898. Bitwise ORs of Subarrays
Description
Given an integer array arr
, return the number of distinct bitwise ORs of all the non-empty subarrays of arr
.
The bitwise OR of a subarray is the bitwise OR of each integer in the subarray. The bitwise OR of a subarray of one integer is that integer.
A subarray is a contiguous non-empty sequence of elements within an array.
Example 1:
Input: arr = [0] Output: 1 Explanation: There is only one possible result: 0.
Example 2:
Input: arr = [1,1,2] Output: 3 Explanation: The possible subarrays are [1], [1], [2], [1, 1], [1, 2], [1, 1, 2]. These yield the results 1, 1, 2, 1, 3, 3. There are 3 unique values, so the answer is 3.
Example 3:
Input: arr = [1,2,4] Output: 6 Explanation: The possible results are 1, 2, 3, 4, 6, and 7.
Constraints:
1 <= arr.length <= 5 * 104
0 <= arr[i] <= 109
Solutions
Solution 1: Hash Table
The problem asks for the number of unique bitwise OR operations results of subarrays. If we enumerate the end position $i$ of the subarray, the number of bitwise OR operations results of the subarray ending at $i-1$ does not exceed $32$. This is because the bitwise OR operation is a monotonically increasing operation.
Therefore, we use a hash table $ans$ to record all the results of the bitwise OR operations of subarrays, and a hash table $s$ to record the results of the bitwise OR operations of subarrays ending with the current element. Initially, $s$ only contains one element $0$.
Next, we enumerate the end position $i$ of the subarray. The result of the bitwise OR operation of the subarray ending at $i$ is the set of results of the bitwise OR operation of the subarray ending at $i-1$ and $a[i]$, plus $a[i]$ itself. We use a hash table $t$ to record the results of the bitwise OR operation of the subarray ending at $i$, then we update $s = t$, and add all elements in $t$ to $ans$.
Finally, we return the number of elements in the hash table $ans$.
The time complexity is $O(n \times \log M)$, and the space complexity is $O(n \times \log M)$. Here, $n$ and $M$ are the length of the array and the maximum value in the array, respectively.
1 2 3 4 5 6 7 8 |
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 |
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 |
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 |
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 |
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