Find the Minimum Possible Sum of a Beautiful Array

Find the Minimum Possible Sum of a Beautiful Array - Problem

Math Greedy Medium

You are given positive integers n and target.

An array nums is beautiful if it meets the following conditions:

nums.length == n.
nums consists of pairwise distinct positive integers.
There doesn't exist two distinct indices, i and j, in the range [0, n - 1], such that nums[i] + nums[j] == target.

Return the minimum possible sum that a beautiful array could have modulo 10⁹ + 7.

Input & Output

Example 1 — Basic Case

$ Input: n = 4, target = 6

› Output: 12

💡 Note: One beautiful array is [1,3,4,6]. The pairs are (1,3), (1,4), (1,6), (3,4), (3,6) and (4,6). None sum to 6, so minimum sum is 1+3+4+6 = 14. But [1,2,4,7] gives 1+2+4+7 = 14, and [1,2,3,6] gives 1+2+3+6 = 12 which is smaller.

Example 2 — Small Target

$ Input: n = 2, target = 3

› Output: 4

💡 Note: We need 2 distinct numbers where no pair sums to 3. We can take [1,3] since 1+3=4≠3, giving sum 4. Or [2,3] since 2+3=5≠3, but [1,3] has smaller sum.

Example 3 — Large Array

$ Input: n = 3, target = 3

› Output: 8

💡 Note: Target is 3, so 1+2=3 is forbidden. We can take [1,3,4] but 1+2=3 so we can't take 2. Best is [1,3,4] giving sum 8.

Constraints

1 ≤ n ≤ 10⁵
1 ≤ target ≤ 10⁹

Visualization

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Asked in

G Google 15 f Meta 12 a Amazon 8

The key insight is to greedily select the smallest available numbers while avoiding pairs that sum to target. The optimal approach uses mathematical analysis: take numbers 1 to target/2, then skip the forbidden range and continue from target. Best approach is mathematical formula. Time: O(1), Space: O(1)

Common Approaches

✓ Brute Force with Set Generation

⏱️ Time: O(∞) Space: O(n)

Try all possible combinations of n distinct positive integers, check if each forms a beautiful array, and track the minimum sum. This explores every possibility but is extremely inefficient.

Greedy Selection

⏱️ Time: O(n) Space: O(n)

Start from 1 and greedily select the smallest available numbers. If adding a number would create a forbidden pair (sum equals target), skip it and continue to the next number.

Mathematical Formula

⏱️ Time: O(1) Space: O(1)

Analyze the pattern mathematically. For target t, we can take all numbers from 1 to t/2, then skip numbers from t/2+1 to t-1 (as they would form pairs), then continue from t onwards.

Brute Force with Set Generation — Algorithm Steps

Generate all combinations of n distinct positive integers
For each combination, check if any pair sums to target
Keep track of minimum sum among valid combinations
Return the minimum sum found

Visualization

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Step-by-Step Walkthrough

Generate

Create all possible arrays of n distinct numbers

Validate

Check if no two elements sum to target

Track Minimum

Keep the array with smallest sum

Code -

solution.c — C

#include <stdio.h>
#include <stdlib.h>
#include <stdbool.h>

int solution(int n, int target) {
    const int MOD = 1000000007;
    
    int result[100000];
    bool used[200000] = {false}; // Assume reasonable upper bound
    int count = 0;
    int current = 1;
    
    while (count < n) {
        int complement = target - current;
        if (complement > 0 && complement < 200000 && !used[complement]) {
            result[count++] = current;
            if (current < 200000) used[current] = true;
        } else if (complement <= 0 || complement >= 200000) {
            result[count++] = current;
            if (current < 200000) used[current] = true;
        }
        current++;
    }
    
    long long sum = 0;
    for (int i = 0; i < n; i++) {
        sum = (sum + result[i]) % MOD;
    }
    return (int) sum;
}

int main() {
    int n, target;
    scanf("%d", &n);
    scanf("%d", &target);
    printf("%d\n", solution(n, target));
    return 0;
}

Time & Space Complexity

Time Complexity

⏱️

O(∞)

Infinite combinations to check, not practically feasible

✓ Linear Growth

Space Complexity

O(n)

Store current combination of n numbers

⚡ Linearithmic Space

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Medium Frequency

~15 min Avg. Time

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Smart Actions

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Input & Output

Constraints

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Related Problems

Common Approaches

Brute Force with Set Generation — Algorithm Steps

Visualization

Code -

Time & Space Complexity

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