Planar Point Location in Sublogarithmic Time

We consider the static planar point location problem in an arbitrary polygonal subdivision given by n segments. We assume points come from the [u]² grid, and consider algorithms for the RAM with words of O(lg u) bits. We give the first solution to the problem which can surpass the traditional query time of O(lgn). Specifically, we can obtain a query time of O(radic(lg u)). Though computational geometry on a grid has been investigated for a long time (including for this problem), it is generally not known how to make good use of a bounded universe in problems of such nonorthogonal flavor. Our result shows this limitation can be surpassed, at least for planar point location. A result by Timothy Chan, appearing independently in FOCS´06, also achieves sublogarithmic query times. Combining the two results, we obtain the following bound. For any S ges 2, the exists a data structure using space O(n middot S) which supports queries in time: O(min {((lg n)/(lg lg n)), (radic((lg u)/(lg lg u))), ((lg u)/(lg S))})