## Introduction

In this article, we will delve deep into the Kth Largest Element in a Stream problem on Leetcode, understand its requirements, and find an optimized Python solution.

## Problem Statement

The problem “Kth Largest Element in a Stream” states:

Design a class to find the `kth`

largest element in a stream. Note that it is the `kth`

largest element in the sorted order, not the `kth`

distinct element. Implement `KthLargest`

class:

`KthLargest(int k, int[] nums)`

Initializes the object with the integer `k`

and the list of integers `nums`

.

`int add(int val)`

Returns the element representing the `kth`

largest element in the stream.

## Understanding the Problem

This problem is about creating a class that can keep track of the `kth`

largest element in a stream of numbers. The class should be initialized with an integer `k`

and a list of integers `nums`

. Then, every time a new number is added through the `add`

method, the class should be able to return the `kth`

largest number seen so far.

It’s important to note that we are not asked to sort all the numbers or to store all the numbers we’ve seen so far. We are only interested in keeping track of the `kth`

largest number.

## Optimized Approach – Using Heaps

An optimal way to solve this problem is by using a min-heap. A heap is a tree-based data structure in which the parent node is either greater than or equal to its child node (max heap) or less than or equal to its child node (min heap).

We use a min heap because it allows us to efficiently track the `kth`

largest number. We maintain a min heap of the `k`

largest numbers. The smallest of these `k`

numbers will then be at the root of the heap and can be accessed in constant time.

This way, when a new number comes in through the `add`

method, we can efficiently determine whether it’s larger than the smallest of the `k`

largest numbers we’ve seen so far, and update our min heap accordingly.

Now, let’s write a Python solution for the problem using the `heapq`

module, which provides an implementation of the heap queue algorithm (priority queue algorithm) or the heap data structure.

```
import heapq
class KthLargest:
def __init__(self, k, nums):
self.k = k
self.heap = nums
heapq.heapify(self.heap)
while len(self.heap) > k:
heapq.heappop(self.heap)
def add(self, val):
if len(self.heap) < self.k:
heapq.heappush(self.heap, val)
elif val > self.heap[0]:
heapq.heapreplace(self.heap, val)
return self.heap[0]
```

In the above Python code:

- We first initialize the
`KthLargest`

class with parameters`k`

and`nums`

. We store`k`

and`nums`

as attributes of the class and transform`nums`

into a min heap using`heapify()`

. We then use`heappop()`

to remove and return the smallest element until only`k`

elements are left. - In the
`add`

method, we add`val`

to the heap if it has fewer than`k`

elements using`heappush()`

. If`val`

is greater than the current`kth`

largest number (the smallest number in the heap), we use`heapreplace()`

to pop the smallest element and push`val`

into the heap in one efficient operation. Finally, we return the`kth`

largest element, which is now the smallest element in the heap.

## Time and Space Complexity Analysis

The time complexity for the `add`

operation is O(log k) because we’re adding an element to and possibly removing an element from a heap of size `k`

– both operations take O(log k) time.

The space complexity is O(k) because we’re storing `k`

elements in the heap.

## Conclusion

In conclusion, the “Kth Largest Element in a Stream” problem is an excellent problem to understand the practical applications of heaps. A good understanding of the properties and operations of heaps can help solve this problem efficiently.

By using a min heap, we are able to track the `kth`

largest number in a stream efficiently, and handle updates in logarithmic time. This is a powerful technique that can be applied to similar problems as well, such as finding the median from a data stream or sorting an almost-sorted array.