Prediction-based compression engines are spoon-fed with sequentially efficiently compressible (SEC) streams of input data that make it possible for the compression engines to more efficiently compress or otherwise compact the incoming data than would be possible with streams of input data accepted on a TV-raster scan basis. Various techniques are disclosed for intentionally forming SEC input data streams. Among these are the tight packing of alike files or fragments into concatenation suitcases and the decomposition of files into substantially predictably consistent (SPC) fragments or segments that are routed to different suitcases according to their type. In a graphics-directed embodiment, image frames are partitioned into segment areas that are internally SPC and multidirectional walks (i.e., U-turning walks) are defined in the segment areas where these defined walks are traced during compression and also during decompression. A variety of pre-compression data transformation methods are disclosed for causing apparently random data sequences to appear more compressibly alike to each other. The methods are usable in systems that permit substantially longer times for data compaction operations than for data decompaction operations.