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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.

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    That seems to do the job. @Karolis Juodelė ca$n$ you post it as an answer so I can mar$k$ it as correct?2012-09-02

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A metric in a set of compacts is Hausdorff distance.