Extending range queries and nearest neighbors

Given an initial rectangular range or k nearest neighbor (k-nn) query (using the L1 metric), we consider the problems of incrementally extending the query by increasing the size of the range, or by increasing k, and reporting the new points incorporated by each extension. Although both problems may be solved trivially by repeatedly applying a traditional range query or L1 k-nn algorithm, such solutions do not minimize the overall time to process all extensions. We present algorithms that obtain e cient overall query times by performing novel searches of multiple range trees and extending k-nn trees, a new data structure introduced here. In two dimensions, when queries eventually incorporate (N) points or require E = (N) extensions, the overall retrieval time of our algorithms is O(E + N), which is optimal. Our extending L1 k-nn algorithm immediately provides a new solution to the traditional L1 k-nn problem, improving upon previous results. Our search techniques and data structures generalize to algorithms for extending xed polytope range queries and extending k-nn using polytope distance functions. In two dimensions, under the same conditions as above, these algorithms also have optimal overall extension time.