An efficient Dijkstra-like labeling method for computing shortest odd/even paths

We show how the problem of determining shortest paths of even or odd length between two specified vertices in a graph G = (V, E) can be reduced to the problem of finding a shortest alternating path with respect to a specific matching in a related graph H. This problem can be solved by a Dijkstra-like labeling procedure of complexity O(|V|²) respectively O(|E|log|V|). Interpreting this procedure appropriately the method can then be applied directly on the given graph G.