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

In this paper, we present an optimal O(n) time algorithm for deciding if a metric space (X, d) on n points can be isometrically embedded into the plane endowed with the l1-metric. It improves the O(n 2 log n) time algorithm of J. Edmonds (2008). Together with some ingredients introduced by Edmonds, our algorithm uses the concept of tight span and the injectivity of the l1-plane. A different O(n ) time algorithm was recently proposed by D. Eppstein (2009).