Question

3. Consider the the following graphs for each of the two subproblems. Each subproblem can be answered (or blank) independently of the other ( subject to the 4 total blank for partial credit rule). s MST algorithm on the graph below and left, starting with vertex all work done so far: al (40 points) You are runing Prim' a. You are about to take vertex g out of the min-ehave not done so yet. Show the order that vertices wer e considered, and weight and π values if they were ever stored with the vertex. Albo, for that instan (a) Already known to be in the MST (b) The edge has not been seen yet. (c) The edge has been seen, and is known to not be in the MST. (d) The edge has been seen, and is stored with a vertex in the priority queue. t in time, categorize each edge (every one of them) of the graph from the following list: Finally, highlight the edges of the tree that has been found so far. b] (10 points) You are in the middle of running Kruskal's Minimum Weight Spanning Tree algorithm, using the Disjoint Set data structure, with union by size and path compression. The graph is given above to the right. You are about to consider the edge with weight 4. Draw the corresponding disjoint set data structure for the graph, before and after the edge with weight 4. Assume that, for each edge of the graph, the edge is called with the alphabetically first vertex first, and that for sets of the same size, the first set joins to the second.

Answer 1

solution question 3) Using Prim's Algorithm for MST starting vertex a: we keep on adding vertices with minimum edge weights with result set.

For figure on left: 1) picking set={a}. 2) adding edge ab with weight 3 set={a, b}. 3) adding edge bf with weight 2 set={a, b, f}. 4) Now about to add vertex g with edge fg weight 3 from minheap of edges.

For figure on right: 1) picking set={a}. 2) adding edge ab with weight 2 set={a, b}. 3) adding edge bf with weight 3 set={a, b, f}. 4) Now about to add vertex g with edge fg weight 1 from minheap of edges.

Kruskal's MST algorithm: order edges in non-decreasing weights. Pick an edge in order if it doesnot form a cycle or loop:

For figure in right:

1) adding edge fg with weight 1

2) adding edge ab with weight 2

3) adding edge bf with weight 3. Here graph is formed with edges {ab, bf, fg}

4)Now about to add edge ae with weight 4. After adding graph is formed with edges {ea, ab, bf, fg}