切比雪夫距离与曼哈顿距离

\[\begin{align*} |x_1 - x_2| + |y_1 - y_2| &= \max\{x_1 - x_2, x_2 - x_1\} + \max\{y_1 - y_2, y_2 - y_1\} \\ &= \max\{x_1 - x_2 + y_1 - y_2, x_1 - x_2 + y_2 - y_1, x_2 - x_1 + y_2 - y_1, x_2 - x_1 + y_1 - y_2\} \\ &= \max\{(x_1 + y_1) - (x_2 + y_2), (x_2 + y_2) - (x_1 + y_1), (x_1 - y_1) - (x_2 - y_2), (y_1 - y_2) - (x_1 - x_2)\} \\ &= \max\{|(x_1 + y_1) - (x_2 + y_2)|, |(x_1 - y_1) - (x_2 - y_2)|\} \end{align*} \]

\((x_1, y_1)\) 与 \((x_2, y_2)\) 的曼哈顿距离等于 \((x_1 + y_1, x_1 - y_1)\) 与 \((x_2 + y_2, x_2 - y_2)\) 的切比雪夫距离；

\((x_1, y_1)\) 与 \((x_2, y_2)\) 的切比雪夫距离等于 \((\frac{x_1 + y_1}{2}, \frac{x_1 - y_1}{2})\) 与 \((\frac{x_2 + y_2}{2}, \frac{x_2 - y_2}{2})\) 的曼哈顿距离。