 
Summary: Optimal External Memory
Planar Point Enclosure
Lars Arge 1# , Vasilis Samoladas 2 , and Ke Yi 1#
1 Department of Computer Science, Duke University, Durham, NC 27708, USA.
{large,yike}@cs.duke.edu
2 Technical University of Crete, Greece. vsam@softnet.tuc.gr
Abstract. In this paper we study the external memory planar point en
closure problem: Given N axisparallel rectangles in the plane, construct
a data structure on disk (an index) such that all K rectangles containing
a query point can be reported I/Oe#ciently. This problem has important
applications in e.g. spatial and temporal databases, and is dual to the
important and wellstudied orthogonal range searching problem. Surpris
ingly, we show that one cannot construct a linear sized external memory
point enclosure data structure that can be used to answer a query in
O(log B N + K/B) I/Os, where B is the disk block size. To obtain this
bound,# (N/B 1# ) disk blocks are needed for some constant # > 0. With
linear space, the best obtainable query bound is O(log 2 N + K/B). To
show this we prove a general lower bound on the tradeo# between the
size of the data structure and its query cost. We also develop a family of
structures with matching space and query bounds.
