In this paper, we use a classical theorem of Hardy, Littlewood and Polya on rearrangements of functions for solving well known Shortest Path problem. We first introduce a special .class of variational problems. To solye this kind of problems, a classical theorem of Hardy, Littlewood and  Polya on rearrangements of functions is used to prove.the equivalence of this class of variational problems. As a consequence, solutions of this class of variational, problems are solution of Shortest Path problem that can be computed numerically via quadratic.programming In this paper we suggest an Algorithm' for the Shortest Path problem when this Path, called (t,F(t)) , starting from given Initial point (0,F(0)) and reaching destination (1,F(1)) while remaining between boundaries defined by (t,A(t)) and (t,B(t)). One extension of the problem can be considering Shortest Path surrounded by the given boundaries while a circle lies in the phase plane between two given boundaries and we should consider this circle. In the solution of Problem A useful application of this problem is optimal, battery interchange policy for solar cars.

For a network with cycle, where at least one cycle exists, the Floyd-War shall Algorithm is probably the most used Algorithm to determine the least cost Path between every pair of nodes on this network, i.e. the solution for the Shortest Path problem with cycle.In this paper, a new Algorithm for this problem which requires less computational effort than the Floyd- War shall Algorithm has been developed Furthermore, it can be shown that the basis of our Algorithm is much easier to be learnt and understood which might be an advantage for educational purposes. A small example validates our Algorithm and shows its implementation.

Shortest-Path problem is one of the well-known optimization problems that has been studied by many scientists in recent years.Applications of this problem such as transportation and communication are generally solved by Dijkstra's Algorithm (Labeling). In this paper, two separate scientific fields, electronics and operation research have been linked to each other and a new Algorithm has been created for to find the optimization solution of a Shortest Path problem by using electric networks and rules. The proposed Algorithm can solve the Shortest-Path problem in directed graphs and no order ones, and also can solve the longest Path problems in directed graphs.In this Algorithm, electrical network are used in a way that the resistance value of each branch is equal to each edge weights in the Shortest-Path problems. Then with using Ohm Law and Kirchhaffs Voltage Law (KVL), the current in each circuit cycle is calculated.Then the branches that contain the most passing current are specified, and according to Ohm's Law, have the lowest resistance or weight.Thus, the Shortest Path in the network is achieved. Advantage of this Algorithm is faster convergence to the answer and less computing time than the conventional method, specially in networks with more nods.The mentioned Algorithm has been described for three examples.

Shortest Path problem is among the most interesting problems in the field of graph and network theory. There are many efficient matrices-based Algorithms for detecting of Shortest Path and distance between all pairs of this problem in the literature of research. In this paper, a new exact Algorithm, named the Cascade Rectangle Algorithm, is presented by using the main structure of previous exact Algorithms and developing some new techniques. In cascade rectangle Algorithm, all mathematical calculations and operations execute in multirectangle structure. This Algorithm is an efficient exact Algorithm with simple procedure and fast running time. Furthermore, a new matrix on the basis of route matrix, called the Transposition Matrix, is defined to apply the sensitivity analysis and reoptimization of the all pairs Shortest Path networks. Finally, one illustrative example is also solved in a step by step procedure to show the details of the cascade rectangle Algorithm, floyd-warshall Algorithm, revised matrix Algorithm, and Transposition Matrix.