K shortest path routing

The K shortest path routing algorithm is an extension algorithm of the shortest path routing algorithm in a given network.

It is sometimes crucial to have more than one path between two nodes in a given network. In the event there are additional constraints, other paths different from the shortest path can be computed. To find the shortest path one can use shortest path algorithms such as Dijkstra’s algorithm or Bellman Ford algorithm and extend them to find more than one path. The K Shortest path routing algorithm is a generalization of the shortest path problem. The algorithm not only finds the shortest path, but also K-1 other paths in non-decreasing order of cost. K is the number of shortest paths to find. The problem can be restricted to have the K shortest path without loops (loopless K shortest path) or with loop.

History

Since 1957 there have been many papers published on the K Shortest path routing algorithm problem. Most of the fundamental works on not just finding the single shortest path between a pair of nodes, but instead listing a sequence of the K shortest paths, were done between the 1960s and up to 2001. Since then, most of the recent research has been about the applications of the algorithm and its variants. In 2010, Michael Gunter et al. published a book on Symbolic calculation of K-shortest paths and related measures with the stochastic process algebra tool CASPA.^[1]

Important works on the K Shortest path problem:

Year of Publication	Author	Title
1971	Jin Y. Yen	Finding the K Shortest Loopless Paths in a Network
1972	M. T. Ardon et al.	A recursive algorithm for generating circuits and related subgraphs
1973	P. M. Camerini et al.	The k shortest spanning trees of a graph
1975	K. Aihara	An approach to enumerating elementary paths and cutsets by Gaussian elimination method
1976	T. D. Am et al.	An algorithm for generating all the paths between two vertices in a digraph and its application
1988	Ravindra K. Ahuja et al.	Faster algorithms for the shortest path problem
1989	S. Anily et al.	Ranking the best binary trees
1990	Ravindra K. Ahuja et al.	Faster algorithms for the shortest path problem
1993	El-Amin et al.	An expert system for transmission line route selection
1997	David Eppstein	Finding the k Shortest Paths
1997	Ingo Althöfer	On the K-best mode in computer chess: measuring the similarity of move proposals
1999	Ingo Althöfer	Decision Support Systems With Multiple Choice Structure
2011	Husain Aljazzar, Stefan Leue	K*: A heuristic search algorithm for finding the k shortest paths

The BibTeX database contains more articles.

Algorithm

The Dijkstra algorithm can be generalized to find the K Shortest paths.

Definitions: G(V, E): weighted directed graph, with set of vertices V and set of directed edges E, w(u, v): cost of directed edge from node u to node v (costs are non-negative). Links that do not satisfy constraints on the shortest path are removed from the graph s: the source node t: the destination node K: the number of shortest paths to find Pu: a path from s to u B is a heap data structure containing paths P: set of shortest paths from s to t countu: number of shortest paths found to node u Algorithm: *P =empty, *countu = 0, for all u in V insert path Ps = {s} into B with cost 0 while B is not empty and countt < K: – let Pu be the shortest cost path in B with cost C – B = B − {Pu }, countu = countu + 1 – if u = t then P = P U Pu – if countu ≤ K then for each vertex v adjacent to u: – if v is not in Pu then – let Pv be a new path with cost C + w(u, v) formed by concatenating edge (u, v) to path Pu – insert Pv into B return P

There are mainly two variants of the K shortest path routing algorithm:

First variant

In the first variant, the paths are not required to be loopless (this is the simple one): David Eppstein's algorithm achieves the best running time complexity.^[2]

Second variant

In the second variant, attributed to Jin Y. Yen, the paths are required to be loopless.^[3] (This restriction introduced another level of complexity.) Yen's algorithm is used where simple paths only are considered, whereas Eppstein's algorithm is used when non-simple paths are allowed (e.g., paths are allowed to revisit the same node multiple times).

Paths are not required to be loopless

In all that follows, m is the number of edges and n is the number of vertices.

Eppstein's algorithm provides the best results.^[2] Eppstein's model finds the K shortest paths (allowing cycles) connecting a given pair of vertices in a digraph, in time O(m+ nlogn + K).

Eppstein's algorithm uses a graph transformation technique.

This model can also find the K shortest paths from a given source s to each vertex in the graph, in total time O(m + n log n + kn).

Loopless K shortest path routing algorithm

The best running time for this model is attributed to Jin. Y. Yen.^[3] Yen's algorithm finds the lengths of all shortest paths from a fixed node to all other nodes in an n-node non negative-distance network. This technique only requires 2n² additions and n² comparisons - which is less than other available algorithms require.

The running time complexity is O(Kn(m + n log n)), which is pseudo-polynomial. m represents the number of edges and n is the number of vertices.

Some examples and description

Example #1

The following example makes use of Yen’s model to find the first K shortest paths between communicating end nodes. That is, it finds the first, second, third, etc. up to the K^th shortest path. More details can be found here. 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:

15-node network containing a combination of bi-directional and uni-directional links

Example #2

Another example is the use of K Shortest 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 "Computer Vision Laboratory – CVLAB" .

Example #3

Another use of K shortest path 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 (2012) studied the K shortest path problems in transit network systems. ^[4]

Applications

The K Shortest path routing is a good alternative for:

• Geographic path planning
• Network routing, especially in optical mesh network where there are additional constraints that cannot be solved by using ordinary shortest path algorithms.
• Hypothesis generation in computational linguistics
• Sequence alignment and metabolic pathway finding in bioinformatics
• Multiple object tracking as described above
• Road Networks: road junctions are the nodes (vertices) and each edge (link) of the graph is associated with a road segment between two junctions.

Variations

There are two main variations of the K Shortest path routing algorithm as mentioned above. Other variations only fall in between these.

• In the first variation, loops are allowed, that is paths are allowed to revisit the same node multiple times. The following papers deal with this variation.^[2]
• The second variation deals with simple paths. It is also called loopless K Shortest path routing problem and is attributed to J. Y. Yen ^[3]

Related problems

• Dijkstra's algorithm solves the single-source shortest path problems.
• The Bellman–Ford algorithm solves the single-source problem if edge weights may be negative.
• The breadth-first search algorithm is used when the search is only limited to two operations.
• The Floyd–Warshall algorithm solves all pairs shortest paths.
• Johnson's algorithm solves all pairs' shortest paths, and may be faster than Floyd–Warshall on sparse graphs.
• Perturbation theory finds (at worst) the locally shortest path.

Cherkassky et al.^[5] provide more algorithms and associated evaluations.

See also

• Shortest path problem
• Constrained shortest path routing
• Diverse Path Routing
• Shared Risk Group (SRG) and Shared Link Risk Group (SRLG)

Notes

1. Michael Günther et al.: “Symbolic calculation of K-shortest paths and related measures with the stochastic process algebra tool CASPA”. In: Int’l Workshop on Dynamic Aspects in Dependability Models for Fault-Tolerant Systems (DYADEM-FTS), ACM Press (2010) 13–18.
2. Eppstein, D. (1998). "Finding the K shortest paths". SIAM J. Comput. 28 (2): 652–673. doi:10.1137/S0097539795290477.
3. Yen, J. Y (1971), "Finding the K-Shortest Loopless Paths in a Network", Management Science 17: 712–716.
4. Xu, W., He, S., Song, R., & Chaudhry, S. (2012). Finding the K shortest paths in a schedule-based transit network. Computers & Operations Research, 39(8), 1812-1826. doi:10.1016/j.cor.2010.02.005
5. Cherkassky, Boris V.; Goldberg, Andrew V.; Radzik, Tomasz (1996). "Shortest paths algorithms: theory and experimental evaluation". Mathematical Programming. Ser. A 73 (2): 129–174.

External links

• Implementation of Yen's algorithm
• http://www.technical-recipes.com/2012/the-k-shortest-paths-algorithm-in-c/#more-2432
• Multiple objects tracking technique using K-shortest path algorithm: http://cvlab.epfl.ch/software/ksp/
• BibTeX database: http://www.ics.uci.edu/~eppstein/bibs/kpath.bib
• Computer Vision Laboratory: http://cvlab.epfl.ch/software/ksp/