E8-MST.pdf - Exercises 8 – minimal spanning trees(Prim...

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Exercises 8 – minimal spanning trees (Prim and Kruskal) Questions 1. Suppose we have an undirected graph with weights that can be either positive or negative. Do Prim’s and Kruskal’s algorithim produce a MST for such a graph? 2. Consider the problem of computing a maximum spanning tree, namely the spanning tree that maximizes the sum of edge costs. Do Prim and Kruskal’s algorithm work for this problem (assuming of course that we choose the crossing edge with maximum cost)? 3. Prove that for any weighted undirected graph such that the weights are distinct (no two edges have the same weight), the minimal spanning tree is unique. (See lecture 8, slide ~15). 4. Cycle Property: Let G be an undirected connected weighted graph. Suppose the graph has at least one cycle (choose one). For that chosen cycle, let edge e be an edge that has strictly greater cost than all other edges in the cycle. (Such an edge might not exist, e.g. there might be two edges that have the same greatest cost). Show that e does not belong to any MST of G. 5. Consider a “reversed” Kruskal’s algorithm for computing a MST. Initialize T to be the set of all edges in the graph. Now, consider edges from largest to smallest cost. For each edge, delete it from T if that edge belongs to a cycle in T. (Never mind how to implement this. Just note that union-find does not allow deletions, so an inefficient implementation of this reversed Kruskal is not obvious.) Assuming all the edge costs are distinct, does this new algorithm correctly compute a MST? 6. (From Sedgewick’s course) Given a connected graph G with positive weights, which of the following will compute an MST of G? a) Change the weight of each edge from c(e) to c(e) + 17.
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  • Fall '19
  • Graph Theory, Planar graph, Spanning tree, sum of edge costs

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