The original CLEAN method is due to [Hogbom 1974]. Several variants exist.
One of the most popular (CLARK) is due to [Clark 1980], and involves minor and major cycles. In Minor cycles, an Hogbom CLEAN is performed, but with a truncated dirty beam, and only on the list of brightest pixels. This search is fast, because of the dirty beam truncation and because of the limited support. The Clean components identified during the minor cycles are removed at once by a FFT during a Major cycle. Because removal is done by FFT, slightly more than the inner map quarter can be cleaned.
A second variant, called MX, due to [Cotton & Schwab 1984], is similar to the CLARK method, except that the Clean components are removed from the uv table at the Major cycle stage (and thus the imaging process is repeated at each major cycle). This avoid aliasing of sidelobes, allows to clean more than the inner quarter, but is relatively slow because of the re-imaging at major cycles. Unless disk storage is a real problem, a faster result of equal is obtained by standard Clean with a twice larger map.
The next variant, called SDI (from [Steer et al 1984]), is again like the CLARK method, but in Minor cycles, no deconvolution is performed, but only a selection of the strongest components down to some threshold. Major cycles are identical to those of the CLARK method. Although the principle is simple, the implementation is not easy because of normalization subtleties in the minor cycle stage. This method is reasonably well suited for more extended structures, but could become unstable if the threshold is inappropriate.
The Multi Resolution Clean (MRC, [Wakker & Schwartz 1988]) separates the problem in a smooth map and a difference map. Since the measurement equation is linear, both maps can be Cleaned (with Hogbom or Clark method) independently. This is faster than the standard CLEAN because the smooth map can be compressed by pixel averaging, and only fine structure left in difference map, so fewer Clean components are required.