Selection by rank in K-dimensional binary search trees

AbstractIn this work we show how to augment general purpose multidimensional data structures, such as K-d trees, to efficiently support search by rank that is, to locate the i-th smallest element along the j-th coordinate, for given i and j and to find the rank of a given item along a given coordinate. To do so, we introduce two simple, practical and very flexible algorithms - Select-by-Rank and Find-Rank - with very little overhead. Both algorithms can be easily implemented and adapted to several spatial indexes, although their analysis is far from trivial. We are able to show that for random K-d trees of size n the expected number of nodes visited by Find-Rank is Pn,i=ï n1-1/K for i=on or i=n-on, and Pn,i=fKi/nï nα+onα for i=xn+on with 0