Succinct Dynamic Dictionaries and Trees

We consider space-efficient solutions to two dynamic data structuring problems. We first give a representation of a set S ⊆ U = {0,..., m-1}, |S| = 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 B+o(B) bits, where B = ⌈lg (m n)⌉ is the information-theoretic minimum space to represent S. This improves upon the O(B)-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+Ɛ) amortised time, for any constant Ɛ > 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(lgc n) time, for some c ≥ 1.