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- 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 algorithmA minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges of a connected, edge-weighted undirected graph that connects all the vertices together, without any cycles and with the minimum possible total edge weight. There are 2 very popular algorthms to calcualte MSTs of a graph. These algorithms are Kruskal's algorithm and Prim's algorithm. 10 Figure 2: A weighted graph and its minimum spanning tree (i) Can you come up with a technique to find the second best MST? Write the algorithm steps and necessary supporting logic. Analyze your algorithms time complexity in terms of the number of vertices V and number of edges E.1 Spanning Trees. Use the Breadth-First Search Algorithm with the vertex ordering a, b, c, d, e to find a spanning tree T for the graph below. Show all steps of the algorithm. b α d e C
- 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.Design a nondeterministic polynomial-time algorithm for following problem: Given a graph G = (V, E), is there a spanning tree with exactly two leaves? Please give an analysis on correctness and running time of your algorithm.Let G = (V, E) be an undirected and connected graph, where each edge (u, v) E E has a weight wt(u, v) > 0. Moreover, assume that all weights are equal. (a) G. The running time of your algorithm must be faster than the running time of Kruskal's algorithm. Design an algorithm to compute a minimum spanning tree (MST) of You must describe your algorithm in plain English (no pseudocode). You must write the running time of your algorithm and explain why you get this running time. (b) In at most 50 words, explain why your algorithm is correct.
- Kruskal's algorithm, which incorporates the union-find data structure, is as follows.algorithm Pre-cond KruskalMST (G): G is an undirected graph.post-cond: The result is a minimum spanning tree.Draw the binary search tree if the in-order traversal of that binary search tree are given as below: In-order traversal: 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 19, 20Name: Degree Constrained Spanning TreeInput: n-node undirected graph G(V,E)Question: What is the smallest integer k such that G has a spanning tree in which no node has degree greater than k?
- Prim's minimal spanning tree algorithm makes use of a priority queue Q of vertices with an associated value on which the queue is sorted. The value on which the priority queue is ordered is the minimum weight of any edge from the vertex to a vertex already removed from Q. O True O FalseThe Floyd-Warshall algorithm is a dynamic algorithm for searching the shortest path in a graph. Each vertex pair has its assigned weight. You are asked to draw the initial directed graph and show the tables for each vertex from Mo to Ms by finding all the shortest paths. Below is the algorithm as a guide. Algorithm 1: Pseudocode of Floyd-Warshall Algorithm Data: A directed weighted graph G(V, E) Result: Shortest path between each pair of vertices in G for each de V do | distance|d][d] «= 0; end for each edge (s, p) € E do | distance[s][p] + weight(s, p); end n = cardinality(V); for k = 1 to n do for i = 1 to n do for j = 1 to n do if distancefi][j] > distance/i][k] + distance/k][j] then | distance i]lj] + distancefi|[k] + distance/k|[j]; end end end end Consider the relation R = {(1,4) =4, (2,1)=3, (2,5)=-3, (3,4)=2, (4,2)=1, (4,3)=1, (5,4)=2 } on A = (1,2,3,4,5) solve the Floyd-Warshall Algorithm.Given an undirected weighted graph G with n nodes and m edges, and we have used Prim’s algorithm to construct a minimum spanning tree T. Suppose the weight of one of the tree edge ((u, v) ∈ T) is changed from w to w′, design an algorithm to verify whether T is still a minimum spanning tree. Your algorithm should run in O(m) time, and explain why your algorithm is correct. You can assume all the weights are distinct. (Hint: When an edge is removed, nodes of T will break into two groups. Which edge should we choose in the cut of these two groups?)