Abstract
We study the following problem: preprocess a set mathcal {O} of objects into a data structure that allows us to efficiently report all pairs of objects from mathcal {O} that intersect inside an axis-aligned query range {Q}. We present data structures of size O(ncdot {{mathrm{polylog,}}}n) and with query time O((k+1)cdot {{mathrm{polylog,}}}n) time, where k is the number of reported pairs, for two classes of objects in {mathbb R}^2: axis-aligned rectangles and objects with small union complexity. For the 3-dimensional case where the objects and the query range are axis-aligned boxes in {mathbb R}^3, we present a data structure of size O(nsqrt{n}cdot {{mathrm{polylog,}}}n) and query time O((sqrt{n}+k)cdot {{mathrm{polylog,}}}n). When the objects and query are fat, we obtain O((k+1)cdot {{mathrm{polylog,}}}n) query time using O(ncdot {{mathrm{polylog,}}}n) storage.
Highlights
The study of geometric data structures is an important subarea within computational geometry, and range searching forms one of the most widely studied topics within this area [4,15]
Theorem 3 Suppose we have a data structure storing a set O of n axis-aligned boxes in R3 that uses s(n) storage and that can decide in t (n) time for a given query axisaligned box Q if there is a pair of boxes from O that intersect inside Q
We presented data structures for finding intersecting pairs of objects inside a query rectangle
Summary
The study of geometric data structures is an important subarea within computational geometry, and range searching forms one of the most widely studied topics within this area [4,15]. Das et al [10] have studied the problem of preprocessing a set H of n horizontal and V of n vertical segments in the plane into a data structure such that given an axis-parallel query rectangle Q and a parameter δ, all the triples (h, v, p) where h ∈ H , v ∈ V , and p is an endpoint of either of the segments and h ∩ v ∩ Q = ∅ and dist (h ∩ v, p) δ can be reported efficiently. Abam et al [1], Gupta [16], and Gupta et al [17] have presented data structures that return the closest pair inside a query range
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