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Chapter 7: Problem 26

Write an algorithm that checks if an essentially complete binary tree is a heap. Analyze your algorithm and show the results using order notation.

Short Answer

Expert verified
The algorithm analyses the tree recursively, checking each node to see if it holds the heap property. In the worst-case scenario, the algorithm will have \(O(n)\) time complexity and \(O(n)\) space complexity, where n is the number of nodes in the tree.

Step by step solution

01

Understand Binary Tree and Heap

A binary tree is a tree-like data structure where each node has at most two children, which are referred to as the left child and the right child. A complete binary tree is a binary tree in which every level, except possibly the last, is completely filled, and all nodes are as far left as possible. A binary heap is a complete binary tree that satisfies the heap property. It is either a Min-Heap: for every node i, \(A[parent(i)] \leq A[i]\) or a Max-Heap: for every node i, \(A[parent(i)] \geq A[i]\).
02

Write the Algorithm

To verify if a binary tree is a heap, start by defining a function IsHeap that requires the binary tree and its size as parameters. The function calls another function CheckHeap that inspects each node to certify it satisfies the heap property. The algorithm runs recursively:
03

Algorithm Description

The CheckHeap function checks if the node at index i is a non-leaf node and the heap property is fulfilled, then recursively call the function for the left and right child of the node. The IsHeap function initially verifies if the tree is complete, which can be done by checking whether the number of nodes is equal to the height of the tree.
04

Analyze the Algorithm

In the worst-case scenario, the algorithm will visit all nodes of the binary tree, which means it runs in \(O(n)\) time complexity, where n is the number of nodes in the binary tree. As the algorithm works by recursively visiting each node of the binary tree once, the space complexity of the algorithm is also \(O(n)\).

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Most popular questions from this chapter

An algorithm called Shell Sort is inspired by Insertion Sort's ability to take advantage of the order of the elements in the list. In Shell Sort, the entire list is divided into noncontiguous sublists whose elements are a distance \(h\) apart for some number \(h\). Each sublist is then sorted using Insertion Sort. During the next pass, the value of \(h\) is reduced, increasing the size of each sublist. Usually the value of each \(h\) is chosen to be relatively prime to: its previous value. The final pass uses the value 1 for \(h\) to sort the list. Write an algorithm for Shell Sort, study its performance, and compare the result with the performance of Insertion Sort.

Implement the Insertion Sort algorithm (Algorithm 7.1 ), run it on your system, and study its best-case, average-case, and worst-case time complexities using several problem instances.

Use the divide-and-conquer approach to write a nonrecursive Mergesort algorithm. Analyze your algorithm, and show the results using order notation. Note that it will be necessary to explicitly maintain a stack in your algorithm.

Show that there are \(n(n-1) / 2\) inversions in a permutation of \(n\) distinct ordered elements with respect to its transpose.

In the process of rebuilding the master list, the Radix Sort algorithm (Algorithm 7.6) wastes a lot of time examining empty sublists when the number of piles (radix) is large. Is it possible to check only the sublists that are not empty?
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