  We define an analog of David Little's algorithm for reduced words in type B,
and investigate its main properties. In particular, we show that our algorithm
preserves the recording tableau of Kra\'{s}kiewicz insertion, and that it
provides a bijective realization of the Type B transition equations in Schubert
calculus. Many other aspects of type A theory carry over to this new setting.
Our primary tool is a shifted version of the dual equivalence graphs defined by
Assaf and further developed by Roberts. We provide an axiomatic
characterization of shifted dual equivalence graphs, and use them to prove a
structure theorem for the graph of Type B Coxeter-Knuth relations.
