Priority Queues1.5. Insert and Delete Operations in 2D Structure

So, we have two basic operations in a priority queue; the first is to extract the next job which in this case means take the job with the highest priority, and remove it from the queue... So, the main operation is delete max; delete the maximum priority item from the queue.

In a priority queue, there are two primary operations: extracting the next job (which is the job with the highest priority) and inserting a new job with a specific priority. The first operation, often called 'delete max', removes the job with the highest priority from the queue.

Imagine a line at a ticket counter where people have different priorities (e.g., VIP customers are served first). Extracting the next job is like serving the person with the highest priority. Meanwhile, inserting a new job with priority is like adding a new VIP customer to the front of the line.

So, the first solution that one could think of to keep such data is to maintain a list, some kind of a linear structure and array or a list... So, whether we use an unsorted list or a sorted list over a sequence of N jobs, supposing N jobs arrive in the system will be process them.

One approach to implementing a priority queue is to use a linear structure, like an array or list. If the list is unsorted, adding a new job is quick, but finding and deleting the maximum priority job takes longer. Conversely, if the list is sorted, deleting the maximum job is quick, but inserting a new job takes longer. The process results in a trade-off, where both methods lead to an overall inefficient time complexity of O(N^2) for processing N jobs.

Think about organizing books in a library. If you just throw them into a basket (unsorted), you can add books quickly, but finding a specific title is slow. If you organize them alphabetically (sorted), finding a book is fast, but putting a new book away takes longer. Thus, both methods have their downsides.

So, we have to go from a one dimensional structure to a two dimensional structure... So, now you can see here an example of such a thing.

To improve efficiency, it is suggested to transition from a one-dimensional structure (like a list) to a two-dimensional structure (like a 5x5 grid). Here, jobs are maintained in rows sorted by priority, allowing for more efficient searching and insertion. This structure facilitates finding the right row for insertion and keeps overall processing costs lower.

Imagine a filing cabinet with multiple drawers (rows), where each drawer is neatly organized by categories (priorities). Instead of searching through a messy pile (one-dimensional list), you quickly check each drawer to find where to place or locate the files (jobs). This organized approach saves time.

So, suppose you want to insert a new job into this list... So, overall it is time O order of square root of n.

When inserting a job into the 2D structure, you first need to locate an empty row. You can systematically check each row for space based on previously tracked sizes, and then find the appropriate place to insert the job within that row. This process allows for faster operations, with a time complexity of O(√N) for the insertion.

Envision a game of Tetris, where you need to place a new block (job) in the right spot within the existing rows (or levels) of blocks. By checking each row for gaps before placing your block, you can quickly ensure it fits well without disrupting the entire structure.

So, we want to delete the maximum element in this array... So, again it is an order root N operation.

To delete the maximum element from the 2D structure, you can quickly identify the maximum value located at the end of each sorted row. By comparing these values, you can efficiently find and remove the maximum priority job. This operation also takes O(√N) time, making it much faster compared to the linear approaches.

Think of a board game where the player with the highest score (maximum element) needs to be identified. Instead of checking every player one by one, you check the last score of each player in separate compartments (the rows) to quickly find out who is winning and remove them from the competitive round.

So, we have now achieved a data structure, which keeps track of elements in a priority queue where insert takes order root N time... So, this is just a sample to explain that a two dimensional structure can give you significant savings over a linear search.

The transition to a 2D structure allows for significant efficiency improvements with both insertion and deletion operations. With both operations now taking O(√N) time, processing N jobs overall takes O(N√N), which is a substantial reduction from the O(N²) time complexity observed in linear structures.

Consider the difference between navigating a complex maze (linear) versus using a map with clearly marked paths (2D structure). The map enables quicker travel from one point (job) to another, as you can easily visualize and choose the best route.

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