  We study an ancient problem that in a static or dynamical system, sought an
optimal path, which the context always means within an extremal condition. In
fact, through those discussions about this theme, we established a universal
essential calculated model to serve for these complex systems. Meanwhile we
utilize the sample space to character the system. These contents in this paper
would involve in several major areas including the geometry, probability, graph
algorithms and some prior approaches, which stands the ultimately subtle linear
algorithm to solve this class problem. Along with our progress, our discussion
would demonstrate more general meaning and robust character, which provides
clear ideas or notion to support our concrete applications, who work in a more
popular complex system.
