On the K shortest path trees problem

We address the problem for finding the K best path trees connecting a source node with any other non-source node in a directed network with arbitrary lengths. The main result in this paper is the proof that the kth shortest path tree is adjacent to at least one of the previous (k−1) shortest path trees. Consequently, we design an O(+Km) time and O(K+m) space algorithm to determine the K shortest path trees, in a directed network with n nodes, m arcs and maximum absolute length , where O() is the best time needed to solve the shortest simple paths connecting a source node with any other non-source node. max (, ,) f n m C max C max (, ,) f n m C The shortest path (SP) problem in a directed network of n nodes an m arcs with arbitrary lengths on the arcs, finds shortest length paths from a source node to all other nodes or detects a cycle of negative length. The SP problem appears in many important real cases and there are numerous algorithms to solve it (see, for example, Ahuja et al. [1]). The mathematical formulation of the SP problem lets the solutions of the SP problem to be characterized by path trees, that is, a tree containing a directed only one path from source node to any non-source node. The determination of the optimal path tree (shortest path tree) can be efficiently determined by Bellman-Ford-Moore (Bellman [3], Ford [7], and Moore [21]) label-correcting algorithm achieving the best strongly polynomial running time of O(nm). In this paper, we consider the K shortest path trees problem as the problem to determine the K best basis trees (solutions) of the classical mathematical formulation of the SP problem. The determination of the K shortest paths in a network has a wide range of applications. Some of them are cited in Eppstein [8]. We are unaware of any previous references to this problem in the literature. A proof is offered which shows that the kth best basis tree is adjacent to at least one of the previous k−1 best basis trees. In other words, the kth best solution is obtained from one of the previous best solutions by exchanging an arc in the basis tree for an arc outside of the basis tree. This results allows an algorithm to be designed running in