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 *(ρ k) where ρ≈1.357 and mention some consequences of this result.
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W. Kern supported by BRICKS. X. Wang supported by Netherlands Organization for Scientific Research (NWO) grant 613.000.322 (Exact Algorithms).
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License ( https://creativecommons.org/licenses/by-nc/2.0 ), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Fuchs, B., Kern, W. & Wang, X. The Number of Tree Stars Is O *(1.357k). Algorithmica 49, 232–244 (2007). https://doi.org/10.1007/s00453-007-9020-y
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DOI: https://doi.org/10.1007/s00453-007-9020-y