Find an Optimal Path in Static System and Dynamical System within Polynomial Runtime

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.