Space-time tradeoffs for emptiness queries (extended 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.

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