Complexity Bounds for Vertical Decompositions of Linear Arrangements in Four Dimensions

Authors:
Vladlen Koltun
Affiliations:
-
Venue:
WADS '01 Proceedings of the 7th International Workshop on Algorithms and Data Structures
Year:
2001

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Abstract

We prove tight and near-tight combinatorial complexity bounds for vertical decompositions of arrangements of linear surfaces 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 [7] 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(n4α(n) log n), improving the best previously known bound [3] by a near-linear factor. We believe that the techniques used for obtaining these results can also be extended to analyze decompositions of arrangements of fixed-degree algebraic surfaces (or surface patches) in four dimensions.