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# HW2 - Homework#2 Due Date Wednesday September 29th start of...

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Homework #2 Due Date: Wednesday, September 29th, start of class 1. Let X be a binary search tree with height indicators located at each node. Let X 1 and X 2 be the subtrees rooted at the children of the root of X . Assuming X 1 and X 2 are height balanced AVL trees of height h 1 and h 2 respectively, give an O ( | h 1 - h 2 | + 1) algorithm to balance the whole tree without increasing its height. 2. Suppose we are given all the elements we wish to insert into an initially empty MaxHeap up front. If we insert one at a time, it takes O ( n log n ) time to build the MaxHeap. Suppose we instead build a “pre-heap” of depth d by forming a complete binary tree with each node in an arbitrary position. We must “heapify” this pre-heap so that each node has value greater than its children. (a) Suppose we take the subtrees of size 3 rooted at depth d - 1. Show how to heapify all subtrees in O ( n ) total time. (b) Assuming that all subtrees rooted at depth j are valid MaxHeaps, show how to heapify all subtrees rooted at depth j - 1 in O ( ( d - j ) n 2 d - j ) total time. (c) Analyze the running time to build a heap, using the inductive design process outlined in parts a and b . Hint: n i =1 i 2 i = O (1). 3. Aaron and Ranjan, being diabolical overlords, design a new data structure, Q, for the express purpose of assigning torturous homework problems to CSCI 570 students at USC. Let ARQ

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