On-the-Fly Array Initialization in Less Space

TL;DR

This paper introduces a space-efficient data structure that supports fast array initialization and access, optimizing memory usage and operation times in word RAM models for various tradeoffs.

Contribution

It presents a novel array representation method that achieves constant-time initialization and access with significantly reduced space compared to prior solutions.

Findings

Supports constant-time array initialization and access with minimal space overhead.

Provides a flexible tradeoff between initialization time, access time, and memory usage.

Achieves near-optimal space complexity for arbitrary access times with fast initialization.

Abstract

We show that for all given $n,t,w \in \{1,2,...\}$ with $n<2^w$, 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 $nw+\lceil n(t/(2 w))^t\rceil$ 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 $nw+\lceil n/w^t\rceil$ bits for arbitrary fixed $t$, to be compared with $nw+\Theta(n)$ bits for the best previous solution, and at the opposite end, still with constant-time initialization, we support $O(\log n)$-time access with just $nw+1$ bits, which is optimal for arbitrary access times if the initialization executes fewer than $n$ steps.