On the scalability of 2-D discrete wavelet transform algorithms

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 on 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) Mesh and Hypercube they are scalable as M(2) = Omega(P-3/3-gamma) (NS-CP), and as M(2) = Omega (P-2/2-gamma) (S-CP), respectively, where M(2) is the number of elements in the input matrix, and gamma is an element of (0, 1) is a parameter relating M to the number of desired octaves J as J = [gamma log M]. 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) = Omega(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.