Sharp Bounds for Vertical Decompositions of Linear Arrangements in Four Dimensions

Authors:
Vladlen Koltun
Affiliations:
Computer Science Division, University of California, Berkeley, CA 94720-1776, USA
Venue:
Discrete & Computational Geometry
Year:
2004

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Abstract

We prove tight and near-tight combinatorial complexity bounds for vertical decompositions of arrangements of hyperplanes and 3-simplices in four dimensions. In particular, we prove a tight upper bound of Θ(n4) for the vertical decomposition of an arrangement of n hyperplanes in four dimensions, improving the best previously known bound [8] by a logarithmic factor. We also show that the complexity of the vertical decomposition of an arrangement of n 3-simplices in four dimensions is O(n4 α (n) log2 n), where α (n) is the inverse Ackermann function, improving the best previously known bound [2] by a near-linear factor.