Question

Implement the Prim's minimum spanning tree algorithm for a weighted undirected graph using a heap data structure. The time complexity of the algorithm should be O((|V| + |E|) log |V|). The heap data structure should be implemented as an abstract data type (a class in C++) on a set of elements, where each element has an id and key, with the following operations: - heap_init(keys, n): initializes the heap with the array keys of n elements indexed from 1, where key[i] is the key of the element whose id is i. - in_heap(id): returns true if the element whose id is id is in the heap. - min_key(): returns the minimum key of the heap. - min_id(): returns the id of the element with the minimum key in the heap. - key(id): returns the key of the element whose id is id in the heap. - delete_min(): deletes the element with the minimum key from the heap. - decrease_key(id, new_key): sets the key of the element whose id is id to new_key if its current key is greater than new_key. An input graph file will be available. The format of the input file is as follows: - The first line of the input file contains an integer indicating the number of vertices of the input graph. - Each of the remaining lines contains a triple "i j w" indicating an edge between vertex i and vertex j with a cost of w. The vertex can be considered as the root. The output of your program should be in the following format: - The input graph in the adjacency list representation format, listing each edge with its weight. - The edges (with their weights) of the minimum spanning tree, in the order in which they are produced by the Prim's algorithm.

          Implement the Prim's minimum spanning tree algorithm for a weighted undirected graph using a heap data structure. The time complexity of the algorithm should be O((|V| + |E|) log |V|). The heap data structure should be implemented as an abstract data type (a class in C++) on a set of elements, where each element has an id and key, with the following operations: 
- heap_init(keys, n): initializes the heap with the array keys of n elements indexed from 1, where key[i] is the key of the element whose id is i.
- in_heap(id): returns true if the element whose id is id is in the heap.
- min_key(): returns the minimum key of the heap.
- min_id(): returns the id of the element with the minimum key in the heap.
- key(id): returns the key of the element whose id is id in the heap.
- delete_min(): deletes the element with the minimum key from the heap.
- decrease_key(id, new_key): sets the key of the element whose id is id to new_key if its current key is greater than new_key.

An input graph file will be available. The format of the input file is as follows: 
- The first line of the input file contains an integer indicating the number of vertices of the input graph.
- Each of the remaining lines contains a triple "i j w" indicating an edge between vertex i and vertex j with a cost of w.

The vertex can be considered as the root. The output of your program should be in the following format:
- The input graph in the adjacency list representation format, listing each edge with its weight.
- The edges (with their weights) of the minimum spanning tree, in the order in which they are produced by the Prim's algorithm.

implement the prim minimum spanning tree algorithm for weighted undirected graph using heap data structure the time complexity of the algorithm should be olv iei log iv the heap data structu 25007

Added by Joshua A.

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