Beautiful Arrangement II

Beautiful Arrangement II

What is this problem about?

The "Beautiful Arrangement II interview question" is a construction problem based on absolute differences. Given two integers $n$ and $k$, you need to create a list of $n$ numbers from $1$ to $n$ such that there are exactly $k$ unique absolute differences between adjacent elements. For example, if $n=3$ and $k=1$, the arrangement [1, 2, 3] has differences |1-2|=1 and |2-3|=1, so only 1 unique difference.

Why is this asked in interviews?

Google and Bloomberg ask the "Beautiful Arrangement II coding problem" to test a candidate's ability to find a mathematical pattern and implement a constructive algorithm. Unlike the first version, this doesn't require backtracking; it requires a clever way to "spend" your $k$ differences greedily. It evaluates "Math interview pattern" and "Array interview pattern" skills.

Algorithmic pattern used

This problem is solved using a Greedy Constructive Pattern.

1. Initial Part: To get $k$ unique differences, we can alternate between the largest and smallest available numbers. For example, if $k=3$, we might do 1, n, 2, n-1.... Each jump creates a new, unique, large difference.
2. Remaining Part: Once we have created $k-1$ unique differences, the remaining numbers should be placed in increasing or decreasing order. Since they are consecutive, they all contribute a difference of 1 (which we either already have or is our $k^{th}$ difference).
3. Logic: Use two pointers (low = 1, high = n) and alternate picking from them $k$ times. Then fill the rest linearly.

Example explanation

$n = 5, k = 2$

1. We need 2 unique differences.
2. Start alternating:
   • Pick low (1). low = 2.
   • Pick high (5). high = 4. Difference is 4.
3. We have used 2 elements and created 1 unique difference. We need 1 more.
4. Fill the rest in descending order (since we last picked from high): 4, 3, 2. Arrangement: [1, 5, 4, 3, 2]. Differences: |1-5|=4, |5-4|=1, |4-3|=1, |3-2|=1. Unique differences: {4, 1}. Total 2. Correct!

Common mistakes candidates make

• Trying Backtracking: Attempting to use a search algorithm for a problem with a $10,000$ limit. This is a construction problem, not a search problem.
• Incorrect Alternation: Getting confused about when to stop alternating and which direction to fill the remaining numbers.
• Off-by-one: Miscounting the number of differences created vs the number of elements placed.

Interview preparation tip

When a problem asks you to "construct" something with a specific property, look for a pattern. Often, placing numbers at the extremes (smallest and largest) is the key to creating distinct properties. Practice identifying when a problem is a "Search" problem ($O(2^N)$) versus a "Construction" problem ($O(N)$).