An Optimal Choice Dictionary
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A choice dictionary is a data structure that can be initialized with a parameter n∈{1,2,...} and subsequently maintains an initially empty subset S of {1,...,n} under insertion, deletion, membership queries and an operation 'choice' that returns an arbitrary element of S. The choice dictionary is fundamental in space-efficient computing and has numerous applications. The best previous choice dictionary can be initialized with n and a second parameter t∈{1,2,...} in constant time and subsequently executes all operations in O(t) time and occupies n+O(n(t/w)^t+ n) bits on a word RAM with a word length of w=Ω( n) bits. We describe a new choice dictionary that, following a constant-time initialization, executes all operations in constant time and, in addition to the space needed to store n, occupies only n+1 bits, which is shown to be optimal if w=o(n). Allowed _2(n+1) bits of additional space, the new data structure also supports iteration over the set S in constant time per element.
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