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100 lines
2.1 KiB
Markdown
100 lines
2.1 KiB
Markdown
---
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title: The XOR Swap
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date: 02-05-2024
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---
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# The XOR Swap
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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.
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## Algorithm
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The algorithm is described as follows:
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$$
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\begin{align*}
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x_1 &= x_0 \oplus y_0 \\
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y_1 &= x_1 \oplus y_0 \\
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x_2 &= x_1 \oplus y_1
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\end{align*}
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$$
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Expanding this:
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$$
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\begin{align*}
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x_1 &= x_0 \oplus y_0 \\
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y_1 &= (x_0 \oplus y_0) \oplus y_0 \\
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x_2 &= (x_0 \oplus y_0) \oplus [(x_0 \oplus y_0) \oplus y_0]
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\end{align*}
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$$
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When changing the order of operations:
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$$
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\begin{align*}
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x_1 &= x_0 \oplus y_0 \\
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y_1 &= (y_0 \oplus y_0) \oplus x_0 \\
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x_2 &= (x_0 \oplus x_0) \oplus (y_0 \oplus y_0) \oplus y_0
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\end{align*}
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$$
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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:
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$$
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\begin{align*}
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y_1 &= x_0 \\
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x_2 &= y_0
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\end{align*}
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$$
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## Practical Example
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Suppose we have two numbers we want to swap:
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$$
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\begin{align*}
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x_0 = 101_2 \\
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y_0 = 010_2
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\end{align*}
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$$
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> [!info] Info
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> 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.
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Applying the XOR operation on these values:
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$$
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\begin{align*}
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x_1 &= 101 \oplus 010 \\
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x_1 &= 111
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\end{align*}
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$$
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Continuing with the process:
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$$
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\begin{align*}
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y_1 &= 111 \oplus 010 \\
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y_1 &= 101
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\end{align*}
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$$
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And finally:
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$$
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\begin{align*}
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x_2 &= 111 \oplus 101 \\
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x_2 &= 010
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\end{align*}
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$$
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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.
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