Guidelines for writing pseudocode:

1. C/Java/Python instructions are fine. But do not write object-oriented additions. Do not declare or use any class. Declare only procedures (if necessary) and explain in words what each procedure does, and what is the use of each parameter.

2. One instruction per line

3. Match the brackets with a horizontal line

4. Number your lines

5. Write down if your array is indexed 0...n1 or 1...n.

6. If you decide to use procedures/functions/methods from the class (such as, say, DELETE- MAX(S) for S a max-heap), explain clearly what they do and in what running time. Explain the data structures, if any.

a.) You are given a tree represented by the following data structure. Nodes have three fields: element, parent, and auxiliary. The auxiliary field is 0 and is there to help during traversals. In this problem your algorithm should leave the tree unchanged. You are given the location (or pointer to) two nodes x and y. The goal is to print the elements on the path from x to y in tree, in time proportional to the length of the path. That is, if the tree has n nodes and this path has length k, the running time of the algorithm should be O(k) (O(n) is not good enough, and the tree does not have to be balanced). Argue that your algorithm algorithm indeed has running time O(k). Dont forget to bring back the tree to its original state. Present pseudocode.

b.) We are given two arrays of size n A and B, of integers, Consider the n closed intervals [A[i],B[i]], which may not be disjoint. The union of these n intervals is another set of intervals, which can be made disjoint and maximal. Present an O(n log n) algorithm to output this new set of disjoint and maximal intervals into arrays C and D. Present pseudocode and analyze the running time.

c.) Suppose that you have a black-box worst-case linear-time median subroutine. Give a simple, linear-time algorithm that, given an array A[1..n] and a positive integer i n finds the i^th smallest element of A. Present pseudocode using procedures from the textbook or notes; give complete specifications and state the running time of the procedure in terms of its parameters.