Rank Selection in Multidimensional Data

Abstract

Abstract. Suppose we have a set of K-dimensional records stored in a general purpose spatial index like a K-d tree. The index eciently supports insertions, ordinary exact searches, orthogonal range searches, nearest neighbor searches, etc. Here we consider whether we can also eciently support search by rank, that is, to locate the i-th smallest element along the j-th coordinate. We answer this question in the armative by developing a simple algorithm with expected cost O(n (1=K) log n), where n is the size of the K-d tree and (1=K) < 1 for any K 2. The only requirement to support the search by rank is that each node in the K-d tree stores the size of the subtree rooted at that node (or some equivalent information). This is not too space demanding. Furthermore, it can be used to randomize the update algorithms to provide guarantees on the expected performance of the various operations on K-d trees. Although selection in multidimensional data can be solved more eciently than with our algorithm, those solutions will rely on ad-hoc data structures or superlinear space. Our solution adds to an existing data structure (Kd trees) the capability of search by rank with very little overhead, and it can be easily adapted to other spatial indexes as well. The simplicity of the algorithm makes it easy to implement, practical and very exible; however, its correctness and eciency are far from self-evident.

A limit process for partial match queries in random quadtrees and $2$-d trees

“…First, these multidimensional trees are data structures of choice for applications that range from collision detection in motion planning to mesh generation [22,41]. Furthermore, the cost of partial match queries also appears in (hence influences) the complexity of a number of other geometrical search questions such as range search [12] or rank selection [11]. For general references on multidimensional data structures and more details about their various applications, see the series of monographs by Samet [38][39][40].In this paper, we provide refined analyses of the costs of partial match queries in some of the most important two dimensional data structures.…”

Partial match queries in random quadtrees

“…We stress that this equation does not only hold true pointwise for fixed s but also as cádlág functions on the unit interval. The relation in (8) is the fundamental equation for us.…”

“…Letting n → ∞ (formally) in (8) suggests that, if n −β C n (s) does converge to a random variable Z(s) in a sense to be precised, then the distribution of the process (Z(s), 0 ≤ s ≤ 1) should satisfy the following fixed point equation…”

“…For general references on multidimensional data structures and more details about their various applications, see the series of monographs by Samet [32,33,34]. The cost of partial match queries also appears in (hence influences) the complexity of a number of other geometrical search questions such as range search [6] or rank selection [8].…”

Partial match queries in random quadtrees

“…For general references on multidimensional data structures and more details about their various applications, see the series of monographs by Samet [32,33,34]. The cost of partial match queries also appears in (hence influences) the complexity of a number of other geometrical search questions such as range search [6] or rank selection [8].…”