Question

Question II - Graph Traversal and Minimum Spanning Trees [40 Points] Consider the following graph: B 10 1 4 1 H 9 4 a) Traverse the graph starting from vertex A, and using the Breadth-First Search algorithm. Show the traversal result and the data structure you are using. [10 Points] b) Traverse the graph starting from vertex A, and using the Depth-First Search (Post-order) algorithm. Show the traversal result and the data structure you are using. [10 Points] c) Apply Kruskal's algorithm to find a minimum spanning tree for the above graph. Show the steps of the algorithm; i.e. show the order in which the edges are added to the minimum spanning tree during the execution of the algorithm. Draw the resulting minimum spanning tree. [15 points) d) What is the weight of the minimum spanning tree you found in part c)? [5 points]

Answer 1

(a) Breadth First Search uses a queue data structure. First store the value for every node as 0 which shows node unvisited. Start entering the vertices into the queue if they are not visited. While removing the element from queue mark it as visited and enter the adjacent nodes if they are not visited.

BF5 TA А D Tolco E 0 3 o Ť clole [ CLO DIE E ELO F Telplan V Hlo G V Gle HI H BFS sequence : A, B, C, D,E,F,G,H N

(d) Depth First Search uses a stack data structure. First store the value for every node as 0 which shows node unvisited. Start pushing the vertices into the stack if they are not visited and mark it as visited and print it then Push the adjacent nodes. If there is no unvisited node available then pop.

Steps:

1. If Unvisited
2. Make visit = 1
3. print node
4. Push Adjacent node

POPA Push A DES(A) II. visita 2. Print A 3. A → B, C, D, E Push D Push B DES(B) DF5(0) I. visit et L. visit al PopB 12. print B 2. print D 3. 8 C 3.D.F.G Pushc Pop D BOL clot ho Orsich EOL F 01 Push DESCE) Gl01 Hoi lt.visitas Pope 12. print C 3. A Il visited 12. Print E Pope 13. EF Push E I. visit 21 12. print F popt به کی (۶)OFs 3. FG.lt Push (G) DESCG) L. visit z 12.print G 3. Gatt popG // Push (H) DES(H) PopH L. visitzt 2. print H 3.HF, CM

DFS Sequence: ABCDEFGH

(c) In Kruskal's Algorithm choose the minimum available edge from the graph and check if it is forming a cycle with the spanning tree formed so far then discard the edge else include the edge in the spanning tree.

and Aitna um edge weight choose BC [D minenum edge weight. I AC has minimum edge weight. 2 c has FG, GH and will son equal minimum weight F G we choose 4

Now we will choose GH, NIE has 1 Now, HE is HF weight I but it the minimum cycle. So So we forming will discard has minimum edge weight. 3 5 AD AB and O AG discard was equal minimum weight 6. creating and choose AD. а cycle. So we will AB 6 5 1 G H the Above tree is tree for the given minimum Spanning Graph.

(d) Weight of Minimum Spanning Tree = 1 + 3 + 6 + 1 + 5 + 4 + 4 = 24

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