Heaps and Array Manipulation in Java

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 length between $1$ to $1000$, we need to create a Java method prefixMedian(). This method 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 method should be {1, 1, 2, 2, 3}.

A heap is a useful tool in Java that helps efficiently organize and retrieve data based on their values.

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. In Java, heaps are implemented using the PriorityQueue class.

For our task, we'll be using these principal operations:

• Adding Elements: You can add a new element to a heap using the add() method. By default, Java's PriorityQueue is a Min Heap. To create a Max Heap, you need to provide a custom comparator like Comparator.reverseOrder(). This ensures that the largest element is always at the top.

  Java

  PriorityQueue<Integer> minHeap = new PriorityQueue<>();
  PriorityQueue<Integer> maxHeap = new PriorityQueue<>(Comparator.reverseOrder());

• Removing Elements: The poll() method removes the top element from the heap, which is the smallest element in the Min Heap or the largest element in the Max Heap.

• Accessing Minimum Element: The peek() method is used to inspect the root element of the heap without removing it. For the Min Heap, peek() will return the smallest element.

These operations ensure that the root element can always be gathered quickly, in constant time $O(1)$, and new elements can be added while maintaining the heap structure in logarithmic time, $O(log n)$.