Efficient Data Structures for Range Searching on a Grid

We consider the 2-dimensional range searching problem in the case where all points lie on an integer grid. A new data structure is presented that solves range queries on a U * U grid in O( k + log log U) time using O( n log n) storage, where n is the number of points and k the number of reported answers. Although the query time is very good the preprocessing time of this method is very high. A second data structure is presented that can be built in time O( n log n) at the cost of an increase in query time to O(k + m). Similar techniques are used for solving the line segment intersection searching problem in a set of axis-parallel line segments on a grid. The methods presented also yield efficient structures for dominance searching and searching with half-infinite ranges that use only O(n) storage. A generalization to multi-dimensional space, using a normalization approach, yields a static solution to the general range searching problem that is better than any known solution when the dimension is at least 3. Q 1988 Academic Press, Inc.