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

TL;DR

This paper analyzes the VC-dimension, combinatorial dimension, and primal shatter dimension of two space decomposition techniques for hyperplane arrangements, revealing significant gaps and improving point location query bounds.

Contribution

It provides a detailed comparison of dimensions for bottom-vertex triangulation and vertical decomposition, and improves point location query time bounds using vertical decomposition.

Findings

Vertical decomposition has a combinatorial dimension of 2d and VC-dimension between 1 + d(d+1)/2 and O(d^3).

Bottom-vertex triangulation's primal shatter dimension and combinatorial dimension are Θ(d^2), with VC-dimension between d(d+1) and 5d^2 log d.

Point location query time can be improved to O(d^3 log n) using vertical decomposition, with tradeoffs in preprocessing and storage.

Abstract

$\renewcommand{\Re}{\mathbb{R}}$ We re-examine parameters for the two main space decomposition techniques---bottom-vertex 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 $O(d^3)$. For bottom-vertex triangulation, both the primal shatter dimension and the combinatorial dimension are $\Theta(d^2)$, but there seems to be a significant gap between them, as the combinatorial dimension is $\frac12d(d+3)$, whereas the primal shatter dimension is at most $d(d+1)$, and the VC-dimension is between $d(d+1)$ and $5d^2 \log{d}$ (for $d\ge 9$). Our main application is to point location in an arrangement of $n$ hyperplanes is $\Re^d$, 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 $O(d^3\log n)$, instead of $O(d^4\log d\log n)$ in Meiser's algorithm. For these bounds to hold, the preprocessing and storage are rather large (super-exponential 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).