9.4. Insertion and Deletion in Heaps

So, recall that our goal is to implement a priority queue. In a priority queue, we have a sequence of jobs that keeps entering the system, each job has a priority. Whenever we are ready to schedule a job to execute, we must pick up not the latest job or the earliest job that we got, but the job which currently has the highest priority among the waiting jobs.

A priority queue is a special type of data structure where each element has a priority assigned to it. When managing tasks represented as jobs, the scheduler selects the job with the highest priority rather than just the next job in line. This requires a system to keep track of these priorities efficiently.

Think of a busy restaurant where customers order food. The chef must prioritize orders based on urgency – for instance, a large party may need their meals first compared to a single diner. Just like the chef prioritizes meal preparation, a priority queue helps manage tasks based on their importance.

So, we saw last time that a linear structure will not allow us to simultaneously optimize these two. We end up with an order N operation for delete max or an order N operation for insert.

Using a simple linear structure like an array, retrieving the job with the highest priority (delete max) or adding a new job (insert) becomes inefficient. Each operation can take time proportional to the number of jobs, denoted as O(N), making it slow as the number of jobs increases.

Imagine trying to find the quickest route through a crowded mall to reach the most important store. If you have to visit every store individually (linear search), it takes a long time. Instead, having a map that shows priority stores can help you navigate quickly, similar to how a heap structure optimizes job processing.

So, the heap is going to be a balance tree whose height is logarithmic in the size that is if flag N nodes in the tree, the height that is the number of edges from the root to any leaf will be log N.

A heap is a specialized tree-based structure where the height is kept logarithmic relative to the number of nodes. This means that as the number of nodes increases, the height grows slowly. The height is crucial because it influences how quickly we can insert or delete jobs in the queue.

Imagine a tall bookshelf where the shelves represent levels of priority. If most of the books (jobs) are organized into only a few shelves (nodes), it’s easy to find or add books. However, if each shelf were overcrowded, it would take longer to search or place books back, just like a poorly structured queue.

The value property says that... So, what does happen in the tree is that we have nodes and each node is the value, so whenever I see a node with value v 1 which has children v 2 and v 3, then what we want is, this is bigger than or equal to v 2 and bigger than or equal to v 3. So, among these three nodes the largest one must be v 1, so this is what is called the max heap property.

In a max heap, every parent node must be greater than or equal to its child nodes. This ensures that the highest priority job is always at the root of the heap, allowing for quick retrieval. In simpler terms, the structure maintains an order based on the values assigned to each node.

Consider a talent show where judges score each contestant. The contestant with the highest score gets to perform first. The scores must ensure no contestant with a lower score is placed before a contestant with a higher score, similar to the max heap maintaining priority.

So, first let us insert 12, so insert 12 means I have to add a value to the heap.... So, I can stop, so this was the result of inserting 12 into this heap.

When inserting a value into the heap, we begin by adding it at the next available position, ensuring the shape of the tree is maintained. After placing the new value, we compare it with its parent node to ensure the max heap property holds, swapping nodes as necessary until no violations exist.

Think of adding a player to a basketball game. You place them in the game (the heap) at the end of the line-up, but then you check to see if they are better than the player currently in front of them. If so, they swap positions, just like swapping values in the heap to maintain the hierarchy.

So, now we have to implement these two operations on heaps, insert and delete max. So, let us see how it works? So, first let us begin... and now I can stop.

The delete max operation removes the highest priority job (the root node) and replaces it with the last node in the heap, followed by a process to restore the heap property. This involves comparing the new root with its children and swapping it with the larger child if necessary, continuing until the tree structure and the heap property are restored.

Imagine removing the fastest car from a race. You replace it with the last car that crossed the finish line, but this may disrupt the ranking. You check which of the remaining cars is faster and swap them as needed until the order is restored, similar to maintaining the heap structure after deleting the max.

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