Experiments with quadtree representation of matrices

Abstract

The quadtrees matrix representation has been recently proposed as an alternative to the conventional linear storage of matrices. If all elements of a matrix are zero, then the matrix is represented by an empty tree; otherwise it is represented by a tree consisting of four subtrees, each representing, recursively, a quadrant of the matrix. Using four-way block decomposition, algorithms on quadtrees accelerate on blocks entirely of zeroes, and thereby offer improved performance on sparse matrices. This paper reports the results of experiments done with a quadtree matrix package implemented in REDUCE to compare the performance of quadtree representation with REDUCE's built-in sequential representation of matrices. Tests on addition, multiplication, and inversion of dense, triangular, tridiagonal, and diagonal matrices (both symbolic and numeric) of sizes up to 100×100 show that the quadtree algorithms perform well in a broad range of circumstances, sometimes running orders of magnitude faster than their sequential counterparts.CR categories and subject descriptorsI.1.2 [Algebraic Manipulation Algorithms]; Algebraic algorithmsE.1 [Data Structures]: TreesG.1.3 [Numerical Linear Algebra]: Sparse and very large systemsGeneral Term: Measurement