On the Scalability of 2-D Discrete Wavelet Transform Algorithms

Abstract

The ability of a parallel algorithm to make efficient use of increasing computational resources is known as its scalability. In this paper, we develop four parallel algorithms for the 2-dimensional Discrete Wavelet Transform algorithm (2-D DWT), and derive their scalability properties on Mesh and Hypercube interconnection networks. We consider two versions of the 2-D DWT algorithm, known as the Standard (S) and Non-standard (NS) forms, mapped onto P processors under two data partitioning schemes, namely checkerboard (CP) and stripped (SP) partitioning. The two checkerboard partitioned algorithms $M^2 = \Omega (P\log P)$ (Non-standard form, NS-CP), and as $M^2 = \Omega (P\log ^2 P)$ (Standard form, S-CP); while on the store-and-forward-routed (SF-routed) Mesh and Hypercube they are scalable as ${{3\; - \;\gamma }}$ (NS-CP), and as ${{2\; - \;\gamma }}$ (S-CP), respectively, where M ² is the number of elements in the input matrix, and γ ∈ (0,1) is a parameter relating M to the number of desired octaves J as $J = \left\lceil {\gamma \;\log M} \right\rceil$. On the CT-routed Hypercube, scalability of the NS-form algorithms shows similar behavior as on the CT-routed Mesh. The Standard form algorithm with stripped partitioning (S-SP) is scalable on the CT-routed Hypercube as M ² = Ω(P ²), and it is unscalable on the CT-routed Mesh. Although asymptotically the stripped partitioned algorithm S-SP on the CT-routed Hypercube would appear to be inferior to its checkerboard counterpart S-CP, detailed analysis based on the proportionality constants of the isoefficiency function shows that S-SP is actually more efficient than S-CP over a realistic range of machine and problem sizes. A milder form of this result holds on the CT- and SF-routed Mesh, where S-SP would, asymptotically, appear to be altogether unscalable.