Costs of quadtree representation of nondense matrices

Abstract

The quadtree representation of matrices is a uniform representation for both sparse and dense matrices which can facilitate shared manipulation on multiprocessors. This paper presents worst-case and average-case resource requirements for storing and retrieving familiar families of patterned matrices: packed, symmetric, triangular, Toeplitz, and banded. Using this representation it compares resource requirements of three kinds of permutation matrices, as examples of nondense, unpatterned matrices. Exact values for the shuffle and bit-reversal permutations (as in the fast Fourier transform) and tight bounds on the expected values from purely random permutations are derived. Two different measures, density and sparsity, are proposed from these values. Analysis of quadtree matrix addition relates density of addends to space bounds on their sum and relates their sparsity to time bounds for computing that sum.