Consider two sets of $N$ $n$-dimensiononal points each:
$\mathcal{X}= \lbrace \mathbf{x}_1,\mathbf{x}_2,\dots,\mathbf{x}_N \rbrace,$
$\mathcal{Y}= \lbrace \mathbf{y}_1,\mathbf{y}_2,\dots,\mathbf{y}_N \rbrace,$
where $\mathbf{x}_i,\mathbf{y}_i\in\mathbb{R}^n$.
Is there a metric $d(\mathcal{X},\mathcal{Y})$ that defines the 'distance' between these two sets of points, assuming that the ordering of the points in the two sets is not necessarily identical? The metric should ideally be $0$ if there is an exact one-to-one correspondence between the points in the two sets, and increase monotonically as the difference (in some sense) between the two sets of points increases.