What is the time complexity of Subarrays with K Different Integers?

The optimal solution has a time complexity of O(n).

What data structures are used to solve Subarrays with K Different Integers?

This problem uses Array, Hash Table, Sliding Window, Counting. Understanding these concepts is essential for solving this Hard problem efficiently.

What are the key insights for solving Subarrays with K Different Integers?

Insight 1: The problem can be solved efficiently using the sliding window technique. Insight 2: The number of subarrays with exactly K distinct integers can be found by subtracting the number of subarrays with at most K-1 distinct integers from the number of subarrays with at most K distinct integers. This simplifies the problem to finding subarrays with 'at most' a certain number of distinct elements.

Which companies ask LeetCode 992. Subarrays with K Different Integers in interviews?

This problem is frequently asked at Morgan Stanley, Roblox.

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Subarrays with K Different Integers - LeetCode 992 Solution

Subarrays with K Different Integers - Complete Solution Guide

Subarrays with K Different Integers is LeetCode problem 992, a Hard level challenge. This complete guide provides step-by-step explanations, multiple solution approaches, and optimized code in python3, java, cpp, c.

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

Detailed Explanation

The problem asks us to find the number of subarrays within a given integer array `nums` that contain exactly `k` distinct integers. A subarray is a contiguous section of the array. For example, given `nums = [1, 2, 1, 2, 3]` and `k = 2`, we need to identify all subarrays that have exactly 2 distinct numbers. The goal is to count how many such subarrays exist. The constraints specify the size of the input array (up to 2 * 10^4) and the range of values within the array (1 to the length of the array), as well as the range for K (1 to the length of the array)

Solution Approach

The provided code utilizes a sliding window approach to count the number of subarrays with 'at most' K distinct integers. It calculates this count for K and K-1, then subtracts the latter from the former to get the desired result. The 'atMostK' function maintains a sliding window and a counter for distinct elements. As the right pointer of the window moves, the counter is updated. If the number of distinct elements exceeds K, the left pointer of the window moves until the condition is met again. The number of valid subarrays ending at each right pointer is added to a running total.

Step-by-Step Algorithm

Step 1: Implement the `atMostK` helper function. This function counts the number of subarrays with at most `K` distinct integers.
Step 2: Initialize a hash map `counts` to store the frequency of each number within the current sliding window.
Step 3: Initialize the left pointer `left` to 0 and the result counter `res` to 0.
Step 4: Iterate through the input array `nums` with the right pointer `right`.
Step 5: For each number `num` at the right pointer, update its frequency in the `counts` map. If the number is encountered for the first time (frequency becomes 1), decrement `K`.
Step 6: While `K` is less than 0, it means the current window has more than `K` distinct integers. Shrink the window from the left by incrementing `left` and decrementing the frequency of the number at the left pointer in the `counts` map. If the frequency of the number at the left pointer becomes 0, increment `K` (as we've removed a distinct integer from the window).
Step 7: After ensuring that the current window has at most `K` distinct integers, add `right - left + 1` to the result counter `res`. This represents the number of valid subarrays ending at the current right pointer.
Step 8: Once the loop finishes, return the `res` from `atMostK` function.
Step 9: In the main function `subarraysWithKDistinct`, call `atMostK(k) - atMostK(k - 1)` to get the desired result. This calculates the number of subarrays with exactly `k` distinct integers.

Key Insights

Insight 1: The problem can be solved efficiently using the sliding window technique.
Insight 2: The number of subarrays with exactly K distinct integers can be found by subtracting the number of subarrays with at most K-1 distinct integers from the number of subarrays with at most K distinct integers. This simplifies the problem to finding subarrays with 'at most' a certain number of distinct elements.
Insight 3: A hash table (or dictionary/map) is essential for efficiently tracking the frequency of elements within the sliding window.

Complexity Analysis

Time Complexity: O(n)

Space Complexity: O(n)

Topics

This problem involves: Array, Hash Table, Sliding Window, Counting.

Companies

Asked at: Morgan Stanley, Roblox.

Subarrays with K Different Integers | LeetCode 992

Hard

Array Hash Table Sliding Window Counting

Time: O(n)Space: O(n)

Solution Code

Available in:

import java.util.HashMap;
import java.util.Map;

class Solution {
    public int subarraysWithKDistinct(int[] nums, int k) {
        
        
        
        return atMostK(nums, k) - atMostK(nums, k - 1);
    }
    
    private int atMostK(int[] nums, int K) {
        Map<Integer, Integer> counts = new HashMap<>();
        int left = 0;
        int res = 0;
        for (int right = 0; right < nums.length; right++) {
            int num = nums[right];
            if (!counts.containsKey(num) || counts.get(num) == 0) {
                K -= 1;
            }
            counts.put(num, counts.getOrDefault(num, 0) + 1);
            
            while (K < 0) {
                int left_num = nums[left];
                counts.put(left_num, counts.get(left_num) - 1);
                if (counts.get(left_num) == 0) {
                    K += 1;
                }
                left += 1;
            }
            
            res += right - left + 1;
        }
        return res;
    }
}

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Problem Info

Description

Problem 992: Subarrays with K Different Integers - Detailed explanation and solution approaches.

Topics

Array Hash Table Sliding Window Counting

Companies

Morgan StanleyRoblox

Stats

Difficulty:Hard

Languages:4

Companies:2

Problem #:992

Problem Explanation

💡

Think Before You Code

🤔 Ask Yourself

→What is the input? What output do I need?
→Can I solve a smaller version of this problem first?
→What patterns have I seen in similar problems?

✏️ Try This First

Before looking at the solution, try to solve it on paper with a simple example!

🚀 Try on LeetCode

Solution Approach

📋

Step-by-Step Solution

9 steps

1
Step 1: Implement the `atMostK` helper function. This function counts the number of subarrays with at most `K` distinct integers.
2
Step 2: Initialize a hash map `counts` to store the frequency of each number within the current sliding window.
3
Step 3: Initialize the left pointer `left` to 0 and the result counter `res` to 0.
4
Step 4: Iterate through the input array `nums` with the right pointer `right`.
5
Step 5: For each number `num` at the right pointer, update its frequency in the `counts` map. If the number is encountered for the first time (frequency becomes 1), decrement `K`.
6
Step 6: While `K` is less than 0, it means the current window has more than `K` distinct integers. Shrink the window from the left by incrementing `left` and decrementing the frequency of the number at the left pointer in the `counts` map. If the frequency of the number at the left pointer becomes 0, increment `K` (as we've removed a distinct integer from the window).
7
Step 7: After ensuring that the current window has at most `K` distinct integers, add `right - left + 1` to the result counter `res`. This represents the number of valid subarrays ending at the current right pointer.
8
Step 8: Once the loop finishes, return the `res` from `atMostK` function.
9
Step 9: In the main function `subarraysWithKDistinct`, call `atMostK(k) - atMostK(k - 1)` to get the desired result. This calculates the number of subarrays with exactly `k` distinct integers.

💡

Pro Tip for Students

Try tracing through each step with a small example on paper before coding. Understanding "why" each step happens is more important than memorizing the code!

⚡

Complexity Analysis

⏱️

Time Complexity

O(n)

👍 Linear - Good! Visits each element once.

💾

Space Complexity

O(n)

📦 Linear space - Memory grows with input size.

📝

Why?

The time complexity is O(n) because the sliding window (left and right pointers) moves linearly through the array. The space complexity is O(n) due to the hash map storing counts of distinct elements, which in the worst case might have to store all elements if they're unique.

📖New to Big O? Click to learn more

Big O Notation - Quick Guide

O(1)Best - Instant

O(log n)Excellent - Binary Search

O(n)Good - One loop

O(n log n)OK - Sorting

O(n²)Slow - Nested loops

O(2ⁿ)Very Slow - Recursion

Alternative Approaches

Brute Force

Generate all possible subarrays and check if each subarray has exactly K distinct integers. Count the subarrays that satisfy the condition.

Time: O(n^3) - Generating all subarrays is O(n^2), and checking distinct elements in each subarray is O(n).Space: O(n) - In the worst case, for creating a hashmap for each subarray.

Trade-offs: This is a very simple approach to understand, but it is not efficient for large input arrays due to the high time complexity. It's good for very small inputs but impractical for the given problem constraints.

✨

Key Insights

1Insight 1: The problem can be solved efficiently using the sliding window technique.
2Insight 2: The number of subarrays with exactly K distinct integers can be found by subtracting the number of subarrays with at most K-1 distinct integers from the number of subarrays with at most K distinct integers. This simplifies the problem to finding subarrays with 'at most' a certain number of distinct elements.
3Insight 3: A hash table (or dictionary/map) is essential for efficiently tracking the frequency of elements within the sliding window.

📚

Key Terms Explained

For Students

Array

A collection of items stored at contiguous memory locations. Like a row of boxes.

Hash Table

A data structure that maps keys to values for quick lookup. Like a dictionary!

Sliding Window

A moving "window" over data to process subsets efficiently.

Edge Cases

Edge case 1: `k = 1`. The solution correctly handles this case by counting subarrays with at most 1 distinct element and subtracting subarrays with at most 0 distinct elements (which would be zero).
Edge case 2: `nums` contains only one distinct element. The algorithm accurately calculates the number of subarrays with exactly k (if the single distinct element equals k).
Edge case 3: `k` is larger than the number of distinct elements in `nums`. The `atMostK` helper function will return the length of the array if k exceeds distinct elements. The whole algorithm works fine with this case.
Edge case 4: Empty Array - While not explicitly stated in the constraints (the lower bound is 1), handling an empty input array gracefully could be considered. However, the problem statement implicitly avoids this through the constraint 1 <= nums.length. The code doesn't explicitly check for an empty array, but it will likely return 0 correctly in most languages.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Not handling the edge case when `k` is equal to 1 correctly. The subtraction `atMostK(k) - atMostK(k - 1)` is crucial for handling this.

✗Common mistake 2: Forgetting to shrink the sliding window when the number of distinct elements exceeds `K`. This would result in an incorrect count of subarrays.

✗Common mistake 3: Incorrectly updating the counts in the hash map, leading to inaccurate tracking of distinct element counts.

✗Common mistake 4: Not understanding the 'atMostK' concept and trying to directly find subarrays with *exactly* K distinct elements, which complicates the logic.

✅

Learning Tip

Every mistake is a learning opportunity! When you make an error, understand why it happened before moving on.

🌍

Where is This Used in Real Life?

💼Real-world scenario 1: Analyzing network traffic to identify patterns of packet flows with a specific number of distinct source IP addresses within a time window. This could help in detecting network anomalies or intrusions.

🏦Real-world scenario 2: In bioinformatics, identifying DNA sequences (subarrays) with a specific number of distinct nucleotides. This can be useful for gene finding or analyzing genetic diversity.

🎮Real-world scenario 3: In market basket analysis, identifying customer purchase patterns where a basket (subarray) contains a specific number of distinct product categories. This can inform targeted marketing campaigns.

💡Understanding real-world applications helps you see the "why" behind algorithms. It's not just about passing interviews—these concepts power the apps and websites you use every day!

Author's Notes

💡 Key Insight

When dealing with arrays, always consider if sorting would simplify the problem. Many array problems have O(n) solutions using hash maps or two-pointer technique.

🎯 Interview Tip

These problems test your ability to optimize. Start with a brute force solution, then improve.

⚠️ Watch Out

Off-by-one errors are the most common mistake. Double-check your loop bounds and array indices.

Written by Saswata Mondal

Software Engineer • Algorithms Enthusiast • Interview Coach

👨‍💻

About the Author

👨‍💻

Saswata MondalAuthor400+ LC Problems

B.Tech CSE, IEM Kolkata | LeetCode Rating: 1672 (Top 15%)

Certified: IBM Java Developer, NPTEL Programming in Java

Last updated: September 1, 2024LeetCode Profile GitHub

This solution has been crafted based on my experience solving 400+ LeetCode problems. Each explanation is designed to help you understand the underlying concepts, not just memorize the code.

Community Tips & Insights

I was asked this exact problem at my Google interview. The interviewer wanted me to optimize space complexity.

@InterviewProJul 14

Try solving this without using extra space first. It's a great exercise for understanding the underlying pattern.

@CodeNinjaJul 7

❓

Frequently Asked Questions

What is the most efficient way to solve array problems?

Most array problems can be solved efficiently using techniques like two-pointer approach, sliding window, or hash maps. The key is identifying whether the array is sorted (enabling binary search) or if tracking seen elements (using hash maps) can help achieve O(n) time complexity.

What are the trade-offs of using a hash table?

Hash tables provide O(1) average-case lookup, insertion, and deletion, making them ideal for frequency counting and duplicate detection. However, they use O(n) extra space and may degrade to O(n) time in worst-case scenarios with hash collisions. Always consider if the space trade-off is acceptable.

How do I approach hard algorithm problems?

Hard problems often require: 1) Breaking down into subproblems, 2) Combining multiple advanced techniques, 3) Careful optimization of brute force. Start with a working solution (even O(n²) or worse), then optimize. Draw examples, identify patterns, and think about what makes this problem unique.

How should I practice algorithm problems effectively?

Focus on understanding patterns rather than memorizing solutions. After solving a problem, review optimal solutions and understand the intuition. Group similar problems to recognize patterns. Practice explaining your approach out loud. Review problems after a few days to test retention.

What is the optimal approach for "Subarrays with K Different Integers"?

For this hard-level Array problem, the key is to identify the core pattern. Analyze the input constraints to determine acceptable time complexity. Consider whether hash table techniques can optimize the solution. Always handle edge cases and validate your approach with examples before coding.

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