De Dictionariis Dynamicis Pauco Spatio Utentibus
Abstract
We develop dynamic dictionaries on the word RAM that use asymptotically optimal space, up to constant factors, subject to insertions and deletions, and subject to supporting perfect-hashing queries and/or membership queries, each operation in constant time with high probability. When supporting only membership queries, we attain the optimal space bound of Theta(n lg(u/n)) bits, where n and u are the sizes of the dictionary and the universe, respectively. Previous dictionaries either did not achieve this space bound or had time bounds that were only expected and amortized. When supporting perfect-hashing queries, the optimal space bound depends on the range {1,2,...,n+t} of hashcodes allowed as output. We prove that the optimal space bound is Theta(n lglg(u/n) + n lg(n/(t+1))) bits when supporting only perfect-hashing queries, and it is Theta(n lg(u/n) + n lg(n/(t+1))) bits when also supporting membership queries. All upper bounds are new, as is the Omega(n lg(n/(t+1))) lower bound.
Keywords
Cite
@article{arxiv.cs/0512081,
title = {De Dictionariis Dynamicis Pauco Spatio Utentibus},
author = {Erik D. Demaine and Friedhelm Meyer auf der Heide and Rasmus Pagh and Mihai Patrascu},
journal= {arXiv preprint arXiv:cs/0512081},
year = {2007}
}
Comments
14 pages. Full version of a paper accepted to LATIN'06