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(n 4) 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(n 4α(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.
This work was supported by a grant from the Israeli Academy of Sciences (center of excellence).
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Koltun, V. (2001). Complexity Bounds for Vertical Decompositions of Linear Arrangements in Four Dimensions. In: Dehne, F., Sack, JR., Tamassia, R. (eds) Algorithms and Data Structures. WADS 2001. Lecture Notes in Computer Science, vol 2125. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44634-6_10
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DOI: https://doi.org/10.1007/3-540-44634-6_10
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