Burrows-Wheeler compression is a three stage process in which the data is transformed with the Burrows-Wheeler Transform, then transformed with Move-To-Front, and finally encoded with an entropy coder. Move-To-Front, Transpose, and Frequency Count are some of the many algorithms used on the List Update problem. In 1985, Competitive Analysis first showed the superiority of Move-To-Front over Transpose and Frequency Count for the List Update problem with arbitrary data. Earlier studies due to Bitner assumed independent identically distributed data, and showed that while Move-To-Front adapts to a distribution faster, incurring less overwork, the asymptotic costs of Frequency Count and Transpose are less. The improvements to Burrows-Wheeler compression this work covers are increases in the amount, not speed, of compression. Best x of 2x-1 is a new family of algorithms created to improve on Move-To-Front's processing of the output of the Burrows-Wheeler Transform which is like piecewise independent identically distributed data. Other algorithms for both the middle stage of Burrows-Wheeler compression and the List Update problem for which overwork, asymptotic cost, and competitive ratios are also analyzed are several variations of Move One From Front and part of the randomized algorithm Timestamp. The Best x of 2x - 1 family includes Move-To-Front, …