Randomized fully dynamic graph algorithms with polylogarithmic time per operation
Journal of the ACM (JACM)
Approximation algorithm for embedding metrics into a two-dimensional space
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Circular partitions with applications to visualization and embeddings
Proceedings of the twenty-fourth annual symposium on Computational geometry
Inapproximability for planar embedding problems
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
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We give a new randomized 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. Our expected time is O(n2 log3 n) (i.e. within a poly-log of the size of the input) beating the previous O(n3) algorithm. In contrast, we prove that detecting l3∞ embeddings is NP-complete. The problem is also NP-complete within l21 or l2∞ 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 l3∞, however, can be if even one point is removed.