# Embedding into the rectilinear plane in optimal $O(n^2)$ time

Abstract : In this paper, 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 $O(n^2\log^2n)$ 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 $l_1$-plane. A different $O(n^2)$ time algorithm was recently proposed by D. Eppstein (2009).
Document type :
Journal articles
Domain :

https://hal.archives-ouvertes.fr/hal-01194732
Contributor : Yann Vaxès <>
Submitted on : Monday, September 7, 2015 - 2:27:43 PM
Last modification on : Friday, April 12, 2019 - 10:18:09 AM

### Identifiers

• HAL Id : hal-01194732, version 1

### Citation

Nicolas Catusse, Victor Chepoi, Yann Vaxès. Embedding into the rectilinear plane in optimal $O(n^2)$ time. Theoretical Computer Science, Elsevier, 2011, 412 (22), pp.2425-2433. ⟨hal-01194732⟩

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