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Chapter 7: Problem 47

Study the idea of designing a sorting algorithm based on a ternary heap. A temary heap is like an ordinary heap except that each internal node has three children.

Short Answer

Expert verified
A sorting algorithm based on a ternary heap leverages the heap property where each parent is greater than or equal to its three children. After building a max ternary heap from the input data, the sort occurs by continually swapping the root with the end of the heap, shrinking the heap by one each time, and then re-heapifying the root, until the entire array is in sorted order.

Step by step solution

01

Understand the Heap Property

The heap property requires that for any given node i, the value of node i is greater than or equal to the values of its children. For a ternary heap, this means that the value of an internal node must be greater than or equal to the values of its three children nodes. This property allows for easy access to the largest (or smallest, in min-heaps) element.
02

Understand the Structure of a Ternary Heap

Analogously to binary heaps, a ternary heap can be represented as an array. The root of the heap corresponds to the first element of the array. Then, moving layer by layer, fill in the array from left to right. With this representation, the child nodes of a parent node at index \(i\) are at indexes \(3i+1\), \(3i+2\), and \(3i+3\). The parent of any given node at position j is at index \((j-1)/3\). This information is crucial for maintaining the heap structure.
03

Implement Heapify

To maintain the heap property, we need a 'Heapify' function. It checks if a node (and its children) obey the heap property. If the node breaks the heap property (i.e., if one of its children is larger), it swaps itself with its largest child. This process is recursively done until the node is in a position where it's larger than its three child nodes or it becomes a leaf node (it has no children).
04

Implement Heapsort

Heapsort using a ternary heap will proceed similar to binary heapsort, just with a heapify that handles three children. You begin by building a max ternary heap from the input data. Then, the largest item is at the root, and you swap it with the last item in the array, effectively shrinking the heap by one. You then heapify the root node and repeat these steps until all nodes are sorted.

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Most popular questions from this chapter

Another way to sort a list by exchanging out-of-order keys is called Bubble Sort. Bubble Sort scans adjacent pairs of records and exchanges those found to have out-of-order keys, After the first time through the list, the record with the largest key (or the smallest key) is moved to its proper position. This process is done repeatedly on the remaining, unsorted part of the list until the list is completely sorted. Write the Bubble Sort algorithm. Analyze your algorithm, and show the results using order notation. Compare the performance of the Bubble Sort algorithm against those of Insertion Sort, Exchange Sort, and Selection Sort.

Write a linear-time sorting algorithm that sorts a permutation of integers 1 through \(n\), inclusive.

Give the transpose of the permutation \([2,5,1,6,3,4],\) and find the number of inversions in both permutations. What is the total number of inversions?

Show that the worst-case time complexity of the number of assignments of records for Heapsort is approximated by \(W(n) \approx n\) lg \(n\)

Write a nonrecursive Quicksort 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.
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