How to Solve Count the Number of Beautiful Subarrays

To solve the 'Count the Number of Beautiful Subarrays' problem, we must identify subarrays whose elements can be reduced to zero after a series of operations. The challenge requires an efficient strategy combining array scanning and hash table lookups, leveraging the XOR operation to check for beautiful subarrays.

Problem Statement

You are given a 0-indexed integer array nums. In one operation, you can subtract a constant k from all elements of a subarray. A subarray is considered beautiful if it is possible to make all of its elements equal to 0 after applying the operation any number of times, including zero.

Return the number of beautiful subarrays in the array nums. A beautiful subarray's elements' XOR should equal zero after a series of operations.

Examples

Example 1

Input: nums = [4,3,1,2,4]

Output: 2

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

There are no beautiful subarrays in nums.

Constraints

• 1 <= nums.length <= 105
• 0 <= nums[i] <= 106

Solution Approach

XOR Subarray Condition

A key observation is that a subarray is beautiful if the XOR of all its elements equals zero. We can scan through the array, keeping track of cumulative XOR results. Using a hash table, we store the frequency of each XOR result encountered. Whenever we encounter the same XOR value again, it means the subarray formed between the previous occurrence and the current index has a zero XOR, indicating a beautiful subarray.

Efficient Array Scanning with Hash Table Lookup

To efficiently solve this problem, iterate through the array while maintaining a running XOR of the elements. Use a hash table to track how many times each XOR result has occurred. If a running XOR value has been seen before, it means that the subarray from the previous occurrence to the current position is beautiful. The algorithm's complexity depends on both the scanning and the hash table operations.

Time and Space Complexity Optimization

The solution has a time complexity of O(n), where n is the length of the input array. This is because we are scanning through the array once and performing constant-time operations for each element using the hash table. The space complexity is also O(n), as we need to store the frequency of XOR results in the hash table.

Complexity Analysis

The algorithm scans the array once, updating the XOR value and checking the hash table. Both the time and space complexities are linear, making the solution efficient for large inputs.

What Interviewers Usually Probe

• Look for familiarity with XOR operations and efficient hash table use.
• Candidates should recognize that this problem relies heavily on array scanning and hash lookups.
• Expect to see a clear understanding of how to use cumulative XOR and hash tables to optimize the solution.

Common Pitfalls or Variants

Common pitfalls

• Forgetting to update the hash table correctly during array scanning.
• Overcomplicating the solution by not recognizing the XOR operation as a key part of the problem.
• Misunderstanding the XOR condition for beautiful subarrays, leading to incorrect subarray identification.

Follow-up variants

• Modify the problem to find the number of subarrays where the XOR is equal to a given target value.
• Extend the problem to multidimensional arrays and find beautiful subarrays in each row or column.
• Allow negative integers in the array and adjust the solution to handle them while still using XOR.

How GhostInterview Helps

• GhostInterview helps by guiding candidates through array scanning and hash table lookup techniques.
• With insights into efficient XOR manipulation, GhostInterview aids in optimizing the solution for large arrays.
• The platform provides targeted feedback on understanding XOR subarray conditions, ensuring precise solution design.

Topic Pages

• Array
• Hash Table
• Bit Manipulation
• Prefix Sum

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FAQ

What is the primary operation used to solve 'Count the Number of Beautiful Subarrays'?

The primary operation is XOR, which is used to identify beautiful subarrays where the XOR of the elements equals zero.

How can hash tables be leveraged to solve this problem?

Hash tables are used to track the cumulative XOR values encountered during array scanning, enabling quick identification of beautiful subarrays.

What is the time complexity of the optimal solution for this problem?

The time complexity of the optimal solution is O(n), where n is the length of the input array, because the array is scanned once while performing constant-time operations using the hash table.

Are negative numbers allowed in the 'Count the Number of Beautiful Subarrays' problem?

The problem assumes non-negative integers, but the solution can be adapted to handle negative numbers by adjusting how XOR operations are applied.

What makes a subarray beautiful in this problem?

A subarray is considered beautiful if, after applying XOR operations to its elements, the XOR result equals zero.