#3209

Hard

auto_awesomeBinary search over the valid answer space

LeetCode Problem Workspace

Number of Subarrays With AND Value of K

The problem asks to find the number of subarrays with a given AND value in an array, utilizing binary search for optimization.

array binary search bit manipulation segment tree

Problem Statement

You are given an array of integers nums and an integer k. Your task is to return the number of subarrays where the bitwise AND of all the elements in the subarray equals k.

For example, if nums = [1,1,1] and k = 1, the output should be 6 because all subarrays of nums have a bitwise AND of 1. A subarray's AND value is the result of ANDing all its elements together.

Examples

Example 1

Input: nums = [1,1,1], k = 1

Output: 6

All subarrays contain only 1's.

Example 2

Input: nums = [1,1,2], k = 1

Output: 3

Subarrays having an AND value of 1 are: [ 1 ,1,2] , [1, 1 ,2] , [ 1,1 ,2] .

Example 3

Input: nums = [1,2,3], k = 2

Output: 2

Subarrays having an AND value of 2 are: [1, 2 ,3] , [1, 2,3 ] .

Constraints

1 <= nums.length <= 105
0 <= nums[i], k <= 109

Solution Approach

Binary Search Over the Valid Answer Space

The problem suggests utilizing binary search to explore the valid subarrays efficiently. By searching over the answer space, we can reduce the time complexity significantly. The goal is to count pairs of indices (l, r) such that the AND operation on the elements in the range from l to r gives the desired result k.

Bitwise AND Manipulation

To solve this problem, understanding bitwise AND operations is crucial. The AND operation filters out elements that would prevent the subarray's AND from matching k. Efficient tracking of the AND of subarrays, while iterating over the elements, enables an optimal solution.

Sliding Window or Prefix Approach

Using a sliding window or prefix sums in combination with the bitwise AND operation can help narrow down the valid subarrays. This approach allows the AND operation to be computed incrementally as the window expands, rather than recalculating the AND for every possible subarray.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity depends on the exact approach. The binary search helps reduce the complexity, but the sliding window or prefix sum strategy also plays a key role in ensuring the solution remains efficient. Depending on the approach, the space complexity may vary based on the data structures used for tracking the AND values.

What Interviewers Usually Probe

Does the candidate identify binary search as a suitable strategy for optimizing the solution?
Can the candidate explain the handling of bitwise AND in the context of subarrays?
Is the candidate able to suggest multiple approaches, such as sliding window or prefix sums?

Common Pitfalls or Variants

Common pitfalls

Not considering the bitwise operation as a way to efficiently handle large numbers.
Failing to optimize the solution by applying binary search over the answer space.
Overlooking the need for efficient data structures to track subarray AND values.

Follow-up variants

Alter the problem by changing the target AND value, k, for different subarray configurations.
Introduce additional constraints, such as limiting the size of the subarrays.
Require handling multiple arrays at once, each with a different k value.

FAQ

What is the optimal approach for solving 'Number of Subarrays With AND Value of K'?

The optimal approach uses binary search over the valid answer space in combination with bitwise AND manipulation for efficient subarray counting.

How does binary search help in 'Number of Subarrays With AND Value of K'?

Binary search helps reduce the complexity by narrowing down the valid subarray ranges, enabling faster identification of subarrays with the desired AND value.

What are the key patterns in 'Number of Subarrays With AND Value of K'?

Key patterns include binary search optimization, bitwise manipulation, and efficient tracking of subarrays via sliding windows or prefix sums.

Can the problem be solved with a brute-force approach?

While a brute-force solution is possible, it is inefficient for large arrays, leading to higher time complexity. Optimizing with binary search is recommended.

What data structures are most useful in solving 'Number of Subarrays With AND Value of K'?

Efficient data structures for tracking subarray AND values include hashmaps and sliding window techniques to avoid redundant calculations.

terminal

Solution

Solution 1: Hash Table + Enumeration

According to the problem description, we need to find the result of the bitwise AND operation of elements from index $l$ to $r$ in the array $\textit{nums}$, that is, $\textit{nums}[l] \land \textit{nums}[l + 1] \land \cdots \land \textit{nums}[r]$, where $\land$ represents the bitwise AND operation.

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class Solution:
    def countSubarrays(self, nums: List[int], k: int) -> int:
        ans = 0
        pre = Counter()
        for x in nums:
            cur = Counter()
            for y, v in pre.items():
                cur[x & y] += v
            cur[x] += 1
            ans += cur[k]
            pre = cur
        return ans

Continue Practicing

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