In-Place Initializable Arrays

TL;DR

This paper introduces an in-place, space-efficient algorithm for initializable arrays that supports constant-time read, write, and initialization operations using only one extra bit, optimizing previous solutions.

Contribution

It presents a novel in-place algorithm for initializable arrays that uses minimal extra space and achieves optimal time complexity for all operations.

Findings

Supports all operations in constant worst-case time

Uses only 1 extra bit beyond normal array storage

Improves space efficiency over previous methods

Abstract

An initializable array is an array that supports the read and write operations for any element and the initialization of the entire array. This paper proposes a simple in-place algorithm to implement an initializable array of length $N$ containing $\ell \in O(w)$ bits entries in $N \ell +1$ bits on the word RAM model with $w$ bits word size, i.e., the proposed array requires only 1 extra bit on top of a normal array of length $N$ containing $\ell$ bits entries. Our algorithm supports the all three operations in constant worst-case time, that is, it runs in-place using at most constant number of words $O(w)$ bits during each operation. The time and space complexities are optimal since it was already proven that there is no implementation of an initializable array with no extra bit supporting all the operations in constant worst-case time [Hagerup and Kammer, ISAAC 2017]. Our algorithm significantly improves upon the best algorithm presented in the earlier studies [Navarro, CSUR 2014] which uses $N + o(N)$ extra bits to support all the operations in constant worst-case time.