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Chapter 7: Problem 27

Show that there are \(2^{j}\) nodes with depth \(j\) for \(j<\mathrm{d}\) in a heap having \(n\) (a power of 2 ) nodes. Here \(d\) is the depth of the heap.

Short Answer

Expert verified
A heap is a complete binary tree. In such a tree, each level \(j\) (where \(j<d\)) will contain \(2^{j}\) nodes. This is an inherent property of a complete binary heap.

Step by step solution

01

Understanding Heap Structure

A heap is a special kind of binary tree in computing. In the context of heaps, 'depth' refers to the number of edges in the longest path from a node to the root node. The root is at depth 0. The depth of the heap, \(d\), will be the longest path from the root to a leaf. In a complete binary heap where \(n\) is a power of 2, the tree is perfectly balanced, i.e., all levels are completely filled.
02

Calculating Number of Nodes per Level

The heap can be divided into levels, each with a depth, \(j\). The root is at level \(j=0\) and possesses one node, which is equivalent to \(2^{j}\) nodes for \(j = 0\). For the second level (\(j=1\)), two children arise from the parent node, which can be expressed as \(2^{j}\) nodes for \(j = 1\). If you continue this pattern, you will notice that on any given level \(j\), there are \(2^{j}\) nodes when \(j<d\), since each parent node must have exactly two children nodes in a complete binary heap.
03

Conclusion

Given that a heap is a complete binary tree, each level \(j\) (where \(j<d\)) will contain \(2^{j}\) nodes. This is an inherent property of a complete binary heap, and thereby the statement is proven.

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Most popular questions from this chapter

Suppose we have a very large list stored in external memory that needs to be sorted. Assuming that this list is too large for internal memory, what major factor(s) should be considered in designing an external sorting algorithm?

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

Suppose we are to find the \(k\) smallest elements in a list of \(n\) elements, and we are not interested in their relative order. Can a linear-time algorithm be found when \(k\) is a constant? Justify your answer.

Give two instances for which Quicksort algorithm is the most appropriate choice.

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 to those of Insertion Sort, Exchange Sort, and Selection Sort.
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