Geometric Searching over the Rationals

We revisit classical geometric search problems under the assumption of rational coordinates. Our main result is a tight bound for point separation, ie, to determine whether n given points lie on one side of a query line. We show that with polynomial storage the query time is Θ(log b/ log log b), where b is the bit length of the rationals used in specifying the line and the points. The lower bound holds in Yao’s cell probe model with storage in nO(1) and word size in bO(1) . By duality, this provides a tight lower bound on the complexity on the polygon point enclosure problem: given a polygon in the plane, is a query point in it?

• Type: Conference
• Year: 1999
• Where: ESA
• Authors: Bernard Chazelle
• Keywords: Algorithms, ESA 1999, Lower Bound, Tight Bound, Tight Lower Bound