Abstract

We consider the following problem: Preprocess a set S of n axis-parallel boxes in Rd so that given a query with an axis-parallel box in Rd, the pairs of boxes of S whose intersection intersects the query box can be reported efficiently. For the case that d=2, we present a data structure of size O(nlog⁡n) supporting O(log⁡n+k) query time, where k is the size of the output. This improves the previously best known result by de Berg et al. which requires O(log⁡n+klog⁡n) query time using O(nlog⁡n) space. There has been no result known for this problem for higher dimensions, except that for d=3, the best known data structure supports O(nlog2⁡n+klog2⁡n) query time using O(nnlog⁡n) space. For a fixed dimension d>2, we present a data structure supporting O(n1−δlogd−1⁡n+klogd−1⁡n) query time for any constant 0<δ<1. The size of the data structure is O(nδd−2δ+1log⁡n).

Highlights

  • Range searching is one of the fundamental problems, which has been studied extensively in computational geometry [2]

  • We present a data structure of size O(n log n) that supports O(log n + k) query time for queries of axis-parallel rectangles

  • We present data structures and their query algorithms to find a set of canonical nodes of (i, j, Q) with I(i, j) ∩ Q = ∅ for a query rectangle Q

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Summary

Introduction

Range searching is one of the fundamental problems, which has been studied extensively in computational geometry [2]. For axis-parallel rectangles in the plane, de Berg et al [6] presented a data structure of size O(n log n) that supports O(log n log∗ n + k log n) query time. Given a set P of points in R2, the 2D orthogonal range reporting problem asks to preprocess them so that given a query of an axis-parallel rectangle, the points of P contained in the query rectangle can be reported To solve this problem using the data structure for our problem, we map each point p in P to two points lying on p (two degenerate boxes). Due to lack of space, some of the proofs and details are omitted

Planar Case
Configurations of Two Intersecting Rectangles
Data Structures
Reporting C5-pairs
Finding all Canonical Nodes for C5-pairs
Handling Each Canonical Node to Find All C5-pairs
Higher Dimensional Case

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