Efficient dynamic algorithms for some geometric intersection problems

Abstract

Efficient dynamic algorithms are presented for four geometric intersection problems. These include: (1) reporting the subset of a set S of nonintersecting segments in the plane that are intersected by a query segment of fixed slope; (2) reporting the segments of S that are intersected by a query segment whose supporting line passes through a fixed point; (3) the orthogonal segment intersection search problem; and (4) the point enclosure problem for isothetic hyper-rectangles in d-dimensional space, d ⩾ 2. These algorithms are the first linear-space solutions to Problems 1–3 and for Problem 4 in the plane. For Problem 3, the update time improves upon the previous best algorithm by a logarithmic factor without affecting the query time. For Problem 4, the update time either matches or improves upon the previous best algorithms, but the query time is larger by a logarithmic factor. No dynamic algorithms were known previously for Problems 1 and 2.