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A Geometric Study of the Split Decomposition

Abstract

This paper sheds new light on split decomposition theory and T-theory from the viewpoint of convex analysis and polyhedral geometry. By regarding finite metrics as discrete concave functions, Bandelt-Dress' split decomposition can be derived as a special case of more general decomposition of polyhedral/discrete concave functions introduced in this paper. It is shown that the combinatorics of splits discussed in connection with the split decomposition corresponds to the geometric properties of a hyperplane arrangement and a point configuration. Using our approach, the split decomposition of metrics can be naturally extended to distance functions, which may violate the triangle inequality, using partial split distances.

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Correspondence to Hiroshi Hirai.

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Hirai, H. A Geometric Study of the Split Decomposition. Discrete Comput Geom 36, 331–361 (2006). https://doi.org/10.1007/s00454-006-1243-1

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Keywords

  • Split Function
  • Submodular Function
  • Hyperplane Arrangement
  • Isolation Index
  • Linearity Domain