#2834

Medium

auto_awesomeGreedy choice plus invariant validation

LeetCode Problem Workspace

Find the Minimum Possible Sum of a Beautiful Array

Find the minimum possible sum of a beautiful array that satisfies the given conditions with the greedy approach.

Problem Statement

You are given two integers, n and target. You need to create an array nums with exactly n distinct positive integers such that the sum of any two distinct integers in the array does not equal the target. The goal is to return the minimum possible sum of nums modulo 10^9 + 7.

A greedy approach should be used to iteratively select the smallest possible number that satisfies the constraints. The solution needs to respect the bounds of 1 <= n <= 10^9 and 1 <= target <= 10^9. The result should be modulo 10^9 + 7.

Examples

Example 1

Input: n = 2, target = 3

Output: 4

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.

Example 2

Input: n = 3, target = 3

Output: 8

We can see that nums = [1,3,4] is beautiful.

The array nums has length n = 3.
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 8 is the minimum possible sum that a beautiful array could have.

Example 3

Input: n = 1, target = 1

Output: 1

We can see, that nums = [1] is beautiful.

Constraints

1 <= n <= 109
1 <= target <= 109

Solution Approach

Greedy Choice

The strategy begins by choosing the smallest number for the array, which is 1, and then repeatedly adding the next smallest valid number that doesn't violate the sum condition with any previously chosen numbers. By doing this, we minimize the sum while ensuring distinct values.

Invariant Validation

The main invariant in this problem is ensuring that no two distinct integers in the array sum up to the target. Every number added to the array must maintain this invariant, and we must validate it at each step to avoid any violation.

Modulo Operation

Once the array is constructed, the final sum must be computed modulo 10^9 + 7, which ensures that large numbers don't cause overflow and keeps the answer within the problem's constraints.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time and space complexity depends on the final approach, specifically how efficiently we can generate the array of distinct integers while checking the sum condition and performing modulo operations. The greedy approach ensures minimal computations, but optimizations may be needed for large values of n.

What Interviewers Usually Probe

The candidate demonstrates understanding of greedy algorithms.
The candidate applies the concept of invariant validation correctly.
The candidate is mindful of the problem's constraints and uses the modulo operation effectively.

Common Pitfalls or Variants

Common pitfalls

Selecting numbers that violate the sum condition.
Failing to ensure all integers are distinct.
Incorrectly handling large values of n and target within the constraints.

Follow-up variants

What if we allow non-distinct integers in the array?
How would the approach change if n was significantly larger?
What if we needed to minimize the product instead of the sum?

FAQ

How do I handle the large values of n and target in this problem?

Use a greedy approach to iteratively select valid numbers while ensuring distinctness and avoiding sums that match the target. The sum should be calculated modulo 10^9 + 7.

What is the greedy approach in this problem?

The greedy approach selects the smallest number that satisfies the problem's constraints and avoids sums matching the target, progressively building the array.

What does it mean to validate the invariant in this problem?

Validating the invariant means ensuring no two distinct numbers in the array sum up to the target at every step of the greedy selection.

What is the final step after constructing the array?

The final step is to calculate the sum of the array modulo 10^9 + 7 to ensure the result fits within the specified range.

How can I optimize my solution for larger values of n?

You can optimize by carefully selecting numbers and ensuring the greedy approach minimizes computational complexity while maintaining the invariant.

terminal

Solution

Solution 1: Greedy + Mathematics

We can greedily construct the array `nums` starting from $x = 1$, choosing $x$ each time and excluding $target - x$.

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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

Continue Practicing

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