Leetcode #526: Beautiful Arrangement
In this guide, we solve Leetcode #526 Beautiful Arrangement in Python and focus on the core idea that makes the solution efficient.
You will see the intuition, the step-by-step method, and a clean Python implementation you can use in interviews.
Problem Statement
Suppose you have n integers labeled 1 through n. A permutation of those n integers perm (1-indexed) is considered a beautiful arrangement if for every i (1 <= i <= n), either of the following is true: perm[i] is divisible by i.
Quick Facts
- Difficulty: Medium
- Premium: No
- Tags: Bit Manipulation, Array, Dynamic Programming, Backtracking, Bitmask
Intuition
The problem breaks into overlapping subproblems, so caching results prevents exponential repetition.
A carefully chosen DP state captures exactly what we need to build the final answer.
Approach
Define the DP state and recurrence, then compute states in the correct order.
Optionally compress space once the recurrence is clear.
Steps:
- Choose a DP state definition.
- Write the recurrence and base cases.
- Compute states in the correct order.
Example
Input: n = 2
Output: 2
Explanation:
The first beautiful arrangement is [1,2]:
- perm[1] = 1 is divisible by i = 1
- perm[2] = 2 is divisible by i = 2
The second beautiful arrangement is [2,1]:
- perm[1] = 2 is divisible by i = 1
- i = 2 is divisible by perm[2] = 1
Python Solution
class Solution:
def countArrangement(self, n: int) -> int:
def dfs(i):
nonlocal ans, n
if i == n + 1:
ans += 1
return
for j in match[i]:
if not vis[j]:
vis[j] = True
dfs(i + 1)
vis[j] = False
ans = 0
vis = [False] * (n + 1)
match = defaultdict(list)
for i in range(1, n + 1):
for j in range(1, n + 1):
if j % i == 0 or i % j == 0:
match[i].append(j)
dfs(1)
return ans
Complexity
The time complexity is O(n·m) (typical). The space complexity is O(n·m) or optimized.
Edge Cases and Pitfalls
Watch for boundary values, empty inputs, and duplicate values where applicable. If the problem involves ordering or constraints, confirm the invariant is preserved at every step.
Summary
This Python solution focuses on the essential structure of the problem and keeps the implementation interview-friendly while meeting the constraints.