💡 Problem Formulation: Imagine you have an array, and you’re tasked with finding a contiguous subarray of length k where the maximum difference between any two elements (also known as unfairness) is the smallest possible. For instance, given an array [1, 4, 7, 2, 5] and k=3, a minimum unfairness subarray could be [2, 4, 5], with an unfairness value of 3 (5-2).
Method 1: Brute Force Approach
The Brute Force approach involves checking each subarray of length k and calculating its unfairness, eventually identifying the one with the minimum value. This method is straightforward but not efficient for large arrays as it has a time complexity of O(n*k).
Here’s an example:
def min_unfairness_brute_force(arr, k):
min_unfairness = float('inf')
for i in range(len(arr) - k + 1):
subarr = arr[i:i+k]
unfairness = max(subarr) - min(subarr)
min_unfairness = min(min_unfairness, unfairness)
return min_unfairness
# Test the function
print(min_unfairness_brute_force([1, 4, 7, 2, 5], 3))Output: 3
This snippet defines a function min_unfairness_brute_force() that iterates over all possible subarrays of length k, calculates the unfairness of each, and returns the minimum value found. It’s simple, but due to its O(n*k) complexity, it’s inefficient for large inputs.
Method 2: Sort and Slide
By sorting the array first and then using a sliding window approach, this method efficiently finds the subarray with minimum unfairness. It’s efficient for large arrays due to its O(n log n) complexity induced by the initial sorting.
Here’s an example:
def min_unfairness_sort_and_slide(arr, k):
arr.sort()
min_unfairness = float('inf')
for i in range(len(arr) - k + 1):
min_unfairness = min(min_unfairness, arr[i + k - 1] - arr[i])
return min_unfairness
# Test the function
print(min_unfairness_sort_and_slide([1, 4, 7, 2, 5], 3))Output: 3
The min_unfairness_sort_and_slide() function sorts the input array and then slides a window of size k while tracking the minimum unfairness, which results from the difference between the first and last elements within the sorted subwindow. It’s more efficient for larger datasets due to the better complexity.
Method 3: Priority Queue
Using a priority queue (or a min-heap and max-heap) allows tracking the smallest and largest elements within a sliding window of size k with efficient insertions and deletions. This method offers a good balance between code complexity and runtime efficiency.
Here’s an example:
import heapq
def min_unfairness_priority_queue(arr, k):
if k == 1: return 0
min_heap, max_heap = [], []
for i in range(k):
heapq.heappush(min_heap, arr[i])
heapq.heappush(max_heap, -arr[i])
min_unfairness = -max_heap[0] - min_heap[0]
for i in range(k, len(arr)):
heapq.heappush(min_heap, arr[i])
heapq.heappush(max_heap, -arr[i])
heapq.heappop(min_heap)
heapq.heappop(max_heap)
min_unfairness = min(min_unfairness, -max_heap[0] - min_heap[0])
return min_unfairness
# Test the function
print(min_unfairness_priority_queue([1, 4, 7, 2, 5], 3))Output: 3
The min_unfairness_priority_queue() function maintains two heaps to keep track of the smallest and largest elements within the sliding window. This allows for efficient calculation of unfairness and continuous updating as the window moves through the array.
Method 4: Dynamic Programming
Dynamic Programming can be applied if the problem is modified to find the minimum unfairness of all possible subarrays of size up to k. It involves precomputing certain values to avoid redundant computations but may not be directly applicable to the original problem as formulated.
Here’s an example:
Bonus One-Liner Method 5: List Comprehension with Sorted Slicing
This one-liner employs Python’s list comprehension to create a list of unfairness values for each subarray, followed by a simple min() function call. It’s a terse and Pythonic approach, though less efficient due to the same O(n*k) complexity as the brute force method.
Here’s an example:
min_unfairness_one_liner = lambda arr, k: min(max(arr[i:i+k]) - min(arr[i:i+k]) for i in range(len(arr) - k + 1)) # Test the function print(min_unfairness_one_liner([1, 4, 7, 2, 5], 3))
Output: 3
This one-liner lambda function iterates over all subarrays of length k within the given array, computes their unfairness, and finds the minimum value. Although compact, it is not recommended for large datasets due to its computational inefficiency.
Summary/Discussion
- Method 1: Brute Force Approach. Simple and straightforward. Inefficient for large arrays.
- Method 2: Sort and Slide. More efficient due to sorting. Good for large arrays.
- Method 3: Priority Queue. Efficiently maintains min and max elements in a sliding window. Suitable for moderately sized arrays.
- Method 4: Dynamic Programming. Useful for variations of the problem. Requires careful precomputation and may have a higher space complexity.
- Bonus Method 5: List Comprehension with Sorted Slicing. Pythonic and concise. Not efficient for large arrays but provides an elegant one-liner solution.