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We consider the problem of maintaining dynamically a set of points in the plane and supporting range queries of the type [a,b]×( ? ? , c]. We assume that the inserted points have their x-coordinates drawn from a class of smooth distributions, whereas the y-coordinates are arbitrarily distributed. The points to be deleted are selected uniformly at random among the inserted points. For the RAM model, we present a linear space data structure that supports queries in O(loglogn + t) expected time with high probability and updates in O(loglogn) expected amortized time, where n is the number of points stored and t is the size of the output of the query. For the I/O model we support queries in O(loglog B n + t/B) expected I/Os with high probability and updates in O(log B logn) expected amortized I/Os using linear space, where B is the disk block size. The data structures are deterministic and the expectation is with respect to the input distribution.

Dynamic 3-Sided Planar Range Queries with Expected Doubly Logarithmic Time