Using Heaps for Prefix Median Calculation in TypeScript

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.

class MinHeap {
  private heap: number[] = [];

  add(value: number): void {
    this.heap.push(value);
    this._heapifyUp();
  }

  poll(): number | null {
    if (this.size() === 0) return null;
    if (this.size() <= 1) return this.heap.pop()!;
    const root = this.heap[0];
    this.heap[0] = this.heap.pop()!;
    this._heapifyDown();
    return root;
  }

  peek(): number | undefined {
    return this.heap[0];
  }

  size(): number {
    return this.heap.length;
  }

  private _heapifyUp(): void {
    let index = this.heap.length - 1;
    while (index > 0) {
      const parentIndex = Math.floor((index - 1) / 2);
      if (this.heap[parentIndex] <= this.heap[index]) break;
      [this.heap[parentIndex], this.heap[index]] = [this.heap[index], this.heap[parentIndex]];
      index = parentIndex;
    }
  }

  private _heapifyDown(): void {
    let index = 0;
    while (2 * index + 1 < this.heap.length) {
      let leftChildIndex = 2 * index + 1;
      let rightChildIndex = 2 * index + 2;
      let swapIndex = leftChildIndex;

      if (rightChildIndex < this.heap.length && this.heap[rightChildIndex] < this.heap[leftChildIndex]) {
        swapIndex = rightChildIndex;
      }

      if (this.heap[index] <= this.heap[swapIndex]) break;

      [this.heap[index], this.heap[swapIndex]] = [this.heap[swapIndex], this.heap[index]];
      index = swapIndex;
    }
  }
}

class MaxHeap extends MinHeap {
  add(value: number): void {
    super.add(-value);
  }

  poll(): number | null {
    const value = super.poll();
    return value !== null ? -value : null;
  }

  peek(): number | undefined {
    const value = super.peek();
    return value !== undefined ? -value : undefined;
  }
}

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.