Embedding into l ∞2 Is Easy, Embedding into l ∞3 Is NP-Complete
Discrete & Computational Geometry
Geometry of Cuts and Metrics
| Hi-index | 5.23 |
In this paper, we present an optimal O(n^2) time algorithm for deciding whether a metric space (X,d) on n points can be isometrically embedded into the plane endowed with the l"1-metric. It improves the O(n^2log^2n) time algorithm of Edmonds (2008) [9]. Together with some ingredients introduced by Edmonds, our algorithm uses the concept of tight span and the injectivity of the l"1-plane. A different O(n^2) time algorithm was recently proposed by Eppstein (2009) [10].