Design & Analysis of Algorithms

1. Draw a single binary tree T such that each of the following properties holds: • each internal node of T stores a single character • a preorder traversal of T yields COMPILE, and • a inorder traversal of T yields PMIOLCE.

2. Give the pseudocode for an O(n)-time algorithm that computes the depth of each node of a tree T, where n is the number of nodes of T. Assume the existence of methods setDepth(v,d) and getDepth(v) that run in O(1)-time.

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3. Design an algorithm, inorderNext(v), which returns the node visited after node v in an inorder traversal of binary tree T of size n. Analyze its worst-case running time. Your algorithm should avoid performing traversals of the entire tree.

4. Let T be a binary tree with n nodes. It is realized with an implementation of the Binary Tree ADT that has O(1) running time for all methods except positions() and elements(), which have O(n) running time. Give the pseudocode for a O(n) time algorithm that uses the methods of the Binary Tree interface to visit the nodes of T by increasing values of the level numbering function p given in Section 2.3.4. This traversal is known as the level order traversal. Assume the existence of an O(1) time visit(v) method (it should get called once on each vertex of T during the execution of your algorithm)

5. (a) Illustrate the execution of the selection-sort algorithm on the following input sequence: (21, 14, 32, 10, 44, 8, 2, 11, 20, 26)

(b) Illustrate the execution of the insertion-sort algorithm on the following input sequence: (21, 14, 32, 10, 44, 8, 2, 11, 20, 26)

6. Let S be a sequence containing pairs (k, e) where e is an element and k is its key. There is a simple algorithm called count-sort that will construct a new sorted sequence from S provided that all the keys in S are different from each other. For each key k, count-sort scans S to count how many keys are less than k. If c is the count for k then (k, e) should have rank c in the sorted sequence.

(a) Give the pseudocode for count-sort as it is described above.

(b) Determine the number of comparisons made by count-sort. What is its running time?

(c) As written, count-sort only works if all of the keys have different values. Explain how to modify count-sort to work if multiple keys have the same value

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