Space-Time Tradeoffs for Emptiness Queries
نویسنده
چکیده
We develop the rst 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 de nes 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 satis es the inequality std = ((n= logn)d (d 1)=(d+1)). Using di erent 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 o ine result, we show that for online hyperplane emptiness queries, (nd=polylog n) space is required to achieve polylogarithmic query time, and (n(d 1)=d=polylogn) query time is required if only O(npolylog n) space is available. These two lower bounds are optimal up to polylogarithmic factors. For two-dimensional queries, we obtain an optimal continuous tradeo st2 = (n2) between these two extremes. Finally, using a lifting argument, we show that the same lower bounds hold for both o ine and online halfspace emptiness queries in Rd(d+3)=2 .
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عنوان ژورنال:
- SIAM J. Comput.
دوره 29 شماره
صفحات -
تاریخ انتشار 2000