How do you solve Beautiful Arrangement on LeetCode?

Beautiful Arrangement (LeetCode #526) can be solved using Brute Force or Optimal Solution. The optimal approach is Optimal Solution with O(n * 2^n) time complexity and O(n) space complexity. Key insight: Backtracking is a powerful technique for generating permutations and combinations efficiently.

What is the brute force solution for Beautiful Arrangement?

The brute force approach for Beautiful Arrangement is Brute Force, with O(n!) time complexity and O(n) space complexity. The brute force approach generates all possible permutations of the numbers from 1 to n and checks each permutation to see if it meets the criteria for being a beautiful arrangement. This is straightforward but inefficient for larger values of n. The optimal Optimal Solution approach improves this to O(n * 2^n).

What are the interview tips for Beautiful Arrangement?

Explain your thought process clearly while coding. Consider both brute force and optimal solutions to show your understanding. Practice backtracking problems to become comfortable with recursion and state management.

What are common mistakes when solving Beautiful Arrangement?

Not considering edge cases where n is small (like 1 or 2). Failing to properly manage the used numbers in backtracking.

#526

Beautiful Arrangement

Medium

ArrayDynamic ProgrammingBacktrackingBit ManipulationBitmaskBacktrackingBit Manipulation

LeetCode ↗

Approaches

Brute ForceOptimal

Complexity Comparison

	Brute Force	Optimal Solution★
Time	O(n!)	O(n * 2^n)
Space	O(n)	O(n)

💡

Intuition

Time O(n * 2^n)Space O(n)

The optimal solution uses backtracking with bit manipulation to efficiently track which numbers have been used in the arrangement. This reduces the overhead of generating all permutations explicitly.

⚙️

Algorithm

4 steps

1Step 1: Use a backtracking function that takes the current position and a bitmask to track used numbers.
2Step 2: For each position, iterate through numbers from 1 to n, checking if they can be placed based on the beautiful arrangement conditions.
3Step 3: If valid, mark the number as used (update the bitmask) and proceed to the next position.
4Step 4: Count valid arrangements when reaching the end of the array.

solution.py10 lines

1def countArrangement(n):
2    def backtrack(pos, used):
3        if pos > n:
4            return 1
5        count = 0
6        for i in range(1, n + 1):
7            if not (used & (1 << i)) and (i % pos == 0 or pos % i == 0):
8                count += backtrack(pos + 1, used | (1 << i))
9        return count
10    return backtrack(1, 0)

ℹ

Complexity note: The time complexity is O(n * 2^n) due to the backtracking with bitmasking, where we explore each number and the number of subsets of used numbers. The space complexity is O(n) for the recursion stack.

1Backtracking is a powerful technique for generating permutations and combinations efficiently.
2Using bit manipulation allows us to track used numbers without needing additional data structures.

Solutions and explanations are original Tejav content. Problem titles © LeetCode — use the LeetCode button above for the full problem statement.