An Extendable Data Structure for Incremental Stable Perfect Hashing

We consider the problem of dynamically assigning.. elements unique indices, known as hashcodes, in the range [( 1 + o (1))n]. This problem is known as perfect hashing and is considered a fundamental building block in the design of more involved data structures. The challenge we address is that of designing a data structure that meets several, seemingly opposing, requirements: (1) the range and the space of the data structure must be, at all times, proportional to the current cardinality n(t) of the input set, and (2) the hashcodes it assigns must be stable in that the hashcode of an element must not change while the element is continuously in the set. A simple argument shows that these two desiderata are impossible to achieve when arbitrary deletions and insertions are allowed. In this paper, we show that one can achieve them when only insertions occur and, more generally, when the hashcode range and the space are allowed to grow as a function of the maximum cardinality N-t of the set until time t. The data structure executes all operations in worst case constant time with high probability and requires space that is within a constant factor of the lower bound. In particular, this leads to a hash table design that does not need to move elements as its size grows. More generally, we present, as an application, a cyclic sequence of reductions between data structures that lead to the following bootstrapping result. Let B(u,n) denote the lower bound on the space of a dictionary for n elements over a universe [u]. Given a compact dynamic dictionary (i.e., space O (B(u,n))), we can use it to build a dynamic dictionary with space B(u,n) + O (n log log n). This reduction increases the time per operation by an additive constant and applies both to the extendable and non-extendable settings (failure probability is 1/poly(n) per insertion).