An efficient time and space K point-to-point shortest simple paths algorithm

We address the problem identifying the K best point-to-point simple paths connecting a given pair of nodes in a directed network with arbitrary lengths. The main result in this paper is the proof that a path tree containing the kth point-to-point shortest simple path can be obtained by using one of the previous (k-1) path trees containing each one of the previous (k-1) best point-to-point shortest simple paths. The proof implies that at most n single-source shortest path computations (re-optimizations) in a network with non-negative length arcs are made in each iteration of the method. In the “optimistic” case, this strategy only needs O(m) time to compute the best “neighbor” associated with a path tree, that is, the second shortest simple path given a shortest simple path. The algorithm runs in O(K max ( , , ) nf n m C ) time and uses O(K+m) space to determine the K point-to-point shortest simple paths in a directed network with n nodes, m arcs and maximum absolute length max C . Here, O( max ( , , ) f n m C ) is the best time needed to determine the shortest simple paths connecting a source node with any other non-source node in a network with non-negative length arcs. We improve required space in Yen’s algorithm by a multiplicative factor of O( 2 n ) for each best solution. Moreover, our algorithm runs in the “optimistic” case in O( max ( , , ) Kf n m C ) time. This affirmation is confirmed by an experimental study where O(K) shortest paths are used to determine the K point-to-point shortest simple paths in two versions of our algorithm.