Low-Distortion Embeddings of Trees

  • Robert Babilon
  • Jiří Matoušek
  • Jana Maxová
  • Pavel Valtr
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2265)

Abstract

We prove that every tree T=(V, E) on n vertices can be embedded in the plane with distortion \( O(\sqrt n )\) that is, we construct a mapping f: VR2 such that \( \rho (u,\upsilon ) \leqslant \parallel f(u) - f(\upsilon )\parallel \leqslant O(\sqrt n ) \cdot \rho (u,\upsilon )\) for every u, υV, where ρ(u, υ) denotes the length of the path from u to υ in T (the edges have unit lengths).The embedding is described by a simple and easily computable formula.This is asymptotically optimal in the worst case. We also prove several related results.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Robert Babilon
    • 1
  • Jiří Matoušek
    • 1
  • Jana Maxová
    • 1
  • Pavel Valtr
    • 1
  1. 1.Department of Applied Mathematics and Institute for Theoretical Computer Science (ITI)Charles UniversityPragueCzech Republic

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