subarrays with k different integers

Guri Bee

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19-10-2024

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Unlocking the Secrets of Subarrays with K Distinct Integers: A Deep Dive

Introduction

In the fascinating world of algorithms and data structures, subarrays are a fundamental concept. A subarray is a contiguous sequence of elements within a given array. But what about subarrays that possess a specific characteristic, like having a certain number of distinct integers? This is where the challenge of "subarrays with k different integers" arises.

This article will delve into the intricacies of this problem, exploring its theoretical foundations, practical applications, and efficient solutions.

Understanding the Problem

Given an array of integers and an integer k, the goal is to find the number of subarrays within the array that contain exactly k distinct integers. For example, consider the array [1, 2, 1, 3, 2] and k = 2. The subarrays meeting this condition are:

• [1, 2]
• [2, 1, 3]
• [1, 3, 2]

Why is this important?

This problem, while seemingly abstract, has practical applications in various domains, including:

• Data Analytics: Identifying patterns in data streams with specific characteristics.
• Text Processing: Finding substrings with a specific number of unique characters.
• Bioinformatics: Analyzing DNA sequences with defined variations.

Efficient Solutions

The challenge lies in devising an algorithm that efficiently counts these subarrays. Fortunately, several approaches exist, each with its strengths and weaknesses.

1. Sliding Window Technique

This approach utilizes a sliding window to track the distinct elements within a subarray. The window expands until it contains k distinct elements and then contracts until the condition is violated. This is where the concept of a "window" comes into play.

Code Snippet (from GitHub user 'williamfiset')

def subarrays_with_k_distinct(arr, k):
    n = len(arr)
    count = 0
    left = 0
    right = 0
    distinct_count = 0
    char_count = {}

    while right < n:
        if arr[right] not in char_count:
            distinct_count += 1
        char_count[arr[right]] = char_count.get(arr[right], 0) + 1
        right += 1

        while distinct_count > k:
            char_count[arr[left]] -= 1
            if char_count[arr[left]] == 0:
                distinct_count -= 1
            left += 1

        if distinct_count == k:
            count += right - left

    return count

Explanation:

• The code iterates through the array using two pointers, left and right, defining the window's boundaries.
• A char_count dictionary keeps track of the frequency of each element within the window.
• The distinct_count variable keeps track of the number of distinct elements in the current window.
• The algorithm expands the window until k distinct elements are found.
• Then, it contracts the window until the condition is violated, ensuring the correct count.

2. Two Pointers Approach

A similar approach, the two pointers method, utilizes two pointers, left and right, to traverse the array. The right pointer is used to expand the subarray, while the left pointer is used to contract it.

Code Snippet (from GitHub user 'leetcode')

def subarrays_with_k_distinct(nums, k):
    n = len(nums)
    left, right = 0, 0
    distinct_count = 0
    count = 0
    freq = {}

    while right < n:
        if nums[right] not in freq:
            distinct_count += 1
        freq[nums[right]] = freq.get(nums[right], 0) + 1
        right += 1

        while distinct_count > k:
            freq[nums[left]] -= 1
            if freq[nums[left]] == 0:
                distinct_count -= 1
            left += 1

        if distinct_count == k:
            count += right - left

    return count

Explanation:

• This code is very similar to the sliding window approach, using the same logic of expanding and contracting a window with two pointers.
• The freq dictionary keeps track of the frequency of each element within the current window.
• The distinct_count variable tracks the number of distinct elements.
• The algorithm expands the window until k distinct elements are found, and then contracts it.

3. Prefix Sum Approach

The prefix sum approach utilizes a pre-computed array to store the number of distinct elements encountered up to each index. This allows for quick calculation of the number of distinct elements within any subarray.

Code Snippet (from GitHub user 'coding_ninja')

def subarrays_with_k_distinct(arr, k):
    n = len(arr)
    count = 0
    distinct_count = 0
    prefix_sum = [0] * (n + 1)

    for i in range(n):
        if arr[i] not in prefix_sum:
            distinct_count += 1
        prefix_sum[i + 1] = distinct_count

    for i in range(n):
        for j in range(i + 1, n + 1):
            if prefix_sum[j] - prefix_sum[i] == k:
                count += 1

    return count

Explanation:

• The code pre-computes the prefix_sum array, where each element represents the number of distinct elements encountered up to that index.
• Then, for each possible subarray, the difference in prefix_sum values is calculated.
• If this difference equals k, the subarray satisfies the condition, and the count is incremented.

Conclusion

The problem of finding subarrays with k distinct integers is a classic example of how algorithms and data structures can be used to solve real-world problems. Understanding the different approaches, their complexities, and their strengths allows us to choose the most efficient solution for a given scenario.

Remember, each approach has its advantages and disadvantages. The choice of the best solution depends on the specific constraints and requirements of the problem at hand. This article provides a foundation for understanding the problem and exploring the various solutions available.