Succinct Dynamic Dictionaries and Trees

Abstract

We consider space-efficient solutions to two dynamic data structuring problems. We first give a representation of a set $S \subseteq U = \left\{ {0,...,m - 1} \right\},\left| S \right| = n$ that supports membership queries in O(1) worst case time and insertions into/deletions from S in O(1) expected amortised time. The representation uses $\mathcal{B} + o\left( \mathcal{B} \right)$ bits, where $\mathcal{B} = \left\lceil {\lg \left( \begin{gathered} m \hfill \\ n \hfill \\ \end{gathered} \right)} \right\rceil$ is the information-theoretic minimum space to represent S. This improves upon the O($\left( \mathcal{B} \right)$)-bit solutions of Brodnik and Munro [2] and Pagh [16], and uses up to a log-factor less space than search trees or hash tables. The representation can also associate satellite data with elements of S.

We also show that a binary tree on n nodes, where each node has b = O(lg n)-bit data stored at it, can be maintained under node insertions while supporting navigation in O(1) time and updates in O((lg lg n)^1+∈) amortised time, for any constant ∈ > 0. The space used is within o(n) bits of the information-theoretic minimum. This improves upon the equally space-efficient structure of Munro et al. [15], in which updates take O(lg^c n) time, for some c ≥ 1.

Supported in part by UISTRF project 2001.04/IT and EPSRC GR L/92150.