Given an input array of integers and a number k, pick exactly k numbers, some from the beginning of the array and some from the end of it, in such a way that their sum is minimized. Return this minimum sum.
Example
Input: nums = [-2, 5, 2, -1, 3, -10, 9, -2], k = 4
Output: -5
Explanation: Pick the first and the last three numbers
(-2) + (-10) + 9 + (-2) = -5
The core challenge of this problem is to find a way to select exactly k numbers from either the beginning or the end of the array such that their sum is minimized. This problem is significant in scenarios where we need to optimize resource allocation or minimize costs under certain constraints.
Potential pitfalls include misunderstanding the requirement to pick numbers from both ends of the array and not considering all possible combinations of selections from the start and end.
To solve this problem, we need to consider all possible ways to pick k numbers from the beginning and the end of the array. A naive solution would involve checking all combinations, but this is not efficient. Instead, we can use a sliding window approach to optimize the solution.
The naive solution involves generating all possible combinations of picking k numbers from the start and end of the array. This approach is not optimal due to its high time complexity.
We can optimize the solution by using a sliding window approach. The idea is to precompute the sums of the first i elements and the last k-i elements for all possible values of i from 0 to k. This allows us to efficiently find the minimum sum.
1. Compute the prefix sums for the first k elements.
2. Compute the suffix sums for the last k elements.
3. Iterate through all possible ways to split the selection between the start and end of the array, and find the minimum sum.
def min_sum(nums, k):
# Compute prefix sums for the first k elements
prefix_sums = [0] * (k + 1)
for i in range(1, k + 1):
prefix_sums[i] = prefix_sums[i - 1] + nums[i - 1]
# Compute suffix sums for the last k elements
suffix_sums = [0] * (k + 1)
for i in range(1, k + 1):
suffix_sums[i] = suffix_sums[i - 1] + nums[-i]
# Find the minimum sum by considering all possible splits
min_sum = float('inf')
for i in range(k + 1):
current_sum = prefix_sums[i] + suffix_sums[k - i]
min_sum = min(min_sum, current_sum)
return min_sum
# Example usage
nums = [-2, 5, 2, -1, 3, -10, 9, -2]
k = 4
print(min_sum(nums, k)) # Output: -5
The time complexity of this approach is O(k) because we compute the prefix and suffix sums in linear time with respect to k. The space complexity is also O(k) due to the storage of prefix and suffix sums.
Potential edge cases include:
To test the solution comprehensively, consider the following test cases:
When approaching such problems, it's essential to break down the problem into smaller parts and think about how to optimize each part. Practice solving similar problems and study different algorithms to improve problem-solving skills.
In this blog post, we discussed how to solve the problem of minimizing the sum of k numbers picked from the beginning and end of an array. We explored a naive solution and an optimized sliding window approach, provided a detailed algorithm, and analyzed the complexity. Understanding and solving such problems is crucial for developing strong problem-solving skills.