I am not sure if you are want to come up with a different algorithm from Boyer-Moore. Why would that be necessary? It is most likely easier to understand the proof of the existing algorithm than to create a new one.I am looking to solve the problem by perhaps finding a fast way of determining if an element is -not- part of the majority and elliminating those.

What exactly do you not understand? Here is the authors' description and proof. (You can convert a PS file to PDF here.)I understand the Boyer-Moore solution but not its proof.