Bitwise XOR Property
One optimized approach leverages the XOR operation’s properties. XORing a number with itself results in zero (a⊕a=0), and XORing zero with any number gives the same number (a⊕0=a). By exploiting these properties, we can XOR consecutive numbers to cancel each other out, leaving only the XOR of the first and last numbers in the range. This method achieves a time complexity of O(1) by finding the XOR of the first L−1 numbers and the first R numbers, then XORing the results to obtain the final answer.
Example: Let’s us consider an example where L=3 and R=6.
The XOR of all numbers from 3 to 6 is calculated as follows:
3⊕4⊕5⊕6=(3⊕4)⊕(5⊕6)
=(7)⊕(3)
=4
Example: To demonstrate computing the bitwise XOR of all numbers in a range using bitwise XOR properties in JavaScript.
function xorRange(L, R) {
// XOR of numbers from 1 to n
const xorN = (n) => {
if (n % 4 === 0) return n;
if (n % 4 === 1) return 1;
if (n % 4 === 2) return n + 1;
return 0;
};
// XOR of numbers from 1 to L-1
const xorLMinusOne = xorN(L - 1);
// XOR of numbers from 1 to R
const xorR = xorN(R);
// XOR of numbers from L to R
const xorLR = xorLMinusOne ^ xorR;
return xorLR;
}
const L = 3;
const R = 6;
const result = xorRange(L, R);
console.log(`Bitwise XOR of all numbers from ${L} to ${R} is ${result}`);
Output:
Bitwise XOR of all numbers from 3 to 6 is 4
Time Complexity: O(1), as it does not depend on the size of the range.
Space Complexity: O(1), as we only use a constant amount of additional space regardless of the input size.
Compute the Bitwise XOR of all Numbers in a Range using JavaScript
The complexity of bitwise XOR computation for all the numbers in a specific range is the most frequent difficulty that many people face during the problem-solving process.
Problem Description: Given two integers, L and R, find the bitwise XOR of all numbers in the inclusive range from L to R. There are several approaches to computing the bitwise XOR of all numbers in a range using JavaScript which are as follows:
Table of Content
- Bitwise XOR Property
- Segment Trees
- Binary Indexed Trees (Fenwick Trees)
- Bitwise Manipulation