Dynamic Dictionaries in Constant Worst-Case Time

We introduce a technique to maintain a set of n elements from a universe of size u with membership and indel operations, so that elements are associated r-bit satellite data. We achieve constant worst-case time for all the operations, at the price of spending u + o(u) + O(nr + n log log log u) bits of space. Only the variant where the space is of the form O(nr + n log u) was exhaustively explored before, yet in that case existing lower bounds prevent achieving constant worst-case times. As a byproduct, we improve a folklore data structure for initializing an array of n elements in constant time, by reducing its space requirement from 2n log n to n+o(n) bits.