Polygonal Intersection Searching

Herbert Edelsbrunner, Hermann A. Maurer, David G. Kirkpatrick

Abstract: We give a lower bound on the following problem, known as simplex range reporting: Given a collection P of n points in d-space and an arbitrary simplex q, find all the points in P ∩ q. It is understood that P is fixed and can be preprocessed ahead of time, while q is a query that must be answered on-line. We consider data structures for this problem that can be modeled on a pointer machine and whose query time is bounded by O(nδ + r), where r is the number of points to be reported and δ is an arbitrary fixed real. We prove that any such data structure of that form must occupy storage Ω(nd(1 − δ)− ε)<math><mtext>Ω(n</mtext><msup><mi></mi><mn>d(1 − δ)− ε</mn></msup><mtext>)</mtext></math>, for any fixed ε > 0. This lower bound is tight within a factor of nε.