K shortest path routing

The k shortest path routing problem is a generalization of the shortest path routing problem in a given network. It asks not only about a shortest path but also about next k−1 shortest paths (which may be longer than the shortest path). A variation of the problem is the loopless k shortest paths. Finding k shortest paths is possible by extending Dijkstra algorithm or Bellman-Ford algorithm. Since 1957 many papers were published on the k shortest path routing problem. Most of the fundamental works were done between 1960s and 2001. Since then, most of the research has been on the problem's applications and its variants. In 2010, Michael Günther et al. published a book on Symbolic calculation of k-shortest paths and related measures with the stochastic process algebra tool CASPA. The Dijkstra algorithm can be generalized to find the k shortest paths. There are two main variations of the k shortest path routing problem. In one variation, paths are allowed to visit the same node more than once, thus creating loops. In another variation, paths are required to be simple and loopless. The loopy version is solvable using Eppstein's algorithm and the loopless variation is solvable by Yen's algorithm. In this variant, the problem is simplified by not requiring paths to be loopless. A solution was given by B. L. Fox in 1975 in which the k-shortest paths are determined in O(m + kn log n) asymptotic time complexity (using big O notation. In 1998, David Eppstein reported an approach that maintains an asymptotic complexity of O(m + n log n + k) by computing an implicit representation of the paths, each of which can be output in O(n) extra time. In 2015, Akuba et al. devised an indexing method as a significantly faster alternative for Eppstein's algorithm, in which a data structure called an index is constructed from a graph and then top-k distances between arbitrary pairs of vertices can be rapidly obtained. In the loopless variant, the paths are forbidden to contain loops which adds an additional level of complexity.