` `int` `compare(node node0, node node1) ` `        ``{ ` `            ``return` `node0.key - node1.key; ` `        ``} ` `    ``} ` ` `  `    ``// method to add an edge ` `    ``// between two vertices ` `    ``void` `addEdge(Graph graph, ``int` `src, ``int` `dest, ``int` `weight) ` `    ``{ ` ` `  `        ``node1 node0 = ``new` `node1(dest, weight); ` `        ``node1 node = ``new` `node1(src, weight); ` `        ``graph.adj[src].addLast(node0); ` `        ``graph.adj[dest].addLast(node); ` `    ``} ` ` `  `    ``// method used to find the mst ` `    ``void` `prims_mst(Graph graph) ` `    ``{ ` ` `  `        ``// Whether a vertex is in PriorityQueue or not ` `        ``Boolean[] mstset = ``new` `Boolean[graph.V]; ` `        ``node[] e = ``new` `node[graph.V]; ` ` `  `        ``// Stores the parents of a vertex ` `        ``int``[] parent = ``new` `int``[graph.V]; ` ` `  `        ``for` `(``int` `o = ``0``; o < graph.V; o++) ` `            ``e[o] = ``new` `node(); ` ` `  `        ``for` `(``int` `o = ``0``; o < graph.V; o++) { ` ` `  `            ``// Initialize to false ` `            ``mstset[o] = ``false``; ` ` `  `            ``// Initialize key values to infinity ` `            ``e[o].key = Integer.MAX_VALUE; ` `            ``e[o].vertex = o; ` `            ``parent[o] = -``1``; ` `        ``} ` ` `  `        ``// Include the source vertex in mstset ` `        ``mstset[``0``] = ``true``; ` ` `  `        ``// Set key value to 0 ` `        ``// so that it is extracted first ` `        ``// out of PriorityQueue ` `        ``e[``0``].key = ``0``; ` ` `  `        ``// PriorityQueue ` `        ``PriorityQueue queue = ``new` `PriorityQueue<>(graph.V, ``new` `comparator()); ` ` `  `        ``for` `(``int` `o = ``0``; o < graph.V; o++) ` `            ``queue.add(e[o]); ` ` `  `        ``// Loops until the PriorityQueue is not empty ` `        ``while` `(!queue.isEmpty()) { ` ` `  `            ``// Extracts a node with min key value ` `            ``node node0 = queue.poll(); ` ` `  `            ``// Include that node into mstset ` `            ``mstset[node0.vertex] = ``true``; ` ` `  `            ``// For all adjacent vertex of the extracted vertex V ` `            ``for` `(node1 iterator : graph.adj[node0.vertex]) { ` ` `  `                ``// If V is in PriorityQueue ` `                ``if` `(mstset[iterator.dest] == ``false``) { ` `                    ``// If the key value of the adjacent vertex is ` `                    ``// more than the extracted key ` `                    ``// update the key value of adjacent vertex ` `                    ``// to update first remove and add the updated vertex ` `                    ``if` `(e[iterator.dest].key > iterator.weight) { ` `                        ``queue.remove(e[iterator.dest]); ` `                        ``e[iterator.dest].key = iterator.weight; ` `                        ``queue.add(e[iterator.dest]); ` `                        ``parent[iterator.dest] = node0.vertex; ` `                    ``} ` `                ``} ` `            ``} ` `        ``} ` ` `  `        ``// Prints the vertex pair of mst ` `        ``for` `(``int` `o = ``1``; o < graph.V; o++) ` `            ``System.out.println(parent[o] + ``" "` `                               ``+ ``"-"` `                               ``+ ``" "` `+ o); ` `    ``} ` ` `  `    ``public` `static` `void` `main(String[] args) ` `    ``{ ` `        ``int` `V = ``9``; ` ` `  `        ``Graph graph = ``new` `Graph(V); ` ` `  `        ``prims e = ``new` `prims(); ` ` `  `        ``e.addEdge(graph, ``0``, ``1``, ``4``); ` `        ``e.addEdge(graph, ``0``, ``7``, ``8``); ` `        ``e.addEdge(graph, ``1``, ``2``, ``8``); ` `        ``e.addEdge(graph, ``1``, ``7``, ``11``); ` `        ``e.addEdge(graph, ``2``, ``3``, ``7``); ` `        ``e.addEdge(graph, ``2``, ``8``, ``2``); ` `        ``e.addEdge(graph, ``2``, ``5``, ``4``); ` `        ``e.addEdge(graph, ``3``, ``4``, ``9``); ` `        ``e.addEdge(graph, ``3``, ``5``, ``14``); ` `        ``e.addEdge(graph, ``4``, ``5``, ``10``); ` `        ``e.addEdge(graph, ``5``, ``6``, ``2``); ` `        ``e.addEdge(graph, ``6``, ``7``, ``1``); ` `        ``e.addEdge(graph, ``6``, ``8``, ``6``); ` `        ``e.addEdge(graph, ``7``, ``8``, ``7``); ` ` `  `        ``// Method invoked ` `        ``e.prims_mst(graph); ` `    ``} ` `} ` `// This code is contributed by Vikash Kumar Dubey `

/div>

## Python

 `# A Python program for Prims's MST for  ` `# adjacency list representation of graph ` ` `  `from` `collections ``import` `defaultdict ` `import` `sys ` ` `  `class` `Heap(): ` ` `  `    ``def` `__init__(``self``): ` `        ``self``.array ``=` `[] ` `        ``self``.size ``=` `0` `        ``self``.pos ``=` `[] ` ` `  `    ``def` `newMinHeapNode(``self``, v, dist): ` `        ``minHeapNode ``=` `[v, dist] ` `        ``return` `minHeapNode ` ` `  `    ``# A utility function to swap two nodes of  ` `    ``# min heap. Needed for min heapify ` `    ``def` `swapMinHeapNode(``self``, a, b): ` `        ``t ``=` `self``.array[a] ` `        ``self``.array[a] ``=` `self``.array[b] ` `        ``self``.array[b] ``=` `t ` ` `  `    ``# A standard function to heapify at given idx ` `    ``# This function also updates position of nodes  ` `    ``# when they are swapped. Position is needed  ` `    ``# for decreaseKey() ` `    ``def` `minHeapify(``self``, idx): ` `        ``smallest ``=` `idx ` `        ``left ``=` `2` `*` `idx ``+` `1` `        ``right ``=` `2` `*` `idx ``+` `2` ` `  `        ``if` `left < ``self``.size ``and` `self``.array[left][``1``] < ` `                                ``self``.array[smallest][``1``]: ` `            ``smallest ``=` `left ` ` `  `        ``if` `right < ``self``.size ``and` `self``.array[right][``1``] < ` `                                ``self``.array[smallest][``1``]: ` `            ``smallest ``=` `right ` ` `  `        ``# The nodes to be swapped in min heap  ` `        ``# if idx is not smallest ` `        ``if` `smallest !``=` `idx: ` ` `  `            ``# Swap positions ` `            ``self``.pos[ ``self``.array[smallest][``0``] ] ``=` `idx ` `            ``self``.pos[ ``self``.array[idx][``0``] ] ``=` `smallest ` ` `  `            ``# Swap nodes ` `            ``self``.swapMinHeapNode(smallest, idx) ` ` `  `            ``self``.minHeapify(smallest) ` ` `  `    ``# Standard function to extract minimum node from heap ` `    ``def` `extractMin(``self``): ` ` `  `        ``# Return NULL wif heap is empty ` `        ``if` `self``.isEmpty() ``=``=` `True``: ` `            ``return` ` `  `        ``# Store the root node ` `        ``root ``=` `self``.array[``0``] ` ` `  `        ``# Replace root node with last node ` `        ``lastNode ``=` `self``.array[``self``.size ``-` `1``] ` `        ``self``.array[``0``] ``=` `lastNode ` ` `  `        ``# Update position of last node ` `        ``self``.pos[lastNode[``0``]] ``=` `0` `        ``self``.pos[root[``0``]] ``=` `self``.size ``-` `1` ` `  `        ``# Reduce heap size and heapify root ` `        ``self``.size ``-``=` `1` `        ``self``.minHeapify(``0``) ` ` `  `        ``return` `root ` ` `  `    ``def` `isEmpty(``self``): ` `        ``return` `True` `if` `self``.size ``=``=` `0` `else` `False` ` `  `    ``def` `decreaseKey(``self``, v, dist): ` ` `  `        ``# Get the index of v in  heap array ` ` `  `        ``i ``=` `self``.pos[v] ` ` `  `        ``# Get the node and update its dist value ` `        ``self``.array[i][``1``] ``=` `dist ` ` `  `        ``# Travel up while the complete tree is not  ` `        ``# hepified. This is a O(Logn) loop ` `        ``while` `i > ``0` `and` `self``.array[i][``1``] < ` `                    ``self``.array[(i ``-` `1``) ``/` `2``][``1``]: ` ` `  `            ``# Swap this node with its parent ` `            ``self``.pos[ ``self``.array[i][``0``] ] ``=` `(i``-``1``)``/``2` `            ``self``.pos[ ``self``.array[(i``-``1``)``/``2``][``0``] ] ``=` `i ` `            ``self``.swapMinHeapNode(i, (i ``-` `1``)``/``2` `) ` ` `  `            ``# move to parent index ` `            ``i ``=` `(i ``-` `1``) ``/` `2``; ` ` `  `    ``# A utility function to check if a given vertex ` `    ``# 'v' is in min heap or not ` `    ``def` `isInMinHeap(``self``, v): ` ` `  `        ``if` `self``.pos[v] < ``self``.size: ` `            ``return` `True` `        ``return` `False` ` `  ` `  `def` `printArr(parent, n): ` `    ``for` `i ``in` `range``(``1``, n): ` `        ``print` `"% d - % d"` `%` `(parent[i], i) ` ` `  ` `  `class` `Graph(): ` ` `  `    ``def` `__init__(``self``, V): ` `        ``self``.V ``=` `V ` `        ``self``.graph ``=` `defaultdict(``list``) ` ` `  `    ``# Adds an edge to an undirected graph ` `    ``def` `addEdge(``self``, src, dest, weight): ` ` `  `        ``# Add an edge from src to dest.  A new node is ` `        ``# added to the adjacency list of src. The node  ` `        ``# is added at the begining. The first element of ` `        ``# the node has the destination and the second  ` `        ``# elements has the weight ` `        ``newNode ``=` `[dest, weight] ` `        ``self``.graph[src].insert(``0``, newNode) ` ` `  `        ``# Since graph is undirected, add an edge from  ` `        ``# dest to src also ` `        ``newNode ``=` `[src, weight] ` `        ``self``.graph[dest].insert(``0``, newNode) ` ` `  `    ``# The main function that prints the Minimum  ` `    ``# Spanning Tree(MST) using the Prim's Algorithm.  ` `    ``# It is a O(ELogV) function ` `    ``def` `PrimMST(``self``): ` `        ``# Get the number of vertices in graph ` `        ``V ``=` `self``.V   ` `         `  `        ``# key values used to pick minimum weight edge in cut ` `        ``key ``=` `[]    ` `         `  `        ``# List to store contructed MST ` `        ``parent ``=` `[]  ` ` `  `        ``# minHeap represents set E ` `        ``minHeap ``=` `Heap() ` ` `  `        ``# Initialize min heap with all vertices. Key values of all ` `        ``# vertices (except the 0th vertex) is is initially infinite ` `        ``for` `v ``in` `range``(V): ` `            ``parent.append(``-``1``) ` `            ``key.append(sys.maxint) ` `            ``minHeap.array.append( minHeap.newMinHeapNode(v, key[v]) ) ` `            ``minHeap.pos.append(v) ` ` `  `        ``# Make key value of 0th vertex as 0 so  ` `        ``# that it is extracted first ` `        ``minHeap.pos[``0``] ``=` `0` `        ``key[``0``] ``=` `0` `        ``minHeap.decreaseKey(``0``, key[``0``]) ` ` `  `        ``# Initially size of min heap is equal to V ` `        ``minHeap.size ``=` `V; ` ` `  `        ``# In the following loop, min heap contains all nodes ` `        ``# not yet added in the MST. ` `        ``while` `minHeap.isEmpty() ``=``=` `False``: ` ` `  `            ``# Extract the vertex with minimum distance value ` `            ``newHeapNode ``=` `minHeap.extractMin() ` `            ``u ``=` `newHeapNode[``0``] ` ` `  `            ``# Traverse through all adjacent vertices of u  ` `            ``# (the extracted vertex) and update their  ` `            ``# distance values ` `            ``for` `pCrawl ``in` `self``.graph[u]: ` ` `  `                ``v ``=` `pCrawl[``0``] ` ` `  `                ``# If shortest distance to v is not finalized  ` `                ``# yet, and distance to v through u is less than ` `                ``# its previously calculated distance ` `                ``if` `minHeap.isInMinHeap(v) ``and` `pCrawl[``1``] < key[v]: ` `                    ``key[v] ``=` `pCrawl[``1``] ` `                    ``parent[v] ``=` `u ` ` `  `                    ``# update distance value in min heap also ` `                    ``minHeap.decreaseKey(v, key[v]) ` ` `  `        ``printArr(parent, V) ` ` `  ` `  `# Driver program to test the above functions ` `graph ``=` `Graph(``9``) ` `graph.addEdge(``0``, ``1``, ``4``) ` `graph.addEdge(``0``, ``7``, ``8``) ` `graph.addEdge(``1``, ``2``, ``8``) ` `graph.addEdge(``1``, ``7``, ``11``) ` `graph.addEdge(``2``, ``3``, ``7``) ` `graph.addEdge(``2``, ``8``, ``2``) ` `graph.addEdge(``2``, ``5``, ``4``) ` `graph.addEdge(``3``, ``4``, ``9``) ` `graph.addEdge(``3``, ``5``, ``14``) ` `graph.addEdge(``4``, ``5``, ``10``) ` `graph.addEdge(``5``, ``6``, ``2``) ` `graph.addEdge(``6``, ``7``, ``1``) ` `graph.addEdge(``6``, ``8``, ``6``) ` `graph.addEdge(``7``, ``8``, ``7``) ` `graph.PrimMST() ` ` `  `# This code is contributed by Divyanshu Mehta `

Output:

```0 - 1
5 - 2
2 - 3
3 - 4
6 - 5
7 - 6
0 - 7
2 - 8```

Time Complexity: The time complexity of the above code/algorithm looks O(V^2) as there are two nested while loops. If we take a closer look, we can observe that the statements in inner loop are executed O(V+E) times (similar to BFS). The inner loop has decreaseKey() operation which takes O(LogV) time. So overall time complexity is O(E+V)*O(LogV) which is O((E+V)*LogV) = O(ELogV) (For a connected graph, V = O(E))

## tags:

Graph Greedy MST Greedy Graph