Efficient Gauss Elimination for Near-Quadratic Matrices with One Short Random Block per Row, with Applications

Abstract

In this paper we identify a new class of sparse near-quadratic random Boolean matrices that have full row rank over $\mathbb{F}_2=\{0,1\}$ with high probability and can be transformed into echelon form in almost linear time by a simple version of Gauss elimination. The random matrix with dimensions $n(1-\varepsilon) \times n$ is generated as follows: In each row, identify a block of length $L=O((\log n)/\varepsilon)$ at a random position. The entries outside the block are 0, the entries inside the block are given by fair coin tosses. Sorting the rows according to the positions of the blocks transforms the matrix into a kind of band matrix, on which, as it turns out, Gauss elimination works very efficiently with high probability. For the proof, the effects of Gauss elimination are interpreted as a ("coin-flipping") variant of Robin Hood hashing, whose behaviour can be captured in terms of a simple Markov model from queuing theory. Bounds for expected construction time and high success probability follow from results in this area. By employing hashing, this matrix family leads to a new implementation of a retrieval data structure, which represents an arbitrary function $f\colon S \to \{0,1\}$ for some set $S$ of $m=(1-\varepsilon)n$ keys. It requires $m/(1-\varepsilon)$ bits of space, construction takes $O(m/\varepsilon^2$) expected time on a word RAM, while queries take $O(1/\varepsilon)$ time and access only one contiguous segment of $O((\log m)/\varepsilon)$ bits in the representation. The method is competitive with state-of-the-art methods. By well-established methods the retrieval data structure leads to efficient constructions of (static) perfect hash functions and (static) Bloom filters with almost optimal space and very local storage access patterns for queries.