Question

Draw an example of a heap whose keys are all the odd numbers from 1 to 59 (with no repeats), such that the insertion of an entry with key 32 would cause up-heap bubbling to proceed all the way up to a child of the root (replacing that child's key with 32 ).

Short answer

Arrange odd numbers in a min-heap such that inserting 32 causes it to bubble up to a child of the root.

Step-by-step solution

- Understand the Problem

We need to draw a heap containing all odd numbers from 1 to 59. Furthermore, inserting the number 32 must cause it to bubble up to the child of the root.

- List Odd Numbers

List out all odd numbers between 1 and 59: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59.

- Arrange Keys in Min-Heap

Ensure that the structure satisfies the min-heap property where the parent key is less than its children. Arrange them such that the heap property is maintained.

- Insert 32 and Force Up-Heap

Insert 32 into the heap. Ensure that it bubbles up to the child of the root. Achieve this by placing 32 in a position where it needs to replace a smaller key as it moves up.

- Finalize the Tree

Finalize the tree ensuring that the insertion of 32 maintains the heap property and causes the appropriate bubbling. An example setup could be: \[ 1 \rightarrow (7, 3) \rightarrow (23, 11) \rightarrow (15, 5) \rightarrow (19, 9, 13, 17) \rightarrow (21, 25, 27, 29) \rightarrow (33, 35, 37, 39) \rightarrow (41, 43, 45, 47) \rightarrow (49, 51, 53, 55) \rightarrow (57, 59). \]

Key concepts

Min-Heap

A Min-Heap is a type of binary heap where the parent node has a smaller key than its children nodes. This structure ensures that the smallest key is always at the root.
Min-heaps are widely used in algorithms that require frequent access to the smallest element, such as Dijkstra's shortest path algorithm and the priority queue data structure.

• The root is the smallest element.
• Each parent node is smaller than its child nodes.

It's clear from the properties of Min-Heap that they are very efficient in scenarios where you need quick access to the minimum element, yet you also need to be cautious because maintaining these properties requires frequent adjustments during insertions and deletions.

Tree Structure

A heap is represented as a nearly complete binary tree, where all levels are fully filled except possibly for the last level, which is filled from the left to right.
The Tree Structure ensures an efficient way of maintaining order in heap operations. Every node has at most two children, making the heap a binary tree.

• Nodes are filled level by level from left to right.
• The height of the tree is always \(\text{O(log n)}\), making operations efficient.

This near-complete nature of the tree keeps the operations time-efficient and space-efficient.

Up-Heap Bubbling

Up-Heap Bubbling, or heapify-up, is an operation that ensures the heap property is maintained after inserting a new entry. When a new key is inserted into a heap, it might violate the min-heap property because the new key could be smaller than its parent.
To fix this, we perform up-heap bubbling, which involves swapping the new key with its parent until the min-heap property is restored.
This operation effectively 'bubbles' the new key up the heap until it reaches a suitable position.

• Continuously swap the key with its parent.
• Stop when the heap property is restored or the key becomes the root.

In our exercise, inserting 32 causes it to bubble up to just below the root.

Key Insertion

The process of inserting a new key into a min-heap involves several steps. First, place the new key in the next available spot, ensuring the heap remains a complete binary tree.
Then, we check if the min-heap property is violated: if the new key is smaller than its parent, we perform up-heap bubbling.
The goal is to place the key such that it maintains the min-heap properties and the structural integrity of the tree.

• Place the key in the next available leaf spot.
• If min-heap property is violated, swap with parent.
• Continue this process until the heap property is restored.

In our exercise, inserting 32 and ensuring it reaches the child of the root involves strategic placement followed by up-heap bubbling to maintain the simple hierarchical structure.