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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 ( nlg 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 ( nlglg u n + nlg n t+1) bits when supporting only perfect-hashing queries, and it is ( nlg u n + nlg n t+1) bits when also supporting membership queries. All upper bounds are new, as is the ( nlg n t+1) lower bound.

De Dictionariis Dynamicis Pauco Spatio Utentibus