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XOR Queries of a Subarray

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ArraysBit Manipulation

You are given an array arr of positive integers. You are also given the array queries where queries[i] = [left_i,right_i].

For each query i compute the XOR of elements from left_i to right_i (that is, arr[left_i] XOR arr[left_i + 1] XOR ... XOR arr[right_i] ).

Return an array answer where answer[i] is the answer to the i^th query.

Example 1:

Input: arr = [1,3,4,8], queries = [[0,1],[1,2],[0,3],[3,3]]
Output: [2,7,14,8] 
Explanation: 
The binary representation of the elements in the array are:
1 = 0001 
3 = 0011 
4 = 0100 
8 = 1000 
The XOR values for queries are:
[0,1] = 1 xor 3 = 2 
[1,2] = 3 xor 4 = 7 
[0,3] = 1 xor 3 xor 4 xor 8 = 14 
[3,3] = 8

Example 2:

Input: arr = [4,8,2,10], queries = [[2,3],[1,3],[0,0],[0,3]]
Output: [8,0,4,4]

Constraints:

1 <= arr.length, queries.length <= 3 * 10⁴
1 <= arr[i] <= 10⁹
queries[i].length == 2
0 <= left_i <= right_i < arr.length

Try coding it in LeetCode

XOR Queries of a Subarray

Medium

Asked by:

13 views

Topics:

ArraysBit Manipulation

You are given an array arr of positive integers. You are also given the array queries where queries[i] = [left_i,right_i].

For each query i compute the XOR of elements from left_i to right_i (that is, arr[left_i] XOR arr[left_i + 1] XOR ... XOR arr[right_i] ).

Return an array answer where answer[i] is the answer to the i^th query.

Example 1:

Input: arr = [1,3,4,8], queries = [[0,1],[1,2],[0,3],[3,3]]
Output: [2,7,14,8] 
Explanation: 
The binary representation of the elements in the array are:
1 = 0001 
3 = 0011 
4 = 0100 
8 = 1000 
The XOR values for queries are:
[0,1] = 1 xor 3 = 2 
[1,2] = 3 xor 4 = 7 
[0,3] = 1 xor 3 xor 4 xor 8 = 14 
[3,3] = 8

Example 2:

Input: arr = [4,8,2,10], queries = [[2,3],[1,3],[0,0],[0,3]]
Output: [8,0,4,4]

Constraints:

1 <= arr.length, queries.length <= 3 * 10⁴
1 <= arr[i] <= 10⁹
queries[i].length == 2
0 <= left_i <= right_i < arr.length

Try coding it in LeetCode

Solution

Clarifying Questions

When you get asked this question in a real-life environment, it will often be ambiguous (especially at FAANG). Make sure to ask these questions in that case:

What is the maximum size of the input array and the queries array? Are there any constraints on the time complexity?
Can the elements in the input array be negative, zero, or non-integer?
Are the query indices guaranteed to be within the bounds of the input array, and are the start and end indices inclusive?
If a query range is invalid (e.g., start index > end index), what should the corresponding result in the output array be?
What data type should the elements of the input array and query results be (e.g., 32-bit integer)? Are we concerned about integer overflow during the XOR operations?

Brute Force Solution

Approach

The brute force approach to this problem means checking every possible subarray for each query. We calculate the result for each subarray directly without trying to be clever. This guarantees we'll find the answer, but might take a long time.

Here's how the algorithm would work step-by-step:

For each question, we look at the starting and ending positions it gives us.
We take all the numbers between those positions, one at a time.
We combine these numbers using the 'XOR' operation, accumulating a running result.
The final accumulated result after combining all numbers in the specified range is the answer for that question.
We repeat the process for every question we're given.

Code Implementation

def xor_queries_brute_force(numbers, queries):
    result = []

    for query in queries:
        start_index = query[0]
        end_index = query[1]
        xor_sum_for_subarray = 0

        # Iterate through the subarray specified by the query.
        for index in range(start_index, end_index + 1):

            #Accumulate the XOR sum for the current subarray.
            xor_sum_for_subarray ^= numbers[index]

        result.append(xor_sum_for_subarray)

    return result

Big(O) Analysis

Time Complexity

O(n*q) – The algorithm iterates through each query in the queries array. For each query, it iterates through a subarray of the input array nums, performing an XOR operation for each element in the subarray. The length of the subarray is at most n, the length of the input array. Since this is done for each of the q queries, the total time complexity is O(n*q), where n is the length of the input array nums and q is the number of queries.

Space Complexity

O(1) – The brute force approach iterates through subarrays defined by the queries, calculating the XOR sum. For each query, a running result (the XOR sum) is accumulated. This approach uses a fixed number of variables, such as the loop index and the XOR result, irrespective of the input array size or the number of queries. Therefore, the space complexity is constant.

Optimal Solution

Approach

The core idea is to precompute some information to answer queries very quickly. Instead of calculating the XOR for each subarray every time a query comes in, we'll prepare a helper dataset beforehand.

Here's how the algorithm would work step-by-step:

Calculate a 'running XOR' for the original set of numbers, where each number in the running XOR set represents the XOR of all elements up to that point in the original set.
When a query comes in asking for the XOR of a particular subarray, use the running XOR values to quickly determine the answer.
The XOR of the subarray is simply the XOR of the running XOR value at the end of the subarray with the running XOR value at the beginning of the subarray (minus one if the subarray starts from the beginning).
This allows answering each query very quickly, since it involves only one XOR operation using the precomputed information.

Code Implementation

def xor_queries(array, queries):
    running_xor_array = []
    current_xor = 0

    # Precompute the running XOR values for each element.
    for number in array:
        current_xor ^= number
        running_xor_array.append(current_xor)

    result = []
    for start_index, end_index in queries:
        # Handle the case where the subarray starts from the beginning.
        if start_index == 0:
            subarray_xor = running_xor_array[end_index]
        else:
            # Calculate XOR using precomputed running XOR values.
            subarray_xor = running_xor_array[end_index] ^ running_xor_array[start_index - 1]

        result.append(subarray_xor)

    return result

Big(O) Analysis

Time Complexity

O(n) – The algorithm first calculates the running XOR of the input array of size n, which takes O(n) time. Then, for each query, it performs a single XOR operation using the precomputed running XOR values. Since the precomputation dominates the cost and the number of queries is independent of n, the overall time complexity is determined solely by the initial running XOR calculation, resulting in O(n).

Space Complexity

O(N) – The algorithm calculates a 'running XOR' and stores it in a new array. This running XOR array has the same number of elements as the original input array. Therefore, the auxiliary space used is directly proportional to the size of the input array, denoted as N. This results in an auxiliary space complexity of O(N).

Edge Cases

Case	How to Handle
Null or empty input array	Return an empty result array, as there are no subarrays to query.
Empty queries array	Return an empty result array, as there are no queries to process.
Single element array with a query range exceeding its bounds.	The code should handle out-of-bounds queries by returning the element XORed with 0, effectively returning the element itself if the end index is within bounds, or 0 if start index is out of bounds.
Large input array size and numerous queries (performance scaling)	Precompute XOR prefix array for efficient XOR calculation of subarrays to avoid repeated calculations.
Query range where start index is greater than end index	Handle this invalid query by either swapping the indices or returning 0 or an error depending on problem specification.
Integer overflow in XOR calculations for large numbers	Ensure the data type used to store the XOR results (usually int) is large enough to accommodate the maximum possible XOR value (e.g., use long if necessary).
Array contains negative numbers.	The XOR operation is well-defined for negative integers in most languages (using two's complement), so no special handling is required but confirm implementation dependent behavior.
Query range is [0, array.length - 1], querying the entire array.	Ensure the solution correctly calculates and returns the XOR of all elements in the input array when the query spans the entire array.

Null or empty input array

How to Handle:

Return an empty result array, as there are no subarrays to query.

Empty queries array

How to Handle:

Return an empty result array, as there are no queries to process.

Single element array with a query range exceeding its bounds.

How to Handle:

The code should handle out-of-bounds queries by returning the element XORed with 0, effectively returning the element itself if the end index is within bounds, or 0 if start index is out of bounds.

Large input array size and numerous queries (performance scaling)

How to Handle:

Precompute XOR prefix array for efficient XOR calculation of subarrays to avoid repeated calculations.

Query range where start index is greater than end index

How to Handle:

Handle this invalid query by either swapping the indices or returning 0 or an error depending on problem specification.

Integer overflow in XOR calculations for large numbers

How to Handle:

Ensure the data type used to store the XOR results (usually int) is large enough to accommodate the maximum possible XOR value (e.g., use long if necessary).

Array contains negative numbers.

How to Handle:

The XOR operation is well-defined for negative integers in most languages (using two's complement), so no special handling is required but confirm implementation dependent behavior.

Query range is [0, array.length - 1], querying the entire array.

How to Handle:

Ensure the solution correctly calculates and returns the XOR of all elements in the input array when the query spans the entire array.

Solution

Clarifying Questions

When you get asked this question in a real-life environment, it will often be ambiguous (especially at FAANG). Make sure to ask these questions in that case:

What is the maximum size of the input array and the queries array? Are there any constraints on the time complexity?
Can the elements in the input array be negative, zero, or non-integer?
Are the query indices guaranteed to be within the bounds of the input array, and are the start and end indices inclusive?
If a query range is invalid (e.g., start index > end index), what should the corresponding result in the output array be?
What data type should the elements of the input array and query results be (e.g., 32-bit integer)? Are we concerned about integer overflow during the XOR operations?

Brute Force Solution

Approach

Here's how the algorithm would work step-by-step:

For each question, we look at the starting and ending positions it gives us.
We take all the numbers between those positions, one at a time.
We combine these numbers using the 'XOR' operation, accumulating a running result.
The final accumulated result after combining all numbers in the specified range is the answer for that question.
We repeat the process for every question we're given.

Code Implementation

def xor_queries_brute_force(numbers, queries):
    result = []

    for query in queries:
        start_index = query[0]
        end_index = query[1]
        xor_sum_for_subarray = 0

        # Iterate through the subarray specified by the query.
        for index in range(start_index, end_index + 1):

            #Accumulate the XOR sum for the current subarray.
            xor_sum_for_subarray ^= numbers[index]

        result.append(xor_sum_for_subarray)

    return result

Big(O) Analysis

Time Complexity

Space Complexity

Optimal Solution

Approach

Here's how the algorithm would work step-by-step:

Calculate a 'running XOR' for the original set of numbers, where each number in the running XOR set represents the XOR of all elements up to that point in the original set.
When a query comes in asking for the XOR of a particular subarray, use the running XOR values to quickly determine the answer.
The XOR of the subarray is simply the XOR of the running XOR value at the end of the subarray with the running XOR value at the beginning of the subarray (minus one if the subarray starts from the beginning).
This allows answering each query very quickly, since it involves only one XOR operation using the precomputed information.

Code Implementation

def xor_queries(array, queries):
    running_xor_array = []
    current_xor = 0

    # Precompute the running XOR values for each element.
    for number in array:
        current_xor ^= number
        running_xor_array.append(current_xor)

    result = []
    for start_index, end_index in queries:
        # Handle the case where the subarray starts from the beginning.
        if start_index == 0:
            subarray_xor = running_xor_array[end_index]
        else:
            # Calculate XOR using precomputed running XOR values.
            subarray_xor = running_xor_array[end_index] ^ running_xor_array[start_index - 1]

        result.append(subarray_xor)

    return result

Big(O) Analysis

Time Complexity

Space Complexity

Edge Cases

Case	How to Handle
Null or empty input array	Return an empty result array, as there are no subarrays to query.
Empty queries array	Return an empty result array, as there are no queries to process.
Single element array with a query range exceeding its bounds.	The code should handle out-of-bounds queries by returning the element XORed with 0, effectively returning the element itself if the end index is within bounds, or 0 if start index is out of bounds.
Large input array size and numerous queries (performance scaling)	Precompute XOR prefix array for efficient XOR calculation of subarrays to avoid repeated calculations.
Query range where start index is greater than end index	Handle this invalid query by either swapping the indices or returning 0 or an error depending on problem specification.
Integer overflow in XOR calculations for large numbers	Ensure the data type used to store the XOR results (usually int) is large enough to accommodate the maximum possible XOR value (e.g., use long if necessary).
Array contains negative numbers.	The XOR operation is well-defined for negative integers in most languages (using two's complement), so no special handling is required but confirm implementation dependent behavior.
Query range is [0, array.length - 1], querying the entire array.	Ensure the solution correctly calculates and returns the XOR of all elements in the input array when the query spans the entire array.

Null or empty input array

How to Handle:

Return an empty result array, as there are no subarrays to query.

Empty queries array

How to Handle:

Return an empty result array, as there are no queries to process.

Single element array with a query range exceeding its bounds.

How to Handle:

The code should handle out-of-bounds queries by returning the element XORed with 0, effectively returning the element itself if the end index is within bounds, or 0 if start index is out of bounds.

Large input array size and numerous queries (performance scaling)

How to Handle:

Precompute XOR prefix array for efficient XOR calculation of subarrays to avoid repeated calculations.

Query range where start index is greater than end index

How to Handle:

Handle this invalid query by either swapping the indices or returning 0 or an error depending on problem specification.

Integer overflow in XOR calculations for large numbers

How to Handle:

Ensure the data type used to store the XOR results (usually int) is large enough to accommodate the maximum possible XOR value (e.g., use long if necessary).

Array contains negative numbers.

How to Handle:

The XOR operation is well-defined for negative integers in most languages (using two's complement), so no special handling is required but confirm implementation dependent behavior.

Query range is [0, array.length - 1], querying the entire array.

How to Handle:

Ensure the solution correctly calculates and returns the XOR of all elements in the input array when the query spans the entire array.