One less = The number = One more?

From the graph we see that:

$\begin{cases} \text{For } x \le -1, & |x-1| - |x| = 1, & |x| - |x+1| = 1 \\ \text{For } x \ge 1, & |x-1| - |x| = -1, & |x| - |x+1| = -1 \end{cases}$

This means that for any $a \le -1$ , any $b \ge 1$ , and for any $a \ge 1$ , any $b \le -1$ satisfies the equation:

$|a-1|+|b-1| = |a|+|b| = |a+1|+|b+1|$

And the least value of |a-b| is when $a = \pm 1$ and $b = \mp 1$ .

$|a-b|_{min} = |1-(-1)| = |-1-1| = \boxed{2}$