Problem Description
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. Return an array answer where answer[i] is the answer to the i-th query.
Key Insights
- The XOR operation is both associative and commutative, which allows us to compute the result in any order.
- Using prefix XOR can significantly reduce the time complexity by allowing us to compute the XOR for any subarray in constant time after an initial preprocessing step.
- The prefix XOR array is built such that
prefix[i]contains the XOR of all elements from the start of the array to indexi.
Space and Time Complexity
Time Complexity: O(n + q), where n is the number of elements in arr and q is the number of queries.
Space Complexity: O(n), for the prefix XOR array.
Solution
To efficiently solve the problem, we will use the prefix XOR approach. The prefix XOR array is constructed where each element at index i contains the XOR of all elements from the start of the array up to i. This allows us to compute the XOR for any subarray arr[left] to arr[right] using the formula:
This approach allows us to answer each query in constant time after the prefix XOR array has been built.