Tree-based Space Efficient Formats for Storing the Structure of Sparse Matrices

I Simecek,D Langr

doi:10.12694/scpe.v15i1.962

Abstract

Sparse storage formats describe a way how sparse matrices are stored in a computer memory. Extensive research has been conducted about these formats in the context of performance optimization of the sparse matrix-vector multiplication algorithms, but memory efficient formats for storing sparse matrices are still under development, since the commonly used storage formats (like COO or CSR) are not sufficient. In this paper, we propose and evaluate new storage formats for sparse matrices that minimize the space complexity of information about matrix structure. The first one is based on arithmetic coding and the second one is based on binary tree format. We compare the space complexity of common storage formats and our new formats and prove that the latter are considerably more space efficient.

Highlights

The paper investigates memory-efficient storage formats for very large sparse matrices (LSMs)
By LSMs, we mean matrices that due to their sizes must be stored and processed by massively parallel computer systems (MPCSs) with distributed memory architecture consisting of tens or hundreds of thousands of processor cores
This paper deals with the design of four new SSFs called arithmetical coding based format, minimal binary tree format, compressed binary tree format, and compressed quadtree format

Summary

Introduction

The paper investigates memory-efficient storage formats for very large sparse matrices (LSMs). Within our previous work [9, 12, 11, 8, 7], we have addressed weaknesses of previously developed solutions for space-efficient formats for storing of large sparse matrices. The space complexity of the representation of sparse matrices depends strongly on the used matrix storage format. A matrix of order n is considered to be sparse if it contains much less nonzero elements than n2. A matrix is considered sparse if the ratio of nonzero elements drops bellow some threshold. The LSM is used repeatedly and the computation of its elements is slow and it takes more time than its later reading from a file system Our research addresses computations with LSMs satisfying at least one of the following conditions: 1. The LSM is used repeatedly and the computation of its elements is slow and it takes more time than its later reading from a file system

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