Smallest Range Covering K Lists 🎯

Welcome to a fascinating journey through the world of Data Structures and Algorithms! Today, we'll dive deep into a problem called the "Smallest Range Covering K Lists." This problem is a great way to understand and practice some essential data structure concepts. Let's get started!

What is the Smallest Range Covering K Lists Problem? 📝

In this problem, we are given a set of N lists, each containing M unique integers, where N > M. The goal is to find the smallest range that covers all the integers in at least K lists.

Here's a simple example to illustrate the problem:

Lists:
[1, 3, 5], [2, 4, 6], [5, 7, 8], [8, 9]

K = 3

Smallest range covering 3 lists:
(3, 8)

In this example, the smallest range covering at least 3 lists is (3, 8).

Algorithm Approach 💡

The problem can be approached by sorting all the lists and iterating through them. At each step, we keep track of the minimum and maximum values for potential ranges.

Here's a step-by-step breakdown:

Sort all the lists in non-decreasing order.
Initialize min_val and max_val as the smallest and largest values among all the lists respectively.
Start iterating through the lists from the first one.
Maintain the current_min and current_max as the smallest and largest values within the current list.
If current_min is greater than min_val, update min_val to current_min.
If current_max is less than max_val, update max_val to current_max.
If max_val - min_val is greater than or equal to K, return the smallest range (min_val, max_val).
Repeat the process for all the lists.

Code Implementation 💻

Here's a Python implementation of the algorithm:

python

def smallest_range(lists, k):
    if not lists:
        return None

    # Sort the lists
    lists.sort()

    min_val = float('inf')
    max_val = float('-inf')

    for list in lists:
        min_val = min(min_val, min(list))
        max_val = max(max_val, max(list))

        if max_val - min_val >= k:
            return (min_val, max_val)

    return None

# Example usage
lists = [[1, 3, 5], [2, 4, 6], [5, 7, 8], [8, 9]]
k = 3
result = smallest_range(lists, k)
print(result)  # Output: (3, 8)

Quiz Time 📝

That's it for today! We've covered the Smallest Range Covering K Lists problem and implemented a solution using Python. Practice the problem with different sets of lists and various values of K to solidify your understanding.

Happy coding, and see you in the next lesson! 👋🏼