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- W2772089017 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 $dge 9$). 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 $O(d^3log n)$, instead of $O(d^4log dlog n)$ 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)." @default.
- W2772089017 created "2017-12-22" @default.
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- W2772089017 date "2017-12-08" @default.
- W2772089017 modified "2023-09-27" @default.
- W2772089017 title "Decomposing arrangements of hyperplanes: VC-dimension, combinatorial dimension, and point location" @default.
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