A Nearly Quadratic Bound for Point-Location in Hyperplane Arrangements, in the Linear Decision Tree Model

Abstract

We consider the point location problem in an arrangement of n arbitrary hyperplanes in any dimension d, in the linear decision tree model, in which we only count linear comparisons involving the query point, and all other operations do not explicitly access the query and are for free. We mainly consider the simpler variant (which arises in many applications) where we only want to determine whether the query point lies on some input hyperplane. We present an algorithm that performs a point location query with $$O(d^2\log n)$$ linear comparisons, improving the previous best result by about a factor of d. Our approach is a variant of Meiser’s technique for point location (Inf Comput 106(2):286–303, 1993) (see also Cardinal et al. in: Proceedings of the 24th European symposium on algorithms, 2016), and its improved performance is due to the use of vertical decompositions in an arrangement of hyperplanes in high dimensions, rather than bottom-vertex triangulation used in the earlier approaches. The properties of such a decomposition, both combinatorial and algorithmic (in the standard real RAM model), are developed in a companion paper (Ezra et al. arXiv:1712.02913 , 2017), and are adapted here (in simplified form) for the linear decision tree model. Several applications of our algorithm are presented, such as the k-SUM problem and the Knapsack and SubsetSum problems. However, these applications have been superseded by the more recent result of Kane et al. (in: Proceedings of the 50th ACM symposium on theory of computing, 2018), obtained after the original submission (and acceptance) of the conference version of our paper (Ezra and Sharir in: Proceedings of the 33rd international symposium on computational geometry, 2017). This result only applies to ‘low-complexity’ hyperplanes (for which the $$\ell _1$$ -norm of their coefficient vector is a small integer), which arise in the aforementioned applications. Still, our algorithm has currently the best performance for arbitrary hyperplanes.