Abstract
We 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 l 1) 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 l 21 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.
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This research is partially supported by NSERC grants. I would like to thank Steven Watson for his extensive help on this paper.
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Edmonds, J. Embedding into l 2∞ Is Easy, Embedding into l 3∞ Is NP-Complete. Discrete Comput Geom 39, 747–765 (2008). https://doi.org/10.1007/s00454-008-9064-z
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DOI: https://doi.org/10.1007/s00454-008-9064-z