Subarrays with K Different Integers

We can solve this problem by using the sliding window technique to find the subarrays with exactly ( k ) distinct integers.

The approach will be to find the number of subarrays with at most ( k ) distinct integers, and the number of subarrays with at most ( k - 1 ) distinct integers. Then, the difference between the two will give the number of subarrays with exactly ( k ) distinct integers.

Here’s how we can implement this approach:

Create a Function for At Most K Distinct: Write a function atMostK that takes nums and k as input and returns the number of subarrays with at most ( k ) distinct integers. Inside this function, use two pointers, left and right, to define a sliding window, and use a dictionary to keep track of the frequency of the numbers inside the window.
Calculate the Number of Subarrays with Exactly K Distinct: Call atMostK(nums, k) and atMostK(nums, k - 1), and take the difference between the two to find the number of subarrays with exactly ( k ) distinct integers.
Return the Result: Return the calculated result.

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class Solution:
    def subarraysWithKDistinct(self, nums: List[int], k: int) -> int:
        def atMostK(nums, k):
            count = collections.Counter()
            left = 0
            result = 0
            for right in range(len(nums)):
                if count[nums[right]] == 0:
                    k -= 1
                count[nums[right]] += 1
                while k < 0:
                    count[nums[left]] -= 1
                    if count[nums[left]] == 0:
                        k += 1
                    left += 1
                result += right - left + 1
            return result

        return atMostK(nums, k) - atMostK(nums, k - 1)

Key Takeaways:

We use the sliding window technique to maintain a window of contiguous elements with at most ( k ) distinct integers.
By finding the number of subarrays with at most ( k ) and at most ( k - 1 ) distinct integers, we can get the number of subarrays with exactly ( k ) distinct integers.
The time complexity of this solution is ( O(n) ), where ( n ) is the length of the array, since we process each element of the array once.

10 Prerequisite LeetCode Problems

“992. Subarrays with K Different Integers” involves sliding window, and hashmap to track the frequency of integers. Here are some simpler problems to understand the concepts:

209. Minimum Size Subarray Sum: This problem introduces the sliding window concept.
904. Fruit Into Baskets: It requires tracking of two different elements in a sliding window.
3. Longest Substring Without Repeating Characters: It is a classic problem for understanding the sliding window technique.
438. Find All Anagrams in a String: This problem applies sliding window with character frequency tracking.
76. Minimum Window Substring: It introduces the concept of maintaining a balance in a sliding window.
567. Permutation in String: This problem involves using a sliding window and checking the frequency of characters.
424. Longest Repeating Character Replacement: It adds more complexity by allowing character replacements in the sliding window.
159. Longest Substring with At Most Two Distinct Characters: It adds a constraint to the number of distinct characters within the sliding window.
340. Longest Substring with At Most K Distinct Characters: It is similar to the previous problem but the ‘K’ distinct characters adds more complexity.
485. Max Consecutive Ones: This problem involves finding maximum consecutive ones in an array which will help you understand the problem at hand.

Problem Classification

Problem Statement: Given an integer array nums and an integer k, return the number of good subarrays of nums. A good array is an array where the number of different integers in that array is exactly k. For example, [1,2,3,1,2] has 3 different integers: 1, 2, and 3. A subarray is a contiguous part of an array.

Example 1:

Input: nums = [1,2,1,2,3], k = 2 Output: 7 Explanation: Subarrays formed with exactly 2 different integers: [1,2], [2,1], [1,2], [2,3], [1,2,1], [2,1,2], [1,2,1,2] Example 2:

Input: nums = [1,2,1,3,4], k = 3 Output: 3 Explanation: Subarrays formed with exactly 3 different integers: [1,2,1,3], [2,1,3], [1,3,4].

Constraints:

1 <= nums.length <= 2 * 104 1 <= nums[i], k <= nums.length

Analyze the provided problem statement. Categorize it based on its domain, ignoring ‘How’ it might be solved. Identify and list out the ‘What’ components. Based on these, further classify the problem. Explain your categorizations.

Visual Model of the Problem

How to visualize the problem statement for this problem?

Problem Restatement

Could you start by paraphrasing the problem statement in your own words? Try to distill the problem into its essential elements and make sure to clarify the requirements and constraints. This exercise should aid in understanding the problem better and aligning our thought process before jumping into solving it.

Abstract Representation of the Problem

Could you help me formulate an abstract representation of this problem?

Given this problem, how can we describe it in an abstract way that emphasizes the structure and key elements, without the specific real-world details?

Terminology

Are there any specialized terms, jargon, or technical concepts that are crucial to understanding this problem or solution? Could you define them and explain their role within the context of this problem?

Problem Simplification and Explanation

Could you please break down this problem into simpler terms? What are the key concepts involved and how do they interact? Can you also provide a metaphor or analogy to help me understand the problem better?

Constraints

Given the problem statement and the constraints provided, identify specific characteristics or conditions that can be exploited to our advantage in finding an efficient solution. Look for patterns or specific numerical ranges that could be useful in manipulating or interpreting the data.

What are the key insights from analyzing the constraints?

Case Analysis

Could you please provide additional examples or test cases that cover a wider range of the input space, including edge and boundary conditions? In doing so, could you also analyze each example to highlight different aspects of the problem, key constraints and potential pitfalls, as well as the reasoning behind the expected output for each case? This should help in generating key insights about the problem and ensuring the solution is robust and handles all possible scenarios.

Identification of Applicable Theoretical Concepts

Can you identify any mathematical or algorithmic concepts or properties that can be applied to simplify the problem or make it more manageable? Think about the nature of the operations or manipulations required by the problem statement. Are there existing theories, metrics, or methodologies in mathematics, computer science, or related fields that can be applied to calculate, measure, or perform these operations more effectively or efficiently?

Problem Breakdown and Solution Methodology

Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.

Inference of Problem-Solving Approach from the Problem Statement

How did you infer from the problem statement that this problem can be solved using ?

Could you please provide a stepwise refinement of our approach to solving this problem?
How can we take the high-level solution approach and distill it into more granular, actionable steps?
Could you identify any parts of the problem that can be solved independently?
Are there any repeatable patterns within our solution?

Solution Approach and Analysis

Thought Process

Explain the thought process by thinking step by step to solve this problem from the problem statement and code the final solution. Write code in Python3. What are the cues in the problem statement? What direction does it suggest in the approach to the problem? Generate insights about the problem statement.

From Brute Force to Optimal Solution

Could you please begin by illustrating a brute force solution for this problem? After detailing and discussing the inefficiencies of the brute force approach, could you then guide us through the process of optimizing this solution? Please explain each step towards optimization, discussing the reasoning behind each decision made, and how it improves upon the previous solution. Also, could you show how these optimizations impact the time and space complexity of our solution?

Coding Constructs

Consider the following piece of complex software code.

What are the high-level problem-solving strategies or techniques being used by this code?
If you had to explain the purpose of this code to a non-programmer, what would you say?
Can you identify the logical elements or constructs used in this code, independent of any programming language?
Could you describe the algorithmic approach used by this code in plain English?
What are the key steps or operations this code is performing on the input data, and why?
Can you identify the algorithmic patterns or strategies used by this code, irrespective of the specific programming language syntax?

Language Agnostic Coding Drills

Your mission is to deconstruct this code into the smallest possible learning units, each corresponding to a separate coding concept. Consider these concepts as unique coding drills that can be individually implemented and later assembled into the final solution.

Dissect the code and identify each distinct concept it contains. Remember, this process should be language-agnostic and generally applicable to most modern programming languages.
Once you’ve identified these coding concepts or drills, list them out in order of increasing difficulty. Provide a brief description of each concept and why it is classified at its particular difficulty level.
Next, describe the problem-solving approach that would lead from the problem statement to the final solution. Think about how each of these coding drills contributes to the overall solution. Elucidate the step-by-step process involved in using these drills to solve the problem. Please refrain from writing any actual code; we’re focusing on understanding the process and strategy.

Targeted Drills in Python

Now that you’ve identified and ordered the coding concepts from a complex software code in the previous exercise, let’s focus on creating Python-based coding drills for each of those concepts.

Begin by writing a separate piece of Python code that encapsulates each identified concept. These individual drills should illustrate how to implement each concept in Python. Please ensure that these are suitable even for those with a basic understanding of Python.
In addition to the general concepts, identify and write coding drills for any problem-specific concepts that might be needed to create a solution. Describe why these drills are essential for our problem.
Once all drills have been coded, describe how these pieces can be integrated together in the right order to solve the initial problem. Each drill should contribute to building up to the final solution.

Remember, the goal is to not only to write these drills but also to ensure that they can be cohesively assembled into one comprehensive solution.