Simple Algorithms for the On-Line Multidimensional Dictionary and Related Problems

Abstract

The on-line multidimensional dictionary problem consists of executing \mbox{on-line} any sequence of the following operations: INSERT($p$), DELETE($p$) and MEMBERSHIP($p$), where $p$ is any (ordered) $d$-tuple (or string with $d$ elements, or points in $d$-space where the dimensions have been ordered). We introduce a clean structure based on balanced binary search trees, which we call multidimensional balanced binary search trees, to represent the set of $d$-tuples. We present algorithms for each of the above operations that take $O(d + \log n)$ time, where $n$ is the current number of $d$-tuples in the set, and each INSERT and DELETE operation requires no more than a constant number of rotations. Our structure requires $dn$ words to represent the input, plus $O(n)$ pointers and data indicating the first component where pairs of $d$-tuples differ. This information, which can be easily updated, enables us to test for MEMBERSHIP efficiently. Other operations that can be performed efficiently in our multidimensional balanced binary search trees are: print in lexicographic order ($O(dn)$ time), find the (lexicographically) smallest or largest $d$-tuple ($O(\log n)$ time), and concatenation ($O(d + \log n)$ time). Finding the (lexicographically) $k^{th}$ smallest or largest $d$-tuple can also be implemented efficiently ($O(\log n)$ time), at the expense of adding an integer value at each node.