quartz/content/Miscellaneous/The XOR Swap.md

date
02-05-2024

The XOR Swap

The \text{XOR} swap algorithm is a clever programming trick used to swap the values of two variables without using a third temporary variable. This method exploits the properties of the \text{XOR} bitwise operation to perform the swap efficiently and in a mathematically elegant manner. The \text{XOR}, or "exclusive or," operation on two bits results in a value of 1 if and only if the bits are different; otherwise, the result is 0.

Algorithm

The algorithm is described as follows:

\begin{align*}

x_1 &= x_0 \oplus y_0 \\
y_1 &= x_1 \oplus y_0 \\
x_2 &= x_1 \oplus y_1

\end{align*}

Expanding this:

\begin{align*}

x_1 &= x_0 \oplus y_0 \\
y_1 &= (x_0 \oplus y_0) \oplus y_0 \\
x_2 &= (x_0 \oplus y_0) \oplus [(x_0 \oplus y_0) \oplus y_0]

\end{align*}

When changing the order of operations:

\begin{align*}

x_1 &= x_0 \oplus y_0 \\
y_1 &= (y_0 \oplus y_0) \oplus x_0 \\
x_2 &= (x_0 \oplus x_0) \oplus (y_0 \oplus y_0) \oplus y_0

\end{align*}

Based on the \text{XOR} properties, where we know that x \oplus x = 0 and that x \oplus 0 = x, 0 \oplus x = x, we arrive at the following conclusions, completing the swap process:

\begin{align*}

y_1 &= x_0 \\
x_2 &=  y_0

\end{align*}

Practical Example

Suppose we have two numbers we want to swap:

\begin{align*}
x_0 = 101_2 \\
y_0 = 010_2
\end{align*}

[!info] Info Let x_0 and y_0 denote the initial values of variables x and y, respectively. Here, the subscript 2 indicates that the numbers are in base-2 (binary) notation.

Applying the XOR operation on these values:

\begin{align*}
x_1 &= 101 \oplus 010 \\
x_1 &= 111
\end{align*}

Continuing with the process:

\begin{align*}
y_1 &= 111 \oplus 010 \\
y_1 &= 101
\end{align*}

And finally:

\begin{align*}
x_2 &= 111 \oplus 101 \\
x_2 &= 010
\end{align*}

Thus, after applying the \text{XOR} swap algorithm, x_0 (originally 101) has been swapped with y_0 (originally 010), demonstrating the algorithm's effectiveness with a practical example.