Beautiful Arrangement II - Practice Coding Problems

Beautiful Arrangement II - Problem

Array Math Medium

Given two integers n and k, construct a list answer that contains n different positive integers ranging from 1 to n and obeys the following requirement:

Suppose this list is answer = [a₁, a₂, a₃, ..., aₙ], then the list [|a₁ - a₂|, |a₂ - a₃|, |a₃ - a₄|, ..., |aₙ₋₁ - aₙ|] has exactly k distinct integers.

Return the list answer. If there are multiple valid answers, return any of them.

Input & Output

Example 1 — Basic Case

$ Input: n = 4, k = 3

› Output: [1,4,2,3]

💡 Note: Differences are |1-4|=3, |4-2|=2, |2-3|=1, giving exactly 3 distinct values {1,2,3}

Example 2 — Small k

$ Input: n = 5, k = 2

› Output: [1,5,2,3,4]

💡 Note: Differences are |1-5|=4, |5-2|=3, |2-3|=1, |3-4|=1, giving exactly 2 distinct values {1,3}

Example 3 — Minimum Case

$ Input: n = 2, k = 1

› Output: [1,2]

💡 Note: Only one difference |1-2|=1, exactly k=1 distinct value

Constraints

1 ≤ k < n ≤ 10⁴

Visualization

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Understanding the Visualization

Input

Given n=4, k=3, need array of [1,2,3,4] with 3 distinct differences

Construction

Alternate between min/max: [1,4,2,3]

Verification

Differences: 3,2,1 - exactly k=3 distinct values

Key Takeaway

🎯 Key Insight: Alternating between minimum and maximum remaining values creates all differences 1,2,3...k efficiently

Asked in

G Google 35 f Facebook 28 a Amazon 22

The key insight is to use an alternating pattern between minimum and maximum values to create exactly k distinct differences efficiently. The mathematical construction approach alternates between low and high values for k+1 positions, then fills remaining positions sequentially. Time: O(n), Space: O(n).

Common Approaches

Approach	Time	Space	Notes
✓ Mathematical Construction	O(n)	O(n)	Build arrangement using alternating pattern to create exactly k distinct differences
Brute Force Permutations	O(n! × n)	O(n)	Generate all permutations and check which ones have exactly k distinct differences

Mathematical Construction — Algorithm Steps

Start with low=1, high=n
Alternate placing low and high to create differences k, k-1, ..., 1
Fill remaining positions sequentially to maintain only those k differences

Visualization

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Step-by-Step Walkthrough

Setup

Initialize low=1, high=4 for n=4, k=3

Alternate

Place 1,4,2,3 creating differences 3,2,1

Result

Achieved exactly k=3 distinct differences

Code -

solution.c — C

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

int main() {
    int n, k;
    scanf("%d", &n);
    scanf("%d", &k);
    
    int* result = (int*)malloc(n * sizeof(int));
    int low = 1, high = n;
    
    // Create k distinct differences using alternating pattern
    for (int i = 0; i <= k; i++) {
        if (i % 2 == 0) {
            result[i] = low++;
        } else {
            result[i] = high--;
        }
    }
    
    // Fill remaining positions sequentially
    for (int i = k + 1; i < n; i++) {
        result[i] = low++;
    }
    
    printf("[");
    for (int i = 0; i < n; i++) {
        if (i > 0) printf(",");
        printf("%d", result[i]);
    }
    printf("]\n");
    
    free(result);
    return 0;
}

Time & Space Complexity

Time Complexity

⏱️

O(n)

Single pass to construct the arrangement

✓ Linear Growth

Space Complexity

O(n)

Result array to store the arrangement

⚡ Linearithmic Space

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Medium Frequency

~15 min Avg. Time

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Ln 1, Col 1

Smart Actions

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Input & Output

Constraints

Visualization

Related Problems

Common Approaches

Mathematical Construction — Algorithm Steps

Visualization

Code -

Time & Space Complexity

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