Embedding into the rectilinear grid

Abstract

Abstract: We show that the embedding of metric spaces into the l 1 -grid Z 2 can be characterized inessentially the same fashion as in the case of the l 1 -plane R 2 . In particular, a metric space can beembedded into Z 2 iff every subspace with at most 6 points is embeddable. Moreover, if such an embed-ding 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 l 1 -metric) is probably the for the embedding in the gridZ 2 (‘‘digital plane’’) en-simplest distance measure in R n . This explains why in dowed with the rectilinear distance (alias city block met-many cases rectilinear spaces are selected as host spaces ric) (see Fig. 1). Again, as in [1],wecantake advantagefor embedding a given metric space; we refer to Hubert of such notions as ‘‘d-split’’ and ‘‘totally decomposableet al. [7] for an application to multidimensional scaling metrics,’’ introduced and investigated in the case of finitewhere one aims at producing a visual display of a given metric spaces by Bandelt and Dress [2].data set and to Deza and Laurent [6] for other applica- Let (X, d) be a metric space. We will say that (X, d)tions. Surprisingly, the problem to characterize the metric satisfies theparity condition if d(u, £) /d(£, w) /d(w,subspaces of the rectilinear space of a given dimension u) is an even integer for any u, £, w ˆX (cf. [10] forseems to be much more difficult than is the analogous a straightforward metric characterization of bipartiteproblem for Euclidean spaces, where suitable criteria are graphs). To every pair S †{A, B} of nonempty subsetsgiven by the classical results of Menger and Schoenberg of X we associate the isolation index a