Kth Largest Element in a Stream

Identifying Problem Isomorphism

“Kth Largest Element in a Stream” has an approximate isomorphic: “First Unique Number”. Both require us to manage a dynamic data set in real-time, but the nature of the operations and data involved make them quite distinct.

In “Kth Largest Element in a Stream”, you are dealing with a stream of integers and your task is to always be able to return the kth largest element. This problem is often solved using a priority queue or heap data structure.

“First Unique Number” asks you to implement the FirstUnique class which has a constructor that takes an array of integers, a method called showFirstUnique that returns the first unique number, and a method called add that appends an integer to the end of the array. This problem may require the usage of a queue along with a map to keep track of the count of each number.

Though both problems deal with real-time data stream management, the “Kth Largest Element in a Stream” is simpler due to its focus on only one operation (finding the kth largest element), while “First Unique Number” requires maintaining the state of uniqueness along with handling insertions.

This problem is a good example of the use of a data structure known as a Heap. In Python, we have the heapq library which provides an implementation of the heap queue algorithm, also known as the priority queue algorithm.

Heaps are binary trees for which every parent node has a value less than or equal to any of its children. This property is called the heap property. This implementation uses arrays for which heap[k] <= heap[2*k+1] and heap[k] <= heap[2*k+2] for all k, counting elements from zero.

Here’s how we can solve the problem:

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import heapq

class KthLargest:
    def __init__(self, k: int, nums: List[int]):
        self.k = k
        # Convert nums into a min heap
        self.nums = nums
        heapq.heapify(self.nums)
        # Pop the smallest elements from the heap until only the k largest elements remain
        while len(self.nums) > k:
            heapq.heappop(self.nums)

    def add(self, val: int) -> int:
        # If there are less than k elements in the heap, we can just add the new element
        if len(self.nums) < self.k:
            heapq.heappush(self.nums, val)
        # If the new element is larger than the smallest element in the heap, we replace the smallest element
        elif val > self.nums[0]:
            heapq.heapreplace(self.nums, val)
        # Return the smallest element in the heap, which is the kth largest element in the stream
        return self.nums[0]

With this code, the heap always contains the k largest elements we have seen so far, and the smallest element in the heap is the kth largest element.

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class KthLargest:
    def __init__(self, k: int, nums: List[int]):
        self.heap = []
        self.k = k
        for i in nums:
            if len(self.heap) < k:
                heapq.heappush(self.heap,i)
            else:
                if i > self.heap[0]:
                    heapq.heappushpop(self.heap,i)

    def add(self, val: int) -> int:
        if len(self.heap) < self.k:
            heapq.heappush(self.heap,val)
        else:
            if val > self.heap[0]:
                heapq.heappushpop(self.heap,val)      
        return self.heap[0]

Problem Classification

This deals with the design and implementation of a custom class that can handle dynamic data in a stream. This problem requires understanding of sorting and maintenance of order statistics in a stream of data.

The ‘What’ components of the problem include:

  1. Class Design: The problem requires the design of a class (KthLargest) with a specific set of methods and behaviors.

  2. Stream of Integers: The class needs to handle a stream of integers. This involves managing dynamic data where new elements can be added over time.

  3. Kth Largest Element: The class must be able to return the kth largest element from the stream of integers at any given time.

The problem can be classified as an “Online Algorithm” problem. Online algorithms are designed to process their inputs piece-by-piece in a serial fashion, i.e., in the order that the input is fed to the algorithm, without having the entire input available from the start. In this case, the algorithm needs to handle the ongoing addition of elements and consistently return the kth largest element from the current set of values. The problem can also be seen as a “Priority Queue” problem, as priority queues can efficiently handle such kth largest or smallest elements scenarios.

The problem involves knowledge of data structures (for designing the class and managing the stream of integers), sorting algorithms or heap data structure (for maintaining and finding the kth largest element), and dynamic data handling. It’s a practical problem that reflects scenarios in real-world systems that need to keep track of certain order statistics in dynamic data.

Language Agnostic Coding Drills

  1. Dissection of code into distinct concepts:

    • Class Definition: This code defines a Python class named KthLargest. This is a foundational concept in object-oriented programming.

    • Instance Variables: The class uses instance variables (self.heap and self.k) to store state between method calls.

    • Constructor Definition: The __init__ method is a special method in Python classes used for initialization. It sets up the initial state of the object.

    • Use of Lists: The class uses a Python list (self.heap) as a basic data structure to store the integers in the stream.

    • Iterating over a List: The code iterates over the list of integers (nums) provided to the constructor.

    • Heap Data Structure: The code uses a heap data structure for managing the kth largest elements. Python’s built-in heapq library is used for this purpose.

    • Conditional Statements: The code uses if statements to conditionally execute blocks of code.

    • Method Definition: The add method is defined to add a new integer to the stream and return the kth largest element.

  2. List of concepts in increasing difficulty:

    • Class Definition: This is a fundamental concept in many modern programming languages. It is essential to understand how to define a class and its methods. It is relatively straightforward with basic knowledge of object-oriented programming.

    • Instance Variables: Another basic concept, understanding instance variables is key to managing state within a class.

    • Constructor Definition: While specific to each language, the idea of a constructor method is relatively simple once you understand classes and methods.

    • Use of Lists: Lists are a fundamental data structure in many languages. However, understanding their manipulation and usage requires a bit more knowledge and practice.

    • Iterating over a List: Iterating over data structures is a foundational skill in programming. It requires understanding of loops and data structure access methods.

    • Conditional Statements: Understanding when and how to use conditionals is a critical skill in programming, but it’s a bit more complex than the previous concepts.

    • Method Definition: Methods require understanding of how to define inputs and outputs and the concept of encapsulation.

    • Heap Data Structure: Understanding heaps requires more advanced knowledge of data structures and algorithms. It’s more complex due to the specific properties and behaviors of a heap.

  3. Problem-solving approach:

The general approach to solving the problem using these concepts involves creating a class that can initialize a state with the stream of integers, while only keeping track of the kth largest elements using a heap. When a new element is added to the stream, the class needs to update the heap if necessary, and always be able to return the kth largest element.

For each new element, if the heap size is less than k, we add it to the heap. If the heap size is already k and the new element is larger than the smallest element in the heap, we remove the smallest element and add the new one. The smallest element in the heap is always the kth largest element in the stream.

This approach requires understanding of the heap data structure and how to use it effectively. It also requires proper use of instance variables and methods to maintain state and provide the required functionality. The use of conditional statements to manage the heap is also critical to the correct operation of the class.

Targeted Drills in Python

  1. Python-based coding drills:

    • Class Definition: A basic drill for defining a class in Python.

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      class MyClass:
    pass

    • Instance Variables: A drill for using instance variables in Python.

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      class MyClass:
    def __init__(self):
        self.my_var = 10

    • Constructor Definition: A basic drill for defining a constructor in Python.

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      class MyClass:
    def __init__(self):
        pass

    • Use of Lists: A basic drill for defining and manipulating a list in Python.

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      my_list = [1, 2, 3]
my_list.append(4)
print(my_list[0])

    • Iterating over a List: A drill for iterating over a list in Python.

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      my_list = [1, 2, 3]
for item in my_list:
    print(item)

    • Conditional Statements: A drill for using conditional statements in Python.

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      x = 10
if x > 5:
    print("x is greater than 5")
else:
    print("x is less than or equal to 5")

    • Method Definition: A drill for defining a method in Python.

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      class MyClass:
    def my_method(self):
        print("Hello, world!")

    • Heap Data Structure: A drill for using the heap data structure in Python.

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      import heapq
h = []
heapq.heappush(h, 1)
heapq.heappush(h, 2)
heapq.heappush(h, 3)
smallest = heapq.heappop(h)

  2. Problem-specific concepts and drills:

    • Heappushpop Function: In our problem, we often add an element to the heap and immediately remove the smallest one. This is exactly what the heappushpop function does. Here’s a simple drill for it:
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      import heapq
h = [1, 2, 3]
largest = heapq.heappushpop(h, 4)

  3. Integration of drills:

The integration of these drills to solve the problem can be explained as follows:

  • First, we define a class KthLargest with the constructor (__init__) method and the add method.

  • In the constructor, we define the instance variables self.heap and self.k. We then iterate over the input list (nums) and for each number, we check if the length of the heap is less than k. If it is, we add the number to the heap. If it’s not, we check if the number is greater than the smallest element in the heap. If it is, we use heappushpop to add the number and remove the smallest element in one operation.

  • The add method is similar to the iteration in the constructor. We take a new number and either add it to the heap if its size is less than k, or use heappushpop if the number is greater than the smallest element in the heap.

  • Finally, we return the smallest element in the heap, which is the kth largest element in the stream.

By breaking the problem down into these smaller drills and then integrating them together, we create a comprehensive solution to the problem.