Dynamic 3-sided Planar Range Queries with Expected Doubly Logarithmic Time⋆ Gerth Stølting Brodal1, Alexis C. Kaporis2, Apostolos N. Papadopoulos4, Spyros Sioutas3, Konstantinos Tsakalidis1, Kostas Tsichlas4 1 MADALGO⋆⋆, Department of Computer Science, Aarhus University, Denmark gerth,tsakalid @madalgo.au.dk 2 { } Computer Engineering and Informatics Department, University of Patras, Greece
[email protected] 3 Department of Informatics, Ionian University, Corfu, Greece
[email protected] 4 Department of Informatics, Aristotle University of Thessaloniki, Greece
[email protected] Abstract. This work studies the problem of 2-dimensional searching for the 3-sided range query of the form [a,b] ( , c] in both main × −∞ and external memory, by considering a variety of input distributions. We present three sets of solutions each of which examines the 3-sided problem in both RAM and I/O model respectively. The presented data structures are deterministic and the expectation is with respect to the input distribution: (1) Under continuous µ-random distributions of the x and y coordinates, we present a dynamic linear main memory solution, which answers 3- sided queries in O(log n + t) worst case time and scales with O(log log n) expected with high probability update time, where n is the current num- ber of stored points and t is the size of the query output. We external- ize this solution, gaining O(logB n + t/B) worst case and O(logB logn) amortized expected with high probability I/Os for query and update operations respectively, where B is the disk block size. (2)Then, we assume that the inserted points have their x-coordinates drawn from a class of smooth distributions, whereas the y-coordinates are arbitrarily distributed.