Embedding into l 2 ∞ is Easy , Embedding into l 3 ∞ is NP - Complete
نویسنده
چکیده
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 R Cartesian space preserving their l∞ (or l1) metric distances. Its expected time is O(n 2 log n) (i.e. within a poly-log of the size of the input) beating the previous O(n) algorithm. In contrast, we prove that detecting l ∞ embeddings is NP-complete. The problem is also NP-complete within l 1 or l ∞ with the added constraint that the locations of two of the points are given or alternatively that the two dimension are curved into a 3-dimensional sphere. We also refute a compaction theorem by giving a metric space that cannot be embedded in l ∞ , however, can be if any single point is removed. ∗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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