A deterministic skip list for k-dimensional range search

Abstract

This paper presents a new data structure for multi-dimensional point data which is based on an extension of the deterministic skip list data structure projected into k dimensions. The structure is labeled the k-d Range Deterministic Skip List and it supports fast insertions, deletions, and range search. The k-d Range Deterministic Skip List is optimal (i.e. $\Theta (\lg^{{\rm k}}n + $ t) to locate and report t of n data points in range) for k-dimensional range search, assuming that our data points are elements of a commutative semigroup with set union as the commutative and associative addition operation. A dynamic data structure is defined to be optimally balanced if the product of its worst case cost functions for k-dimensional range search, insertion, deletion, storage, and preprocessing is minimal. The k-d Range Deterministic Skip List is found to be optimally balanced based on this definition.