Embedding into the rectilinear plane in optimal O*(n^2)

Nicolas Catusse (1) , Victor Chepoi (1) , Yann Vaxès (1)

1 LIF - Laboratoire d'informatique Fondamentale de Marseille

We present an optimal O*(n^2) time algorithm for deciding if 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^2 log^2 n) time algorithm of J. Edmonds (2008). Together with some ingredients introduced by J. 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 D. Eppstein (2009).

Domaines: Géométrie algorithmique [cs.CG]; Algorithme et structure de données [cs.DS]

Nicolas Catusse, Victor Chepoi, Yann Vaxès. Embedding into the rectilinear plane in optimal O*(n^2). Theoretical Computer Science, 2011, 412 (22), pp.2425-2433. ⟨10.1016/j.tcs.2011.01.038⟩. ⟨hal-00949183⟩