Alcom-FT Technical Report Series ALCOMFT-TR-01-35 New Data Structures for Orthogonal Range Searching Stephen Alstrup∗ The IT University of Copenhagen Glentevej 67, DK-2400, Denmark
[email protected] Gerth Stølting Brodal† BRICS ‡ Department of Computer Science University of Aarhus Ny Munkegade, DK-8000 Arhus˚ C, Denmark
[email protected] Theis Rauhe§ The IT University of Copenhagen Glentevej 67, DK-2400, Denmark
[email protected] 5th April 2001 Abstract We present new general techniques for static orthogonal range searching problems in two and higher dimensions. For the general range reporting problem in R3, we achieve ε query time O(log n + k) using space O(n log1+ n), where n denotes the number of stored points and k the number of points to be reported. For the range reporting problem on ε an n n grid, we achieve query time O(log log n + k) using space O(n log n). For the two-dimensional× semi-group range sum problem we achieve query time O(log n) using space O(n log n). ∗Partially supported by a grant from Direktør Ib Henriksens Fond. †Partially supported by the IST Programme of the EU under contract number IST-1999-14186 (ALCOM- FT). ‡Basic Research in Computer Science, Center of the Danish National Research Foundation. §Partially supported by a grant from Direktør Ib Henriksens Fond. Part of the work was done while the author was at BRICS. 1 1 Introduction Let P be a finite set of points in Rd and Q a query range in Rd. Range searching is the problem of answering various types of queries about the set of points which are contained within the query range, i.e., the point set P Q.