Dual-Network Model and Its Application in Algorithms for Solving the Designated Vertex Shortest Path Problem

In this paper, we presented a dual-network model to solve the designated vertex shortest path (SP) problem. First we made bi-directional search using Bellman-Ford algorithm on the undirected network graph and labeled the vertexes with the shortest distance to build a dual-network. Then we proposed and proved the sum value theorem and the difference value theorem of vertexes. The theorems show that the length of the shortest path from source to destination passing an arbitrary vertex is equal to the sum value of the vertex, while the shortest path passing two vertexes will pass the vertex with smaller difference value firstly. Based on the theorems, we presented two procedures for solving the shortest path problem from source to destination passing one or two designated vertexes. The complexity analysis showed that our algorithms are better than Bellman-Ford algorithm in these cases. Finally we presented a computational example.