How much work is required to combine the solutions

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How big is each subproblem? How much work is required to combine the solutions? Use the Master Theo- 1
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rem to solve the resulting recurrence relation, and get that the running time is O( n log 3 2 ). 5. For this problem, a tree is an undirected graph without cycles. Show that for every pair of nodes x, y in the tree, there is exactly one path between x and y . Solution Hint: What would it mean if there were more than one path from x to y ? 6. Suppose you have a weighted, undirected tree. Show that if you run Di- jkstra’s algorithm on this tree starting from the root, then at each iteration, the known region also consists of a tree. What if you run Dijkstra’s algorithm from a different node? Solution Hint: Show that the known region is always a connected set. What would it mean if it were not a tree? 7. You are given an array A of length n . Each value is 1. However, you do not know what the value of n is. You are allowed to access an element i of the array by calling A [ i ] If i < n , this will return 1, and if i n , this will return a message saying that the location does not exist (you can think of this as an error message, but it will not crash your program). Describe an O(log n ) algorithm to determine the value of n . Solution Hint: This is a variation on binary search! 8. Suppose you want to implement the merge operation from MergeSort, but instead of merging 2 arrays, you want to merge k arrays, each of length n . a. You can do this by first merging the first array and the second array, and then merging the result with the third array, and then merging the result with the fourth array, etc. What is the running time of the described algorithm?
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