On the Scalability of 2-D Discrete Wavelet Transform Algorithms

• Authors: José Fridman;Elias S. Manolakos
• Venue: Multidimensional Systems and Signal Processing
• Year: 1997

The ability of a parallel algorithmto make efficient use of increasing computational resources isknown as its scalability. In this paper, we develop four parallelalgorithms for the 2-dimensional Discrete Wavelet Transform algorithm(2-D DWT), and derive their scalability properties on Mesh andHypercube interconnection networks. We consider two versionsof the 2-D DWT algorithm, known as the Standard (S) and Non-standard(NS) forms, mapped onto P processors under two datapartitioning schemes, namely checkerboard (CP) and stripped(SP) partitioning. The two checkerboard partitioned algorithmson the cut-through-routed (CT-routed) Mesh are scalable as 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) Meshand Hypercube they are scalable as M^2=\Omega(P^{\frac{3}{3-\gamma}})(NS-CP), and as M^2=\Omega(P^{\frac{2}{2-\gamma}})(S-CP), respectively, where M^{2} is the numberof elements in the input matrix, and \gamma\in (0,1)is a parameter relating M to the number of desiredoctaves J as J=\lceil \gamma \log M \rceil.On the CT-routed Hypercube, scalability of the NS-form algorithmsshows similar behavior as on the CT-routed Mesh. The Standardform algorithm with stripped partitioning (S-SP) is scalableon the CT-routed Hypercube as M^{2}=\Omega(P^{2}),and it is unscalable on the CT-routed Mesh. Although asymptoticallythe stripped partitioned algorithm S-SP on the CT-routed Hypercubewould appear to be inferior to its checkerboard counterpart S-CP,detailed analysis based on the proportionality constants of theisoefficiency function shows that S-SP is actually more efficientthan 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.