Chapter 4: Problem 8
Do you think it is possible for a minimum spanning tree to have a cycle? Justify your answer.
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Use induction to prove the correctness of Dijkstra's Algorithm (Algorithm 4.3).
Use a greedy approach to construct an optimal binary search tree by considering the most probable key, \(K e y_{2}\), for the root, and constructing the left and right subtrees for \(K e y_{1}, K e y_{2}, \ldots, K e y_{2}-1,\) and \(K e y_{2}+\ldots, K e y_{2}+2 \ldots \ldots, K e y_{2}\) recursively in the same way. (a) Assuming the keys are already sorted, what is the worst-case time complexity of this approach? Justify your answer. (b) Use an example to show that this greedy approach does not always find an optimal binary search tree.
Implement Prim's Algorithm (Algorithm 4,1) on your system, and study its performance using different graphs.
Implement Kruskal's Algorithm (Algorithm 4.2) on your system, and study its performance using different graphs
Show that in the refined dynamic programming algorithm for the 0 - 1 Knapsack Problem, the total number of entries computed is about \((W+1) \times\) \((n+1) / 2,\) when \(n=W+1\) and \(w_{i}=1\) for all \(i\).
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