Showing papers by "Jeff Erickson published in 1997"

Erratum to Better Lower Bounds on Detecting Affine and Spherical Degeneracies.

Abstract: We show that in the worst case, /spl Omega/(n/sup d/) sidedness queries are required to determine whether a set of n points in R/sup d/ is affinely degenerate, i.e., whether it contains d+1 points on a common hyperplane. This matches known upper bounds. We give a straightforward adversary argument, based on the explicit construction of a point set containing /spl Omega/(n/sup d/) \"collapsible\" simplices, any one of which can be made degenerate without changing the orientation of any other simplex. As an immediate corollary, we have an /spl Omega/(n/sup d/) lower bound on the number of sidedness queries required to determine the order type of a set of n points in R/sup d/. Using similar techniques, we also show that /spl Omega/(n/sup d+1/) in-sphere queries are required to decide the existence of spherical degeneracies in a set of n points in R/sup d/. >

Space-time tradeoffs for emptiness queries

Abstract: We develop the first nontrivial lower bounds on the complexity of online hyperplane and halfspace emptiness queries. Our lower bounds apply to a general class of geometric range query data structures called partition graphs. Informally, a partition graph is a directed acyclic graph that describes a recursive decomposition of space. We show that any partition graph that supports hyperplane emptiness queries implicitly defines a halfspace range query data structure in the Fredman/Yao semigroup arithmetic model, with the same asymptotic space and time bounds. Thus, results of Bronnimann, Chazelle, and Pach imply that any partition graph of size s that supports hyperplane emptiness queries in time t satisfies the inequality $st^d = \Omega((n/\log n)^{d - (d-1)/(d+1)})$. Using different techniques, we improve previous lower bounds for Hopcroft's problem---Given a set of points and hyperplanes, does any hyperplane contain a point?---in dimensions four and higher. Using this offline result, we show that for online hyperplane emptiness queries, $\Omega(n^d/{\mbox{ polylog }} n)$ space is required to achieve polylogarithmic query time, and $\Omega(n^{(d-1)/d}/{\mbox{ polylog }} n)$ query time is required if only O(n polylog n) space is available. These two lower bounds are optimal up to polylogarithmic factors. For two-dimensional queries, we obtain an optimal continuous tradeoff $st^2=\Omega(n^2)$ between these two extremes. Finally, using a lifting argument, we show that the same lower bounds hold for both offline and online halfspace emptiness queries in ${\mathbb{R}}^{d(d+3)/2}$.

Space-time tradeoffs for emptiness queries (extended abstract)

Abstract: We present the fist nontrivial spat-time tradeoff lower bounds for hyperplane and halfspaceemptinessqueries. Our lower bounds apply to a general class of geometric range query data structures called partition graphs. Informally, a partition graph is a directed acyclic graph that describes a recursive decomposition of space. We show that any partition graph that supports hyperplane emptiness queries implicitly deiines a halfspace range query data structure in the Fredman/Yao semigroup arithmetic model, with the same space and time bounds. Thus, results of Bronnimann, Chazelle, and Path imply that any partition graph of size s that supports hyperplane emptiness queries in time t must satisfy the inequality st d = ~((n/logn)*–~d–Tl/ld+l)]. Using difterent techniques, we show that fl(nd/ polylog n) preprocessing time is required to achieve polylogarithmic query time, and that Cl(n('–1 " d/ polylogn) query time is required if only O(n polylog n) preprocessing time is used. These two lower bounds are optimal up to polylogarithrnic factors. For twcdimensional queries, we obtain an optimal continuous tradeoff between these two extremes. Finally, using a reduction argument, we show that the same lower bounds hold for halfspace emptiness queries in R.d('+3)'2 on a restricted class of partition graphs.