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Chapter 1: Problem 8

Write an algorithm that determines whether or not an almost complete binary tree is a heap.

Short Answer

Expert verified
The algorithm will go through the binary tree, checking each node to ensure that it meets the properties of a heap (value priority compared to its children and completeness). If any node fails this test, then the binary tree is not a heap.

Step by step solution

01

- Initialization

Start by initializing an empty queue, say \( q \), and push the root of the binary tree into it.
02

- Heap Validity Check

Next, we dequeue an element from \( q \) and perform the following validations: A. Check whether the dequeued node has a higher priority than its children. In a min heap, node value is less than or equal to children and in a max heap, node value is greater than or equal to children. B. Check that dequeued node preserves completeness of the binary tree. For this, after encountering the first non-full node (a node that does not have both the children), every following node in the level order traversal should be a leaf node.
03

- Continue Traversal

If the dequeued element has a left child, then push it into the queue. Do the same for its right child if it exists.
04

- Repeat

Repeat steps 2 and 3 until the queue is empty. If a step ever results in the tree failing the heap validation (step 2), then the tree is not a heap.

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Most popular questions from this chapter

Suppose you have a computer that requires I minute to solve problem in stances of size \(n=1000 .\) What instance sizes can be run in 1 minute if you buy a new computer that runs 1000 times faster than the old one, assuming the following time complexities \(T(n)\) for our algorithm? (a) \(T(n) \in \Theta(n)\) (b) \(T(n) \in \Theta\left(n^{3}\right)\) (c) \(T(n) \in \Theta\left(10^{n}\right)\)

Let \(p(n)=a_{4} n^{4}+a_{k-1} n^{k-1}+\dots a_{1} n+a_{0},\) where \(a_{k}>0 .\) Using the Properties of Order in Section \(1.4 .2,\) show that \(p(n) \in \Theta\left(n^{k}\right)\)

Write an Insertion Sort algorithm that uses Binary Search to find the position where the next insertion should take place.

Write an algorithm that finds both the smallest and largest numbers in a list of \(n\) numbers, Try to find a method that does at most about \(1.5 n\) comparisons of array items.

Write an algorithm that finds the greatest common divisor of two integers.
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