For the following graph give BFS results of steps of the algorithm by giving sequence of states of the queue (list of vertices) before next iteration of the while loop. Each vertex should be given in the form (vertex number, distance, parent). So for example queue element (s, 3, t) should mean that vertex s was discovered at a distance 3 from the root and it was discovered from vertex t. Assume that the graph is given by its adjacency list representation in which edges of every vertex are ordered by the numbers of end vertices. а. start with vertex 3 b. start with vertex 4 6. 8 5 10 3

Consider the following graph. Start from vertex x, use the Depth First Search (DFS) algorithm (described in the textbook) to compute the DFS tree(s). Show intermediate results similar to the textbook example and show the final DFS tree(s). It is assumed that in the adjacency list representation, all vertices are stored according to their alphabetical order. In other words, a vertex’s neighbors should be explored by their relative alphabetical order.

Write algorithm ShortestPath (G, s) pre-cond: G is a (directed or undirected) graph, and s is one of its nodes. post-cond: π specifies a shortest path from s to each node of G, and d specifiestheir lengths.

Think about the disadvantages of an adjacency list representation for a weighted graph.

a.Given the steps of Kruskal’s algorithm, analyze each loop and find its running time cost in terms of V and E.   MST-KRUSKAL(G,w) A ←  for each vertex v  V MAKE-SET(v) sort E into non-decreasing order by w for each (u, v) taken from the sorted list if FIND-SET(u)  FIND-SET(v) then A ← A  {(u, v)} UNION(u, v) return A b. What is the total asymptotic running time of Kruskal’s algorithm