Weighted Graph: A graph is termed as weighted graph if each edge of the graph is assigned a weight. The weighted edges stored in the weighted graphs can be stored in adjacency lists. Weighted edges can be represented using a two-dimensional array. An weighted edge can be represented as “WeightedEdge(u,v,w)”, where “u” and “v” are edges and “w” represents the weight between them. Example of storing edge in a weighted graph: Object[][] edges = { new Integer(0), new Integer(1), new SomeTypeForWeight(8) }; Prim’s Algorithm : Prim’s Algorithm is a greedy algorithm that finds a minimum spanning tree for a weighted undirected graph by finding a subset of the edges that forms a tree that includes every vertex, where the total weight of all the edges in the tree is minimized. Time Complexity of Prim’s algorithm: Prim’s Algorithm Steps using adjacency matrix to store the weighted edges: Step 1: Select a random vertex v, add v to S, assign an array A where A[i] = d{v, i} Step 2: While there are vertices that belong to G and not to S do: 2.1. Iterate through A, select the minimum value A[i], add vertex i to S 2.2. for each edge e={i, j} connected to vertex i do: 2.2.1. if d{i, j} < A[j] then A[j] = d{i ,j} Given Graph:

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Chapter 29.4, Problem 29.4.1CP

Program Plan Intro

Weighted Graph:

A graph is termed as weighted graph if each edge of the graph is assigned a weight. The weighted edges stored in the weighted graphs can be stored in adjacency lists.

Weighted edges can be represented using a two-dimensional array. An weighted edge can be represented as “WeightedEdge(u,v,w)”, where “u” and “v” are edges and “w” represents the weight between them.

Example of storing edge in a weighted graph:

Object[][] edges =

{ new Integer(0), new Integer(1), new SomeTypeForWeight(8) };

Prim’s Algorithm:

Prim’s Algorithm is a greedy algorithm that finds a minimum spanning tree for a weighted undirected graph by finding a subset of the edges that forms a tree that includes every vertex, where the total weight of all the edges in the tree is minimized.

Time Complexity of Prim’s algorithm:

Prim’s Algorithm Steps using adjacency matrix to store the weighted edges:

Step 1: Select a random vertex v, add v to S, assign an array A where A[i] = d{v, i}

Step 2: While there are vertices that belong to G and not to S do:

2.1. Iterate through A, select the minimum value A[i], add vertex i to S

2.2. for each edge e={i, j} connected to vertex i do:

2.2.1. if d{i, j} < A[j] then A[j] = d{i ,j}

Given Graph:

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