Embedding into the rectilinear plane in optimal O*(n^2)
Nicolas Catusse, Victor Chepoi, Yann Vax\`es
TL;DR
This paper introduces an optimal O*(n^2) time algorithm to determine if a finite metric space can be embedded into the l_1-plane, improving previous algorithms in efficiency.
Contribution
It presents a new optimal algorithm for embedding metric spaces into the l_1-plane, utilizing the concepts of tight span and injectivity, surpassing prior methods in speed.
Findings
Achieves optimal O*(n^2) time complexity
Improves upon previous O*(n^2 log^2 n) algorithm
Utilizes tight span and injectivity concepts
Abstract
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).
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