Decomposing arrangements of hyperplanes: VC-dimension, combinatorial dimension, and point location

This work is motivated by several basic problems and techniques that rely on space decomposition of arrangements of hyperplanes in high-dimensional spaces, most notably Meiser’s 1993 algorithm for point location in such arrangements. A standard approach to these problems is via random sampling, in which one draws a random sample of the hyperplanes, constructs a suitable decomposition of its arrangement, and recurses within each cell of the decomposition with the subset of hyperplanes that cross the cell. The efficiency of the resulting algorithm depends on the quality of the sample, which is controlled by various parameters. One of these parameters is the classical VC-dimension, and its associated primal shatter dimension, of a suitably defined corresponding range space. Another parameter, which we refer to here as the combinatorial dimension, is the maximum number of hyperplanes that are needed to define a cell that can arise in the decomposition of some sample of the input hyperplanes; this parameter arises in Clarkson’s (and later Clarkson and Shor’s) random sampling technique. We re-examine these parameters for the two main space decomposition techniques—bottomvertex triangulation, and vertical decomposition, including their explicit dependence on the dimension d, and discover several unexpected phenomena, which show that, in both techniques, there are large gaps between the VC-dimension (and primal shatter dimension), and the combinatorial dimension. For vertical decomposition, the combinatorial dimension is only 2d, the primal shatter dimension is at most d(d+1), and the VC-dimension is at least 1+d(d+1)/2 and at most O(d3). For bottomvertex triangulation, both the primal shatter dimension and the combinatorial dimension are Θ(d2), ∗Work by Esther Ezra was partially supported by NSF CAREER under grant CCF:AF-1553354 and by Grant 824/17 from the Israel Science Foundation. Work by Sariel Har-Peled was partially supported by NSF AF awards CCF-1421231 and CCF-1217462. Work by Haim Kaplan was supported by Grant 1841/14 from the Israel Science Fund and by Grant 1161/2011 from the German Israeli Science Fund (GIF). Work by Micha Sharir has been supported by Grant 2012/229 from the U.S.-Israel Binational Science Foundation, and by Grant 892/13 from the Israel Science Foundation. Work by Haim Kaplan and Micha Sharir was also supported by the Israeli Centers for Research Excellence (I-CORE) program (center no. 4/11), by the Blavatnik Computer Science Research Fund at Tel Aviv University, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. †Dept. of Computer Science, Bar-Ilan University, Ramat Gan, Israel and School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA. [email protected]. ‡Department of Computer Science, University of Illinois, 201 N. Goodwin Avenue, Urbana, IL, 61801, USA. [email protected]. §Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv 69978 Israel. [email protected]. ¶Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv 69978 Israel. [email protected]. 1 ar X iv :1 71 2. 02 91 3v 1 [ cs .C G ] 8 D ec 2 01 7 but there seems to be a significant gap between them, as the combinatorial dimension is 12d(d+ 3), whereas the primal shatter dimension is at most d(d+1), and the VC-dimension is between d(d+1) and 5d2 log d (for d ≥ 9). Our main application is to point location in an arrangement of n hyperplanes is Rd, in which we show that the query cost in Meiser’s algorithm can be improved if one uses vertical decomposition instead of bottom-vertex triangulation, at the cost of some increase in the preprocessing cost and storage. The best query time that we can obtain is O(d3 log n), instead of O(d4 log d log n) in Meiser’s algorithm. For these bounds to hold, the preprocessing and storage are rather large (superexponential in d). We discuss the tradeoff between query cost and storage (in both approaches, the one using bottom-vertex trinagulation and the one using vertical decomposition). Our improved bounds rely on establishing several new structural properties and improved complexity bounds for vertical decomposition, which are of independent interest, and which we expect to find additional applications. The point-location methodology presented in this paper can be adapted to work in the linear decision-tree model, where we are only concerned about the cost of a query, and measure it by the number of point-hyperplane sign tests that it performs. This adaptation is presented in the companion paper [ES17], where it yields an improved bound for the linear decision-tree complexity of k-SUM. A very recent breakthrough by Kane et al. [KLM17] further improves the bound, but only for special, “low-complexity” classes of hyperplanes. We show here how their approach can be extended to yield an efficient and improved point-location structure in the RAM model for such collections of hyperplanes.