Question

Consider an unweighted, undirected graph G = 〈V, E). The neighbourhood of a node u E V in the graph is the set of all nodes that are adjacent (or directly connected) to v. Subsequently, we can define the neighbourhood degree of the node v as the sum of the degrees of all its neighbours (those nodes that are directly connects to v) (a) Design an algorithm that returns a list containing the neighbourhood degree for each node v V, assuming that the input is the graph, G, represented using an adjacency list. Each item i in the list that you generate will correspond to the correct value for the neighbourhood degree of node v, Your algorithm should be presented in unambiguous pseudocode. Your algorithm should have a time complexity value O(V E) (b) If an adjacency matrix was used to represent the graph instead of an adjacency list, what is the new value for the time complexity? Justify your answer by explicitly referring to the changes that would be necessary to your algorithm from part (a)

Answer 1

a. Inorder to compute the neighborhood degree of a vertex, firstly let us compute degree of every vertex and hence we will create a list L1 of size V| which contains degree of each vertex.So to compute degree of a vertex, weneed to check it's adjacency list to find the number of edges of the vertex in adjacency list.Once degree of a vertex is computed then we will create another list L2 of size V which will store the desired result by totaling the degree of all its neighborhood Below is the pseudo code NEIGHBORHOOD_DEGREE(G (V,E)) 1. Create list L 1 of size lvl with Li [v] = 0 for all vertex v in v 2. For each vertex v in V 3.For each edge e in adjacency list of v:- LIlv+-1; increase the value of Liv] by 5. Create list L2 of size V with L2[v]- for all vertex v in V 6. For each vertex u in V For each edge (u.,v) in adjacency list of u 敥. 9. Retum list L2/List L2 contains desired result L2 [u]L1[v] //add degree of neighbors of u We can see that we have performed to times traversal of entire vertex adjacency list to compute neighborhood degree and hence time complexity will be O(V+E) b. In adjacency list, to compute degree of vertex v, we have to go through the entire adjacency list of vertex v which is equal to degree ofvertex v and hence computing degree ofv can be in O(degree(v)) time. However in case of adjacency matrix, we have to go though entire row of a matrix to get the degree of a particular vertex which will take O(V) time. Hence to compute degree of all vertices will take O(V2) time and then again to compute the neighborhood degree of all node, we have to visit entire adjacency matrix once, which will take again O(V2) time. In case ofadjacency matrix total time complexity willbe O(V2)