#3133

Medium

auto_awesomeBit Manipulation-driven solution strategy

LeetCode Problem Workspace

Minimum Array End

Construct an array where elements are greater than the previous one, and the bitwise AND of all elements equals a given number.

bit manipulation

Problem Statement

You are given two integers n and x. Your task is to construct an array nums of size n such that each element nums[i] is greater than nums[i-1], for all 1 <= i < n. Additionally, the result of the bitwise AND operation on all elements of the array must equal x.

Return the minimum possible value of nums[n - 1].

Examples

Example 1

Input: n = 3, x = 4

Output: 6

nums can be [4,5,6] and its last element is 6.

Example 2

Input: n = 2, x = 7

Output: 15

nums can be [7,15] and its last element is 15.

Constraints

1 <= n, x <= 108

Solution Approach

Understanding the Bitwise AND Condition

The key observation is that the bitwise AND of all elements must equal x. This suggests that each element in the array must be carefully chosen such that its bits do not conflict with x when combined. Specifically, elements need to be constructed by merging x with a value that increases with each subsequent element.

Constructing the Array Elements

Start by setting the first element equal to x. For each subsequent element, merge x with increasing values (starting from 0 upwards) until the last element is found. This approach guarantees the array's elements are greater than the previous one and satisfy the bitwise AND condition.

Optimizing the Last Element

The goal is to minimize the value of the last element. By merging x with values incrementing by one, you ensure the smallest possible last element that still satisfies the conditions. The merge operation is done bitwise to preserve the required bit pattern while minimizing the final value.

Complexity Analysis

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

The time complexity is O(log n) due to the need to manipulate the bits of the integers. The space complexity is O(1) as the solution only requires a constant amount of extra space for the array and computation.

What Interviewers Usually Probe

Tests understanding of bitwise operations and their application in problem-solving.
Evaluates the candidate’s ability to construct an array with constraints using bit manipulation.
Assesses knowledge of optimization techniques when dealing with bitwise AND operations.

Common Pitfalls or Variants

Common pitfalls

Not properly understanding the bitwise AND operation and how it impacts the selection of elements.
Incorrectly merging values without ensuring that the array remains strictly increasing.
Failing to minimize the last element while maintaining the required bitwise AND condition.

Follow-up variants

Change the number of elements in the array and observe how the merging process scales.
Modify the bitwise AND condition to use a different value and test for edge cases.
Introduce additional constraints on the array elements, such as limits on individual values or their bitwise characteristics.

FAQ

What is the main strategy to solve the Minimum Array End problem?

The strategy involves using bitwise operations to construct an array where the AND of all elements equals a given value x, while ensuring the array is strictly increasing.

How does bit manipulation factor into the Minimum Array End problem?

Bit manipulation is essential to ensure the bitwise AND of the array elements equals the target value x. Elements are constructed by merging x with incrementing values.

What is the time complexity of the Minimum Array End problem?

The time complexity is O(log n), where n is the number of elements, due to the bitwise manipulation required to solve the problem.

What is the role of the last element in the Minimum Array End problem?

The last element of the array is the key to minimizing the output while still satisfying the bitwise AND condition across all elements.

Can the strategy used in the Minimum Array End problem apply to other bitwise problems?

Yes, the strategy of merging values using bitwise operations is a common technique used in other bitwise manipulation problems to minimize or meet specific constraints.

terminal

Solution

Solution 1: Greedy + Bit Manipulation

According to the problem description, to make the last element of the array as small as possible and the bitwise AND result of the elements in the array is $x$, the first element of the array must be $x$.

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class Solution:
    def minEnd(self, n: int, x: int) -> int:
        n -= 1
        ans = x
        for i in range(31):
            if x >> i & 1 ^ 1:
                ans |= (n & 1) << i
                n >>= 1
        ans |= n << 31
        return ans

Continue Practicing

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