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

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## Acknowledgements

We are very grateful to Seth Pettie, who triggered this research by asking an insightful question regarding "one block" while discussing the two-block solution from [Martin Dietzfelbinger and Stefan Walzer, 2019]. (This discussion took place at the Dagstuhl Seminar 19051 "Data Structures for the Cloud and External Memory Data".) Thanks are also due to the reviewers, whose comments helped to improve the presentation.

## Cite As

Martin Dietzfelbinger and Stefan Walzer. Efficient Gauss Elimination for Near-Quadratic Matrices with One Short Random Block per Row, with Applications. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 39:1-39:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ESA.2019.39

## Abstract

In this paper we identify a new class of sparse near-quadratic random Boolean matrices that have full row rank over 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-epsilon) x n is generated as follows: In each row, identify a block of length L = O((log n)/epsilon) 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. They readily extend to larger finite fields in place of F_2. By employing hashing, this matrix family leads to a new implementation of a retrieval data structure, which represents an arbitrary function f: S -> {0,1} for some set S of m = (1-epsilon)n keys. It requires m/(1-epsilon) bits of space, construction takes O(m/epsilon^2) expected time on a word RAM, while queries take O(1/epsilon) time and access only one contiguous segment of O((log m)/epsilon) bits in the representation (O(1/epsilon) consecutive words on a word RAM). The method is readily implemented and highly practical, and it is competitive with state-of-the-art methods. In a more theoretical variant, which works only for unrealistically large S, we can even achieve construction time O(m/epsilon) and query time O(1), accessing O(1) contiguous memory words for a query. 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.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Data structures design and analysis
##### Keywords
• Random Band Matrix
• Gauss Elimination
• Retrieval
• Hashing
• Succinct Data Structure
• Randomised Data Structure
• Robin Hood Hashing
• Bloom Filter

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