Fillable arrays with constant time operations and a single bit of redundancy

TL;DR

This paper introduces a data structure for fillable arrays that achieves constant-time read and fill operations with only one bit of redundancy, and efficient write operations, improving performance for this problem.

Contribution

It presents a novel fillable array data structure with minimal redundancy that supports constant-time read and fill, and efficient write operations, both deterministic and randomized.

Findings

Constant-time read and fill operations with 1-bit redundancy.

Efficient write operations in amortized or expected constant time.

Use of almost pairwise independent permutations for randomized implementation.

Abstract

In the fillable array problem one must maintain an array A[1..n] of $w$-bit entries subject to random access reads and writes, and also a $\texttt{fill}(\Delta)$ operation which sets every entry of to some $\Delta\in\{0,\ldots,2^w-1\}$. We show that with just one bit of redundancy, i.e. a data structure using $nw+1$ bits of memory, $\texttt{read}/\texttt{fill}$ can be implemented in worst case constant time, and $\texttt{write}$ can be implemented in either amortized constant time (deterministically) or worst case expected constant (randomized). In the latter case, we need to store an additional $O(\log n)$ random bits to specify a permutation drawn from an $1/n^2$-almost pairwise independent family.