Abstract
We present optimal deterministic algorithms for constructing shallow cuttings in an arrangement of lines in two dimensions or planes in three dimensions. Our results improve the deterministic polynomial-time algorithm of Matousek (Comput Geom 2(3):169---186, 1992) and the optimal but randomized algorithm of Ramos (Proceedings of the Fifteenth Annual Symposium on Computational Geometry, SoCG'99, 1999). This leads to efficient derandomization of previous algorithms for numerous well-studied problems in computational geometry, including halfspace range reporting in 2-d and 3-d, k nearest neighbors search in 2-d, $$({\le }k)$$(≤k)-levels in 3-d, order-k Voronoi diagrams in 2-d, linear programming with k violations in 2-d, dynamic convex hulls in 3-d, dynamic nearest neighbor search in 2-d, convex layers (onion peeling) in 3-d, $$\varepsilon $$?-nets for halfspace ranges in 3-d, and more. As a side product we also describe an optimal deterministic algorithm for constructing standard (non-shallow) cuttings in two dimensions, which is arguably simpler than the known optimal algorithms by Matousek (Discrete Comput Geom 6(1):385---406, 1991) and Chazelle (Discrete Comput Geom 9(1):145---158, 1993).
Highlights
Shallow cuttings were introduced by Matoušek [25] as a tool for range searching, halfspace range reporting
Matoušek [24] improved the deterministic time bound to O(nr) for d = 2, which is optimal if the algorithm is required to output the conflict lists of all the cells (since the worst-case total size of the conflict lists is Matoušek’s later paper described a deterministic O(nrd−1)-time algorithm for any constant dimension d, which is again optimal if we need to output all conflict lists, but this result holds under the restriction that r is not too big, i.e., r < n1−δ for some constant δ > 0
Tsakalidis hierarchy is useful in certain applications.) Recently, at SODA’14, Afshani and Tsakalidis [3] managed to achieve the same bound deterministically, albeit only for an orthogonal variant of the problem where the input objects are orthants in R3; subsequently, Afshani et al [2] improved the time bound for a single shallow cutting to O(n log log n) in the word RAM model
Summary
Shallow cuttings were introduced by Matoušek [25] as a tool for range searching, halfspace range reporting. They have since found applications to numerous other central problems in computational geometry, including (≤ k)-levels in arrangements of hyperplanes, order-k Voronoi diagrams, linear programming with k violations, dynamic convex hulls, and dynamic nearest neighbor search (see Section 1.4 for more information). At SoCG’99, Ramos [29] presented an optimal randomized algorithm for constructing shallow cuttings in two and three dimensions. Chan and Konstantinos Tsakalidis; licensed under Creative Commons License CC-BY 31st International Symposium on Computational Geometry (SoCG’15). Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
