Dempster-Shafer theory, which is a way to manage uncertain information, is based upon that one has evidences for and/or against one or more different propositions. Dempster-Shafer theory contains a rule for the fusion of these evidences, Dempster´s rule. This report presents a new algorithm with lower computational complexity for Dempster´s rule in the case of evidence ordered in a directed acyclic graph than for the classic step by step application of Dempster´s rule. This is the problem when we for every pair of evidences have an evidence against the simultaneous belief in both propositions. The original evidences are situated on the vertices and the new ones are situated on the intermediate edges. The new algorithm which reasons about complete paths through the graph is O(n.4n), compared to O(2n2) of the classic algorithm. After having made a detailed presentation of the reasoning behind the new algorithm we conclude that it is feasible to reason without any approximation about complete paths through a directed acyclic graph not only for the smallest problems.