## Abstract

Given*n* hyperplanes in*E*
^{d}, a (1/*r*)-*cutting* is a collection of simplices with disjoint interiors, which together cover*E*
^{d} and such that the interior of each simplex intersects at most*n/r* hyperplanes. We present a deterministic algorithm for computing a (1/*r*)-cutting of*O*(*r*
^{d}) size in*O*(*nr*
^{d−1}) time. If we require the incidences between the hyperplanes and the simplices of the cutting to be provided, then the algorithm is optimal. Our method is based on a hierarchical construction of cuttings, which also provides a simple optimal data structure for locating a point in an arrangement of hyperplanes. We mention several other applications of our result, e.g., counting segment intersections, Hopcroft's line/point incidence problem, linear programming in fixed dimension.

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This research was supported in part by the National Science Foundation under Grant CCR-9002352.

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Chazelle, B. Cutting hyperplanes for divide-and-conquer.
*Discrete Comput Geom* **9**, 145–158 (1993). https://doi.org/10.1007/BF02189314

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DOI: https://doi.org/10.1007/BF02189314