ECE 242 HOMEWORK 5. In the figure below, one can go from node B to A but not from A to B.

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1 ECE 242 HOMEWORK 5 Question 1: Define the following terms. For lines with multiple terms,differentiate between the terms. Also draw a figure illustrating each term. (a) Directed graph and undirected graph Directed graph is a graph in which all the edges have a specific direction from one node to another. In the figure below, one can go from node B to A but not from A to B. Undirected graph is a graph in which the edges do not have a specific direction, one can go from node A to E in either of the direction.

2 (b) Cycle Cycle is a graph in which all the nodes are connected in a closed chain. In the figure shown below, is a cycle and is a cycle. (c) Spanning tree and minimum spanning tree Spanning tree: A tree form of a graph consisting of all the vertexes and few or all the edges such that there are no cycles and all the vertexes are connected.

3 Minimum spanning tree: A tree form of a graph which consists of all the vertexes with minimum number of edges necessary to connect them. In a weighted graph, minimum spanning tree is a spanning tree with weight less than that compared to every other possible spanning trees possible in that graph. Figure (a) Figure(b) For the graph shown in the figure (a) above, figure (b) shows the minimum spanning tree with cost = 6. Question 2: Suppose G is an undirected graph with vertices labeled 1 through 8. Adjacent vertices for each vertex are listed as follows Vertex Adjacency 1 2,4 2 1,3,4 3 2,4 4 1,2,3,6 5 6,8 6 4,5,7 7 6,8 8 5,7 (a) Order the vertices as they are visited in a depth-first traversal starting at vertex (b) Order the vertices as they are visited in a breadth-first traversal starting at vertex

4 Question 3: (a) Please compare adjacency list and adjacency matrix representations of a graph in terms of i) Memory space required to store the list/matrix. An adjacency matrix is a two-dimensional array in which the elements indicate whether a edge is present between the vertexes or not. For a n sized graph, the adjacency matrix has n*n elements to be stored. An adjacency list is an array of lists in which each individual list shows what vertexes the given vertex is adjacent to. Each edge appears twice if a undirected graph is considered. To represent E edges, we need to store 2E values. It can be seen that Adjacency list requires way less memory than a adjacency matrix. ii)running time to determine if there is an edge between two vertexes. In an adjacency matrix, if we want to determine if there is an edge between two vertexes, we need to inspect the value of the appropriate matrix element. This operation takes a constant amount of time regardless of the size of the network. The complexity is given by O(1). In an adjacency list, to determine an edge between vertex A and vertex B, we need to do through the list of neighbors of A to check if B appears in that list. In worst case,we will have to check all the elements to find the edge between A and B. This complexity is O(N). In this case, using an adjacency matrix might give us a better efficiency. iii)running time to enumerate all neighbors of a specific vertex. In an adjacency matrix, to enumerate the neighbors of a specific vertex, we need to check the entire row searching for an edge with the other nodes in the matrix. This will be of complexity O(N). In an adjacency list, to enumerate the neighbors, we need to run through the list of neighbors of a vertex. The complexity is given by O(N). This will reach N in the worst case scenario. However, this is better than adjacency matrix, to enumerate the neighbors of a vertex. (b)the adjacency matrix representation of an unweighted graph G is given below. Each vertex is labeled with a number and edges are indicated with a 1 (similar to the example from lecture). Please draw the graph G. Vertex

5 (c) Please give the adjacency list representation of the graph in (a). Vertex List containing adjacent vertexes Question 4: A computer network is represented in the form of the graph whose details are provided below. The set V consists of all router nodes and the set E consists of the path costs for the respective links: (a) Construct a graph from the details given below.

6 V = {A, B, C, D, E, F} E = { A-->B = 7 B-->C = 8 A-->E = 4 B-->D = 6 C-->D = 3 D-->E = 5 D-->F = 1 C-->F = 2 } A 7 B 8 C E 5 1 D F (b) Compute the shortest path from A to F using Dijkstra's shortest path algorithm. Write the resulting path. Show all steps for full credit. DIRECTED GRAPH A 7 B 8 C E 5 1 D F

7 STEPS Node Considered B E D C F STEP1 A 7 4 infinity infinity infinity STEP2 E 7 4 infinity infinity infinity STEP3 B infinity (AB) (AB) STEP4 D (AB) (AB) (ABD) STEP5 C (AB) (AB) (ABD) SHORTEST PATH : ABDF(14) UNDIRECTED GRAPH A 7 B 8 C E 5 1 D F

8 STEPS Node Considered B E D C F STEP1 A 7 4 infinity infinity infinity STEP2 E infinity infinity (AE) STEP3 B infinity (AE) (AB) STEP4 D (AE) (AB) (AED) STEP5 C SHORTEST PATH : AEDF (10) (AE) (AB) (AED) (c) Construct the minimum spanning tree for the graph using Kruskal's algorithm. A B C E 5 1 D F