The Number of Tree Stars Is O*(1.357k)

  • Authors:

  • Bernhard Fuchs;Walter Kern;Xinhui Wang

  • Affiliations:

  • University of Cologne, Center for Applied Computer Science Cologne, Group AFS, Weyertal 80, 50931, Köln, Germany;University of Twente, Department of Applied Mathematics, Faculty of EEMCS, P.O. Box 217, Weyertal 80, 7500 AE, Enschede, The Netherlands;University of Twente, Department of Applied Mathematics, Faculty of EEMCS, P.O. Box 217, Weyertal 80, 7500 AE, Enschede, The Netherlands

  • Venue:
  • Algorithmica
  • Year:
  • 2007

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Abstract

Every rectilinear Steiner tree problem admits an optimal tree T* which is composed of tree stars. Moreover, the currently fastest algorithms for the rectilinear Steiner tree problem proceed by composing an optimum tree T* from tree star components in the cheapest way. The efficiency of such algorithms depends heavily on the number of tree stars (candidate components). Fößmeier and Kaufmann (Algorithmica 26, 68–99, 2000) showed that any problem instance with k terminals has a number of tree stars in between 1.32k and 1.38k (modulo polynomial factors) in the worst case. We determine the exact bound O*(ρk) where ρ≈1.357 and mention some consequences of this result.