TRUE OR FALSE: In an undirected weighted graph the heaviest edge is never in the MST.
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False. Dijkstra’s algorithm will use the heaviest edge of a cycle if it is on the shortest path from the start s to a node t.
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TRUE OR FALSE: In an undirected weighted graph the heaviest edge is never in the MST.
Consider the following weighted undirected graph. (a) Explain why edge (B, D) is safe. In other words, give a cut where the edge is the cheapest edge crossing the cut. (b) We would like to run the Kruskal's algorithm on this graph. List the edges appearing in the Minimum Spanning Tree (MST) in the order they are added to the MST. For simplicity, you can refer to each edge as its weight. (c) 1We would like to run the Prim's algorithm on this...
MST For an undirected graph G = (V, E) with weights w(e) > 0 for each edge e ∈ E, you are given a MST T. Unfortunately one of the edges e* = (u, z) which is in the MST T is deleted from the graph G (no other edges change). Give an algorithm to build a MST for the new graph. Your algorithm should start from T. Note: G is connected, and G − e* is also connected. Explain...
You are given an undirected graph G with weighted edges and a minimum spanning tree T of G. Design an algorithm to update the minimum spanning tree when the weight of a single edge is increased. The input to your algorithm should be the edge e and its new weight: your algorithm should modify T so that it is still a MST. Analyze the running time of your algorithm and prove its correctness.
You are given an undirected graph G with weighted edges and a minimum spanning tree T of G. Design an algorithm to update the minimum spanning tree when the weight of a single edge is decreased. The input to your algorithm should be the edge e and its new weight; your algorithm should modify T so that it is still a MST. Analyze the running time of your algorithm and prove its correctness.
Indicate whether the following is True or False. Consider a simple undirected graph G = (V, E), where |E| < |VI – 1. Then G has at least one cycle. True False
You are given an undirected graph G = (V, E) with positive weights on the edges. If the edge weights are distinct, then there is only one MST, so both Prim’s and Kruskal’s algorithms will find the same MST. If some of the edge weights are the same, then there can be several MSTs and the two algorithms could find different MSTs. Describe a method that forces Prim’s algorithm to find the same MST of G that Kruskal’s algorithm finds.
How much work must Kruskal's MST algorithm do before it starts choosing edges for its MST? Assume the undirected graph has n vertices and m edges. Explain the necessary preliminary work and its big-O cost if done efficiently What are the best case and worst case for Kruskal's MST algorithm with parameters n and/or m? Explain your answer. How much work must Kruskal's MST algorithm do before it starts choosing edges for its MST? Assume the undirected graph has n...
Updating an MST when an edge weight changes. You have a graph G= (V, E) with edge weights given in the graph (whatever they are). In addition, a minimum spanning tree T= (V, E′) of this graph has also been given to you. Now, say we need to increase the weight of one particular edge e. Does the MST change? If so, show how to compute the new MST in linear time. You should consider two cases: 1). when e∈E′and...
Let G = (V, E) be a weighted undirected connected graph that contains a cycle. Let k ∈ E be the edge with maximum weight among all edges in the cycle. Prove that G has a minimum spanning tree NOT including k.
Input a simple undirected weighted graph G with non-negative edge weights (represented by w), and a source node v of G. Output: TDB. D: a vector indexed by the vertices of G. Q: priority queue containing the vertices of G using D[] as key D[v]=0; for (all vertex ut-v) [D[u]-infinity:) while not Q. empty() 11 Q is not empty fu - Q.removein(); // retrieve a vertex of Q with min D value for (all vertex : adjacent to u such...