Beyond Compare uses a number of different algorithms depending on the file type and configuration. In v4 the line alignment algorithms are explicitly named in the interface:
Standard alignment - This is a proprietary algorithm; we haven't made the details publicly available.
Myers O(ND) alignment - This is the same one that the GNU diff
utility and most other applications use. It's based on the paper "An O(ND) difference algorithm and its variations" by Eugene Myers (1986).
Patience Diff alignment - This is the "Patience Diff" algorithm that Bram Cohen originally developed for Bazaar, which he talks about here.
The character alignment to highlight differences within lines is based on the Myers O(ND) algorithm with some post-processing to clean up the results.
The alignment setting can be configured by clicking the Rules button
which brings up this dialog, which you can then configure for session scope or with a persistent default against the file association.