2588. Count the Number of Beautiful Subarrays
Description
You are given a 0-indexed integer array nums
. In one operation, you can:
- Choose two different indices
i
andj
such that0 <= i, j < nums.length
. - Choose a non-negative integer
k
such that thekth
bit (0-indexed) in the binary representation ofnums[i]
andnums[j]
is1
. - Subtract
2k
fromnums[i]
andnums[j]
.
A subarray is beautiful if it is possible to make all of its elements equal to 0
after applying the above operation any number of times.
Return the number of beautiful subarrays in the array nums
.
A subarray is a contiguous non-empty sequence of elements within an array.
Example 1:
Input: nums = [4,3,1,2,4] Output: 2 Explanation: There are 2 beautiful subarrays in nums: [4,3,1,2,4] and [4,3,1,2,4]. - We can make all elements in the subarray [3,1,2] equal to 0 in the following way: - Choose [3, 1, 2] and k = 1. Subtract 21 from both numbers. The subarray becomes [1, 1, 0]. - Choose [1, 1, 0] and k = 0. Subtract 20 from both numbers. The subarray becomes [0, 0, 0]. - We can make all elements in the subarray [4,3,1,2,4] equal to 0 in the following way: - Choose [4, 3, 1, 2, 4] and k = 2. Subtract 22 from both numbers. The subarray becomes [0, 3, 1, 2, 0]. - Choose [0, 3, 1, 2, 0] and k = 0. Subtract 20 from both numbers. The subarray becomes [0, 2, 0, 2, 0]. - Choose [0, 2, 0, 2, 0] and k = 1. Subtract 21 from both numbers. The subarray becomes [0, 0, 0, 0, 0].
Example 2:
Input: nums = [1,10,4] Output: 0 Explanation: There are no beautiful subarrays in nums.
Constraints:
1 <= nums.length <= 105
0 <= nums[i] <= 106
Solutions
Solution 1: Prefix XOR + Hash Table
We observe that a subarray can become an array of all $0$s if and only if the number of $1$s on each binary bit of all elements in the subarray is even.
If there exist indices $i$ and $j$ such that $i \lt j$ and the subarrays $nums[0,..,i]$ and $nums[0,..,j]$ have the same parity of the number of $1$s on each binary bit, then we can turn the subarray $nums[i + 1,..,j]$ into an array of all $0$s.
Therefore, we can use the prefix XOR method and a hash table $cnt$ to count the occurrences of each prefix XOR value. We traverse the array, for each element $x$, we calculate its prefix XOR value $mask$, then add the number of occurrences of $mask$ to the answer. Then, we increase the number of occurrences of $mask$ by $1$.
Finally, we return the answer.
The time complexity is $O(n)$, and the space complexity is $O(n)$. Here, $n$ is the length of the array $nums$.
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