On two-dimensional indexability and optimal range search indexing

Abstract

In this paper we settle several longstanding open problems in theory of indexability and external orthogonal range searching. In the rst part of the paper, we apply the theory of indexability to the problem of two-dimensional range searching. We show that the special case of 3-sided querying can be solved with constant redundancy and access overhead. From this, we derive indexing schemes for general 4-sided range queries that exhibit an optimal tradeo between redundancy and access overhead. In the second part of the paper, we develop dynamic external memory data structures for the two query types. Our structure for 3-sided queries occupies O(N=B) disk blocks, and it supports insertions and deletions in O(log B N) I/Os and queries in O(log B N + T=B) I/Os, where B is the disk block size, N is the number of points, and T is the query output size. These bounds are optimal. Our structure for general (4-sided) range searching occupies O (N=B)(log(N=B))= log log B N disk blocks and answers queries in O(log B N + T=B) I/Os, which are optimal. It also supports updates in O (log B N)(log(N=B))= log log B N I/Os. Center for Geometric Computing, Department of Computer Science, Duke University, Box 90129, Durham, NC 27708{0129. Supported in part by the U.S. Army Research O ce through MURI grant DAAH04{96{1{0013 and by the National Science Foundation through ESS grant EIA{9870734. Part of this work was done while visiting BRICS, Department of Computer Science, University of Aarhus, Denmark. Email: large@cs.duke.edu. yDepartment of Computer Sciences, University of Texas at Austin, Austin, TX 78712-1188. Email vsam@cs.utexas.edu zCenter for Geometric Computing, Department of Computer Science, Duke University, Box 90129, Durham, NC 27708{0129. Supported in part by the U.S. Army Research O ce through MURI grant DAAH04{96{1{0013 and by the National Science Foundation through grants CCR{9522047 and EIA{9870734. Part of this work was done while visiting BRICS, Department of Computer Science, University of Aarhus, Denmark and I.N.R.I.A., Sophia Antipolis, France. Email: jsv@cs.duke.edu.