#667

Medium

LeetCode Problem Workspace

Beautiful Arrangement II

Construct a beautiful arrangement of integers from 1 to n with k distinct integers, optimizing for time and space efficiency.

array math

Problem Statement

You are given two integers, n and k, and need to construct a list of n distinct integers ranging from 1 to n. The goal is to generate the arrangement such that there are exactly k distinct integers in the list, with no repetition. Return any valid arrangement that satisfies the condition.

You can approach this problem with a mathematical strategy. Depending on the value of k, the list should be formed in a way that the arrangement contains exactly k distinct integers. For each valid configuration, multiple answers may exist.

Examples

Example 1

Input: n = 3, k = 1

Output: [1,2,3]

The [1,2,3] has three different positive integers ranging from 1 to 3, and the [1,1] has exactly 1 distinct integer: 1

Example 2

Input: n = 3, k = 2

Output: [1,3,2]

The [1,3,2] has three different positive integers ranging from 1 to 3, and the [2,1] has exactly 2 distinct integers: 1 and 2.

Constraints

1 <= k < n <= 104

Solution Approach

Mathematical Construction

Start by selecting numbers from both the smallest and largest parts of the range [1, n] to form the array. Then, place the integers in a manner that maintains the k distinct values while ensuring no repetition. This leads to an O(n) solution in terms of time complexity.

Two Pointer Technique

Use two pointers, one starting from the lowest number and the other from the highest, alternating between them to form the desired arrangement. This approach ensures that you construct the array in an optimal way, ensuring k distinct values while reducing unnecessary computations.

Greedy Approach

A greedy approach can be used to fill the array by selecting numbers in a manner that quickly satisfies the k distinct condition. This method focuses on choosing values that maximize the range of distinct integers, ensuring the solution is obtained in linear time.

Complexity Analysis

Metric	Value
Time	O(n)
Space	O(1)

The time complexity is O(n) because the list is constructed in a single pass, while the space complexity is O(1) as the solution does not require additional space besides the array being returned.

What Interviewers Usually Probe

The candidate should demonstrate a solid understanding of how to manipulate arrays using mathematical methods.
Look for the candidate's ability to optimize both time and space complexity, focusing on an efficient O(n) solution.
Check if the candidate can identify and avoid common pitfalls in generating the array, such as unnecessary calculations.

Common Pitfalls or Variants

Common pitfalls

Failing to account for distinct values: Ensure the arrangement does not have repeated numbers.
Overcomplicating the solution: Aim for a simple yet efficient construction of the array without unnecessary operations.
Incorrect handling of edge cases: Pay attention to how different values of k impact the arrangement and its validity.

Follow-up variants

A variation of this problem could involve ensuring the arrangement meets specific conditions related to the positions of the integers.
A challenge could involve generating multiple valid arrangements that satisfy different values of k.
You might be tasked with modifying this problem to handle large values of n and k, pushing for further optimization.

FAQ

How do I approach solving the Beautiful Arrangement II problem?

Start by using a mathematical approach that alternates between the smallest and largest values to construct the array. This ensures distinct values while maintaining the required length.

What is the expected time complexity for this problem?

The time complexity is O(n) since the array is constructed in a linear pass, with no extra loops or nested iterations.

Can the Beautiful Arrangement II problem have multiple valid solutions?

Yes, multiple valid solutions are possible as long as the arrangement satisfies the condition of having exactly k distinct integers.

How can I optimize my solution for space complexity?

Optimize space by using in-place array construction with two pointers or a greedy approach, avoiding extra data structures.

What are common pitfalls to avoid when solving this problem?

Avoid overcomplicating the solution with unnecessary steps or failing to account for edge cases like incorrect repetition of integers.

terminal

Solution

Solution 1

#### Python3

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class Solution:
    def constructArray(self, n: int, k: int) -> List[int]:
        l, r = 1, n
        ans = []
        for i in range(k):
            if i % 2 == 0:
                ans.append(l)
                l += 1
            else:
                ans.append(r)
                r -= 1
        for i in range(k, n):
            if k % 2 == 0:
                ans.append(r)
                r -= 1
            else:
                ans.append(l)
                l += 1
        return ans

Continue Practicing

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