Embedding into the rectilinear grid

We show that the embedding of metric spaces into the l1-grid Z can be characterized in essentially the same fashion as in the case of the l1-plane R . In particular, a metric space can be embedded into Z iff every subspace with at most 6 points is embeddable. Moreover, if such an embedding exists, it can be constructed in polynomial time (for finite spaces) . q 1998 John Wiley & Sons, Inc. Networks 32: 127–132, 1998 The rectilinear metric (alias l1-metric) is probably the for the embedding in the grid Z 2 (‘‘digital plane’’) endowed with the rectilinear distance (alias city block metsimplest distance measure in R n . This explains why in many cases rectilinear spaces are selected as host spaces ric) (see Fig. 1) . Again, as in [1] , we can take advantage of such notions as ‘‘d-split’’ and ‘‘totally decomposable for embedding a given metric space; we refer to Hubert et al. [7] for an application to multidimensional scaling metrics,’’ introduced and investigated in the case of finite metric spaces by Bandelt and Dress [2] . where one aims at producing a visual display of a given data set and to Deza and Laurent [6] for other applicaLet (X , d) be a metric space. We will say that (X , d) satisfies the parity condition if d(u , £) / d(£, w) / d(w , tions. Surprisingly, the problem to characterize the metric subspaces of the rectilinear space of a given dimension u) is an even integer for any u , £, w √ X (cf. [10] for a straightforward metric characterization of bipartite seems to be much more difficult than is the analogous problem for Euclidean spaces, where suitable criteria are graphs) . To every pair S Å {A , B} of nonempty subsets of X we associate the isolation index aS ( if it exists) with given by the classical results of Menger and Schoenberg (see [4]) . However, at least the two-dimensional case respect to d , as follows: can be approached. Recently, by sharpening a result of Malitz and Malitz [8] , we proved the following MengeraS Å 12r min a ,a =√A b ,b =√B (max{d(a , b) / d(a *, b *) , d(a , b *) type theorem for the rectilinear plane R 2 : A metric space is embeddable in the plane if and only if every subspace / d(a *, b) , d(a , a *) / d(b , b *)} with at most 6 points is such [1] . It was the purpose of this paper to demonstrate that the same result holds true 0 d(a , a *) 0 d(b , b *)) . Evidently, if (X , d) satisfies the parity condition, then all Correspondence to: V. Chepoi, SFB343 Diskrete Strukturen in der distances are integers, and, hence, for every pair S Å {A , Mathematik, Universität Bielefeld, D-33615 Bielefeld, Germany. 2 On leave from the Universitatea de stat din Moldova, Chişinǎu B}, the minimum is attained, so that aS ¢ 0 is well q 1998 John Wiley & Sons, Inc. CCC 0028-3045/98/020127-06