On-the-Fly Array Initialization in Less Space

We show that for all given n, t, w ∈ {1, 2, . . .} with n < 2, an array of n entries of w bits each can be represented on a word RAM with a word length of w bits in at most nw + dn(t/(2w))e bits of uninitialized memory to support constant-time initialization of the whole array and O(t)-time reading and writing of individual array entries. At one end of this tradeoff, we achieve initialization and access (i.e., reading and writing) in constant time with nw+dn/we bits for arbitrary fixed t, to be compared with nw+Θ(n) bits for the best previous solution, and at the opposite end, still with constant-time initialization, we support O(logn)-time access with just nw + 1 bits, which is optimal for arbitrary access times if the initialization executes fewer than n steps.