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 Güntheret al. published a book on Symbolic calculation of k-shortest paths and related measures with the stochastic process algebra tool CASPA.
Algorithm
The Dijkstra algorithm can be generalized to find the k shortest paths.
Definitions: Algorithm:
Variations
There are two main variations of the k shortest path routing problem. In on 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.
Loopy variant
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 asymptotic time complexityextra time. In 2007, John Hershberger and Subhash Suri proposed a replacement paths algorithm, a more efficient implementation of Eppstein's algorithm with O improvement in 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.
Loopless variant
In the loopless variant, the paths are forbidden to contain loops which adds an additional level of complexity. It can be solved using Yen's algorithm to find the lengths of all shortest paths from a fixed node to all other nodes in an n-node non negative-distance network, a technique requiring only 2n2 additions and n2 comparison, fewer than other available shortest path algorithms need. The runningtime complexity is pseudo-polynomial, being .
Some examples and description
Example #1
The following example makes use of Yen’s model to find k shortest paths between communicating end nodes. That is, it finds a shortest path, second shortest path, etc. up to the Kth shortest path. More details can be found . The code provided in this example attempts to solve the k shortest path routing problem for a 15-nodes network containing a combination of unidirectional and bidirectional links:
Example #2
Another example is the use of k shortest paths algorithm to track multiple objects. The technique implements a multiple object tracker based on the k shortest paths routing algorithm. A set of probabilistic occupancy maps is used as input. An object detector provides the input. The complete details can be found at " – CVLAB".
Example #3
Another use of k shortest paths algorithms is to design a transit network that enhances passengers' experience in public transportation systems. Such an example of a transit network can be constructed by putting traveling time under consideration. In addition to traveling time, other conditions may be taken depending upon economical and geographical limitations. Despite variations in parameters, the k shortest path algorithms finds the most optimal solutions that satisfies almost all user needs. Such applications of k shortest path algorithms are becoming common, recently Xu, He, Song, and Chaudry studied the kshortest path problems in transit network systems.
Applications
The k shortest path routing is a good alternative for:
