Leetcode #2834: Find the Minimum Possible Sum of a Beautiful Array
In this guide, we solve Leetcode #2834 Find the Minimum Possible Sum of a Beautiful Array in Python and focus on the core idea that makes the solution efficient.
You will see the intuition, the step-by-step method, and a clean Python implementation you can use in interviews.
Problem Statement
You are given positive integers n and target. An array nums is beautiful if it meets the following conditions: nums.length == n.
Quick Facts
- Difficulty: Medium
- Premium: No
- Tags: Greedy, Math
Intuition
A locally optimal choice leads to a globally optimal result for this structure.
That means we can commit to decisions as we scan without backtracking.
Approach
Sort or preprocess if needed, then repeatedly take the best available local choice.
Maintain the minimal state necessary to validate the greedy decision.
Steps:
- Sort or preprocess as needed.
- Iterate and pick the best local option.
- Track the current solution.
Example
Input: n = 2, target = 3
Output: 4
Explanation: We can see that nums = [1,3] is beautiful.
- The array nums has length n = 2.
- The array nums consists of pairwise distinct positive integers.
- There doesn't exist two distinct indices, i and j, with nums[i] + nums[j] == 3.
It can be proven that 4 is the minimum possible sum that a beautiful array could have.
Python Solution
class Solution:
def minimumPossibleSum(self, n: int, target: int) -> int:
mod = 10**9 + 7
m = target // 2
if n <= m:
return ((1 + n) * n // 2) % mod
return ((1 + m) * m // 2 + (target + target + n - m - 1) * (n - m) // 2) % mod
Complexity
The time complexity is , and the space complexity is . The space complexity is .
Edge Cases and Pitfalls
Watch for boundary values, empty inputs, and duplicate values where applicable. If the problem involves ordering or constraints, confirm the invariant is preserved at every step.
Summary
This Python solution focuses on the essential structure of the problem and keeps the implementation interview-friendly while meeting the constraints.