Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 23.2, Problem 1E
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To show that there exists a path to sort the edges of G in Kruskal’s
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Kruskal's method may generate multiple spanning trees for the same input graph G \sdepending on how it breaks ties when the edges are sorted into order. Show that for any least spanning tree T of G , there is a technique to arrange the edges of G in Kruskal's algorithm such that the algorithm produces T .
Hey,
Kruskal's algorithm can return different spanning trees for the input Graph G.Show that for every minimal spanning tree T of G, there is an execution of the algorithm that gives T as a result.
How can i do that?
Thank you in advance!
Kruskal's algorithm can return different spanning trees for the same input G.
Use task part 1 to show that there is only one minimum spanning tree for a graph in which all the edge weights are different.
Does the inverse also hold?
Part 1 was:
Show that for each minimal spanning tree T of G, there is an execution of the algorithm that returns T as a result.
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- Prof also said that the fact that there is a minimum spanning tree T of G (G is input) for every application of the kruskal algorithm that delivers T as a result can be used to show that there is a minimal spanning tree for a Graph with only different edge weights. how is that?arrow_forwardGiven a graph G = (V, E), let us call G an almost-tree if G is connected and G contains at most n + 12 edges, where n = |V |. Each edge of G has an associated cost, and we may assume that all edge costs are distinct. Describe an algorithm that takes as input an almost-tree G and returns a minimum spanning tree of G. Your algorithm should run in O(n) time.arrow_forwardGiven 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?)arrow_forward
- We recollect that Kruskal's Algorithm is used to fi nd the minimum spanning tree in a weighted graph. Given a weighted undirected graph G = (V , E, W), with n vertices/nodes, the algorithm will fi rst sort the edges in E according to their weights. It will then select (n-1) edges with smallest weights that do not form a cycle. (A cycle in a graph is a path along the edges of a graph that starts at a node and ends at the same node after visiting at least one other node and not traversing any of the edges more than once.) Use Kruskal's Algorithm to nd the weight of the minimum spanning tree for the following graph.arrow_forwardLet G = (V, E) be an undirected graph and each edge e ∈ E is associated with a positive weight ℓ(e).For simplicity we assume weights are distinct. Is the following statement true or false? Prove by contradiction or counterexample. Let T be a minimum spanning tree for the graph with the original weight. Suppose we replace eachedge weight ℓ(e) with ℓ(e)^2, then T is still a minimum spanning tree.arrow_forwardGiven an undirected tree T = (V, E) and an integer k. Give a polynomial time algorithm which outputs ”yes” if T has a vertex cover of size at most k and ”no” otherwise. Prove that your algorithm is correct and that it runs in polynomial time.arrow_forward
- Given the graph below, what should be the souce node such that in finding the shortest path tree, the result would be the same whe the minimum spanning tree is searched?arrow_forwardLet G = (V, E) be an undirected graph and each edge e ∈ E is associated with a positive weight ℓ(e).For simplicity we assume weights are distinct. Is the following statement true or false? Let T be a minimum spanning tree for the graph with the original weight. Suppose we replace eachedge weight ℓ(e) with ℓ(e)^2, then T is still a minimum spanning tree.arrow_forwardLet G = (V, E) be a connected, undirected graph, and let s be a fixed vertex in G. Let TB be the spanning tree of G found by a bread first search starting from s, and similarly TD the spanning tree found by depth first search, also starting at s. (As in problem 1, these trees are just sets of edges; the order in which they were traversed by the respective algorithms is irrelevant.) Using facts, prove that TB = TD if and only if TB = TD = E, i.e., G is itself a tree.arrow_forward
- Show all the steps of Kruskal''s minimum cost spanning tree algorithm for a complete graph of 6 vertices where the weight of the edge between the distinct vertices i and j is |i-j-1|, for 1 <= i, j <= 6.arrow_forwardLet the graph G be a cycle of n nodes in which x edges have the weight 100 and y edges have weight 200. How many minimum spanning trees does G have?arrow_forwarda) Draw the connected subgraph of the given graph above which contains only four nodes ACGB and is also a minimum spanning tree with these four nodes. What is its weighted sum? Draw the adjacency matrix representation of this subgraph (use boolean matrix with only 0 or 1, to show its adjacency in this case).b) Find the shortest path ONLY from source node D to destination node G of the given graph above, using Dijkstra’s algorithm. Show your steps with a table as in our course material, clearly indicating the node being selected for processing in each step.c) Draw ONLY the shortest path obtained above, and indicate the weight in each edge in the diagram. Also determine the weighted sumarrow_forward
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