Embedding into l∞2 Is Easy, Embedding into l∞3 Is NP-Complete

AbstractWe give a new algorithm for enumerating all possible embeddings of a metric space (i.e., the distances between every pair within a set of n points) into ℝ2 Cartesian space preserving their l∞ (or l1) metric distances. Its expected time is $\mathcal {O}(n^{2}\log^{2}n)$ (i.e., within a poly-log of the size of the input) beating the previous $\mathcal {O}(n^{3})$ algorithm. In contrast, we prove that detecting l∞3 embeddings is NP-complete. The problem is also NP-complete within l12 or l∞2 with the added constraint that the locations of two of the points are given or alternatively that the two dimensions are curved into a three-dimensional sphere. We also refute a compaction theorem by giving a metric space that cannot be embedded in l∞3; however, it can be embedded if any single point is removed.