2568. Minimum Impossible OR
Description
You are given a 0-indexed integer array nums
.
We say that an integer x is expressible from nums
if there exist some integers 0 <= index1 < index2 < ... < indexk < nums.length
for which nums[index1] | nums[index2] | ... | nums[indexk] = x
. In other words, an integer is expressible if it can be written as the bitwise OR of some subsequence of nums
.
Return the minimum positive non-zero integer that is not expressible from nums
.
Example 1:
Input: nums = [2,1] Output: 4 Explanation: 1 and 2 are already present in the array. We know that 3 is expressible, since nums[0] | nums[1] = 2 | 1 = 3. Since 4 is not expressible, we return 4.
Example 2:
Input: nums = [5,3,2] Output: 1 Explanation: We can show that 1 is the smallest number that is not expressible.
Constraints:
1 <= nums.length <= 105
1 <= nums[i] <= 109
Solutions
Solution 1: Enumerate Powers of 2
We start from the integer $1$. If $1$ is expressible, it must appear in the array nums
. If $2$ is expressible, it must also appear in the array nums
. If both $1$ and $2$ are expressible, then their bitwise OR operation $3$ is also expressible, and so on.
Therefore, we can enumerate the powers of $2$. If the currently enumerated $2^i$ is not in the array nums
, then $2^i$ is the smallest unexpressible integer.
The time complexity is $O(n + \log M)$, and the space complexity is $O(n)$. Here, $n$ and $M$ are the length of the array nums
and the maximum value in the array nums
, respectively.
1 2 3 4 |
|
1 2 3 4 5 6 7 8 9 10 11 12 13 |
|
1 2 3 4 5 6 7 8 9 10 11 |
|
1 2 3 4 5 6 7 8 9 10 11 |
|
1 2 3 4 5 6 7 8 9 10 11 |
|