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Efficient Heuristics for Orientation Metric and Euclidean Steiner Tree Problems

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  • Y.Y. Li
  • K.S. Leung
  • C.K. Wong

Abstract

We consider Steiner minimum trees (SMT) in the plane, where only orientations with angle $${\sigma }$$ , 0 ≤ i ≤ σ − 1 and σ an integer, are allowed. The orientations define a metric, called the orientation metric, λσ, in a natural way. In particular, λ2 metric is the rectilinear metric and the Euclidean metric can beregarded as λ∞ metric. In this paper, we provide a method to find an optimal λσ SMT for 3 or 4 points by analyzing the topology of λσ SMT's in great details. Utilizing these results and based on the idea of loop detection first proposed in Chao and Hsu, IEEE Trans. CAD, vol. 13, no. 3, pp. 303–309, 1994, we further develop an O(n2) time heuristic for the general λσ SMT problem, including the Euclidean metric. Experiments performed on publicly available benchmark data for 12 different metrics, plus the Euclidean metric, demonstrate the efficiency of our algorithms and the quality of our results.

Suggested Citation

  • Y.Y. Li & K.S. Leung & C.K. Wong, 2000. "Efficient Heuristics for Orientation Metric and Euclidean Steiner Tree Problems," Journal of Combinatorial Optimization, Springer, vol. 4(1), pages 79-98, March.
    • Handle: RePEc:spr:jcomop:v:4:y:2000:i:1:d:10.1023_a:1009837006569
    DOI: 10.1023/A:1009837006569
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    References listed on IDEAS

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    1. Beasley, J. E., 1992. "A heuristic for Euclidean and rectilinear Steiner problems," European Journal of Operational Research, Elsevier, vol. 58(2), pages 284-292, April.
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