Lower bounds for intersection searching and fractional cascading in higher dimension

Authors:
Bernard Chazelle;Ding Liu
Affiliations:
NEC Research Institute, Princeton University, Princeton, NJ and Department of Computer Science, Princeton University, Princeton, NJ;Department of Computer Science, Princeton University, Princeton, NJ
Venue:
Journal of Computer and System Sciences - STOC 2001
Year:
2004

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Abstract

Given an n-edge convex subdivision of the plane, is it possible to report its k intersections with a query line segment in O(k + polylog(n)) time, using subquadratic storage? If the query is a plane and the input is a polytope with n vertices, can one achieve O(k + polylog(n)) time with subcubic storage? Does any convex polytope have a boundary dominant Dobkin-Kirkpatrick hierarchy? Can fractional cascading be generalized to planar maps instead of linear lists? We prove that the answer to all of these questions is no, and we derive near-optimal solutions to these classical problems.