This paper presents a new method to quickly extract geodesic paths on images and 3D meshes. We use a heuristic to drive the front propagation procedure of the classical Fast Marching. This results in a modification of the Fast Marching algorithm that is similar to the A algorithm used in artificial intelligence. In order to find very quickly geodesic paths between any given couples of points, we advocate for the initial computation of distance maps to a set of landmark points and make use of these distance maps through a relevant heuristic. We show that our method brings a large speed up for large scale applications that require the extraction of geodesics on images and 3D meshes. We introduce two distortion metrics in order to find an optimal seeding of landmark points for the targeted applications. We also propose a compression scheme to reduce the memory requirement without impacting the quality of the extracted paths.