1310. XOR Queries of a Subarray

In the realm of competitive programming and data structures, understanding bitwise operations and their applications is paramount. This article dives deep into the "XOR Queries of a Subarray" problem, exploring its essence, various solution approaches, and practical implementations.

Introduction

The problem, aptly named "1310. XOR Queries of a Subarray," is a captivating challenge on LeetCode. It presents us with an array of integers and a series of queries. Each query asks for the XOR (exclusive OR) of all elements within a specified subarray. Let's delve into its specifics and unravel the strategies to tackle it effectively.

Problem Statement

Given an array of integers arr , you are tasked with processing queries. Each query is represented by two integers left and right , denoting the starting and ending indices (inclusive) of a subarray within arr . The goal is to determine the XOR of all elements within the specified subarray.

For instance, if arr = [1, 3, 4, 8] and the query is left = 0 and right = 2 , the subarray is [1, 3, 4] . The XOR of this subarray is: 1 ^ 3 ^ 4 = 2 .

Significance

This problem is not merely a mathematical puzzle. It embodies fundamental concepts in data structures and algorithms, particularly:

• **Bitwise Operations:** Understanding XOR and its properties is crucial for efficient solutions.
• **Prefix XOR Array:** A key data structure used to optimize query processing.
• **Time Complexity:** The ability to design solutions with optimal time complexity is a core skill in competitive programming.

Core Concepts

Before diving into the solutions, let's lay a foundation by understanding the core concepts involved.

Bitwise XOR

The exclusive OR (XOR) operator (represented by the symbol ^ ) is a binary operation. It yields a 1 in the output bit position if and only if the corresponding input bits are different.

For example:

 1011  (11 decimal)
^ 0101  (5 decimal)
-------
 1110  (14 decimal)

Here, the output is obtained by comparing each corresponding bit position. The XOR operation possesses a vital property: it's commutative and associative . This means the order in which XOR operations are performed doesn't affect the outcome.

Prefix XOR Array

A prefix XOR array is a powerful tool for efficiently calculating XORs of subarrays. It's constructed as follows:

1. Initialize an array prefixXOR with the same size as the input array arr .

2. Set prefixXOR[0] = arr[0] .

3. For each subsequent element i , calculate prefixXOR[i] = prefixXOR[i-1] ^ arr[i] .

In essence, the prefix XOR array at index i stores the XOR of all elements from index 0 to i inclusive.

Example

Consider the array arr = [1, 3, 4, 8] . The corresponding prefix XOR array would be:

prefixXOR = [1, 4, 0, 8]

Let's break down how this array is constructed:

• prefixXOR[0] = arr[0] = 1 .
• prefixXOR[1] = prefixXOR[0] ^ arr[1] = 1 ^ 3 = 4 .
• prefixXOR[2] = prefixXOR[1] ^ arr[2] = 4 ^ 4 = 0 .
• prefixXOR[3] = prefixXOR[2] ^ arr[3] = 0 ^ 8 = 8 .

Solution Approaches

Now, let's explore different solutions to the "XOR Queries of a Subarray" problem. We'll focus on two common approaches:

1. Brute Force

The most straightforward approach is brute force. For each query, we iterate through the subarray and calculate the XOR of all elements.

Code Example (Python)

def xorQueries(arr, queries):
    results = []
    for left, right in queries:
        xor_sum = 0
        for i in range(left, right + 1):
            xor_sum ^= arr[i]
        results.append(xor_sum)
    return results

Time Complexity: O(N * Q)

This brute force approach has a time complexity of O(N * Q), where N is the length of the array and Q is the number of queries. For each query, we iterate through the subarray, making the overall time complexity proportional to the product of N and Q.

Space Complexity: O(Q)

The space complexity is O(Q) due to the storage of the results in the results array.

2. Prefix XOR Array

Utilizing a prefix XOR array, we can significantly optimize the solution. Here's how:

1. **Calculate the Prefix XOR Array:** Construct the prefix XOR array as described earlier.

2. **Process Queries:** For each query ( left , right ), the XOR of the subarray is calculated as: prefixXOR[right] ^ prefixXOR[left - 1] .

Code Example (Python)

def xorQueries(arr, queries):
    n = len(arr)
    prefixXOR = [0] * (n + 1)
    for i in range(n):
        prefixXOR[i + 1] = prefixXOR[i] ^ arr[i]
    results = []
    for left, right in queries:
        xor_sum = prefixXOR[right + 1] ^ prefixXOR[left]
        results.append(xor_sum)
    return results

Time Complexity: O(N + Q)

This optimized approach has a time complexity of O(N + Q). The calculation of the prefix XOR array takes O(N) time, and processing each query takes constant time (O(1)).

Space Complexity: O(N)

The space complexity is O(N) due to the storage of the prefix XOR array.

Example Illustration

Let's illustrate the prefix XOR approach with an example:

Suppose arr = [1, 3, 4, 8] and we have the following queries:

1. left = 0, right = 2
2. left = 1, right = 3

1. **Calculate Prefix XOR Array:**

prefixXOR = [0, 1, 4, 0, 8]

2. **Process Queries:**

• **Query 1:** (left = 0, right = 2) * xor_sum = prefixXOR[right + 1] ^ prefixXOR[left] = prefixXOR[3] ^ prefixXOR[0] = 0 ^ 0 = 0 .
• **Query 2:** (left = 1, right = 3) * xor_sum = prefixXOR[right + 1] ^ prefixXOR[left] = prefixXOR[4] ^ prefixXOR[1] = 8 ^ 4 = 12 .

Therefore, the results for the queries are [0, 12] . The XOR of the subarray [1, 3, 4] is 0, and the XOR of the subarray [3, 4, 8] is 12.

Code Implementation (Python)

Here's a complete Python implementation of the solution using the prefix XOR array:

def xorQueries(arr, queries):
    n = len(arr)
    prefixXOR = [0] * (n + 1)
    for i in range(n):
        prefixXOR[i + 1] = prefixXOR[i] ^ arr[i]
    results = []
    for left, right in queries:
        xor_sum = prefixXOR[right + 1] ^ prefixXOR[left]
        results.append(xor_sum)
    return results

Conclusion

The "XOR Queries of a Subarray" problem highlights the significance of understanding bitwise operations and their application in data structures. The prefix XOR array provides an efficient solution to efficiently process XOR queries on a subarray, significantly reducing the time complexity compared to brute force. This problem demonstrates the value of choosing appropriate data structures and algorithms for optimizing code performance.

Remember, mastering bitwise operations and prefix sums are fundamental skills in competitive programming and software development. As you continue your journey into the world of algorithms and data structures, these concepts will prove invaluable in tackling more complex challenges.