Two tagless variations on the Deutsch-Schorr-waite algorithm

Two variations are presented of the Deutsch-Schorr-Waite algorithm for marking finite, rooted, binary, directed graphs that dispense with the tag bit while retaining the same overall structure. These variations illustrate the gains that can be made through careful analysis of loop invariants. Also, by keeping the overall structure of a well-known algorithm, they represent an increase in clarity over previous tag-free marking algorithms. One result demonstrated is that a 2nd variation that works only on acyclic graphs can achieve linear time as well as constant workspace. It is suggested that linear time and constant workspace may not be simultaneously attainable by any algorithm on all graphs.