Hello there, budding programmer! I hope you're ready because today we're going to dive deep into high-level data manipulation and increase our understanding of heaps. Heaps are fundamental data structures commonly used in algorithms. We're going to leverage their potential today in an interesting algorithmic problem. Are you ready for the challenge? Let's get started!

We have a task at hand related to array manipulation and the use of heaps. The task is as follows: given an array of unique integers with elements ranging from $1$ to $10^6$ and lengths between $1$ to $1000$, we need to create a TypeScript function prefixMedian(). This function will take the array as input and return a corresponding array, which consists of the medians of all the prefixes of the input array.

Remember that a prefix of an array is a contiguous subsequence that starts from the first element. The median of a sequence of numbers is the middle number when the sequence is sorted. If the length of the sequence is even, the median is the element in the position length / 2 - 1.

For example, consider an input array [1, 9, 2, 8, 3]. The output of your function should be [1, 1, 2, 2, 3].

A heap is a specialized binary tree-based data structure that satisfies the heap property: for a Min Heap, every parent node is less than or equal to its child nodes; for a Max Heap, every parent node is greater than or equal to its child nodes. This property makes heaps particularly efficient for operations like finding the minimum or maximum value.

In practice, heaps can be implemented using arrays. For a given node at index i in an array:

• Its left child is at index 2 * i + 1
• Its right child is at index 2 * i + 2
• Its parent is at index Math.floor((i - 1) / 2)

In our context, we use two specific types of heaps: a Min Heap and a Max Heap. The Min Heap is used to store the larger half of the numbers seen so far, while the Max Heap stores the smaller half. TypeScript allows us to define classes with explicit type annotations for clarity and type safety.

To make the functions easier to understand, let's define our MinHeap and MaxHeap classes first. We will use negation to simulate a MaxHeap with MinHeap operations.

The MinHeap class defines a min-heap with methods to add elements (add), remove and return the smallest element (poll), get the smallest element without removing it (peek), and get the heap size (size). The methods _heapifyUp and _heapifyDown maintain the heap property after adding or removing elements, respectively. The MaxHeap class extends MinHeap by using negation to maintain a max-heap; it overrides the add, poll, and peek methods to negate the values, effectively turning the min-heap operations into max-heap operations.

Alright, let's break our approach down into manageable steps. To begin with, we're going to need two heaps: a Min Heap to store the larger half of the numbers seen so far and a Max Heap to store the smaller half. We'll also need an array to store the median for each prefix. Now, let's initialize these.

This snippet initializes two heap instances, minHeap and maxHeap, to store the larger and smaller halves of the numbers, respectively. It also creates an array medians of the same length as the input array to store the medians of each prefix.

As the next step, we will sequentially take each number from the array and, depending on its value, push it into the minHeap or the maxHeap. If it is smaller than the maximum of the lower half, it will go into the maxHeap. Otherwise, it will go into the minHeap.

Next, we need to balance the two heaps to ensure that the difference between their sizes is never more than one. This way, we can always have quick access to the median. If the maxHeap size becomes larger than the minHeap, we poll the maxHeap's top element and add it to the minHeap. If the minHeap becomes more than one element larger than the maxHeap, we do the reverse.

Let's take an input array [1, 9, 2, 8, 3] and walk through how the heaps are manipulated to find the medians of the prefixes:

• Initial State:

  • MaxHeap: []
  • MinHeap: []
  • Medians: []

• Add 1:

  • MaxHeap: [1] (since it's the first element)
  • MinHeap: []
  • Medians: [1] (median is the only element, 1)

• Add 9:

  • MaxHeap: [1]
  • MinHeap: [9] (9 is greater than 1, so it goes to MinHeap)
  • Medians: [1, 1] (median is still 1, the element in MaxHeap)

• Add 2:

  • MaxHeap: [1]
  • MinHeap: [2, 9] (2 is greater than 1, so it goes to MinHeap)
  • Balance: Move 2 from MinHeap to MaxHeap
  • MaxHeap: [2, 1] (MaxHeap is rearranged after balancing)
  • MinHeap: [9]
  • Medians: [1, 1, 2] (median is 2, the element at the root of MaxHeap)

• Add 8:

  • MaxHeap: [2, 1]
  • MinHeap: [8, 9] (8 is greater than 2, so it goes to MinHeap)
  • Medians: [1, 1, 2, 2] (median is 2, the root of MaxHeap)

• Add 3:

  • MaxHeap: [2, 1]
  • MinHeap: [3, 9, 8] (3 goes to MinHeap because it's greater than 2)
  • Balance: Move 3 from MinHeap to MaxHeap
  • MaxHeap: [3, 1, 2] (MaxHeap maintains balance with MinHeap)
  • MinHeap: [8, 9]
  • Medians: [1, 1, 2, 2, 3] (median is 3, the root of MaxHeap)

Having balanced the heaps, we've set ourselves up for the effortless retrieval of the median. We compute the median based on the elements at the top of the maxHeap and minHeap, and then append it to our array of medians.

After balancing the heaps, this snippet computes the median for each prefix and stores it in the medians array. If the sizes of both heaps are equal, the median is the root of the maxHeap; otherwise, it's the root of the minHeap. The final medians array is returned and tested with an example input.

Congratulations! You've successfully tackled an interesting algorithmic problem that required the use of heaps for array manipulation in TypeScript. The solution you've created not only uses heaps but also demonstrates your understanding of array hierarchies and the meaningful interpretation of numerical values.

In the next session, you'll be given more similar problems to solve. This will encourage you to use heaps and array manipulations fluently, helping you consolidate your understanding of today's lesson. Keep practicing, and remember — practice makes perfect. Happy coding!