Kshort

Kshort, often seen as K-shortest paths, refers to the problem in graph theory of finding the K distinct s–t paths with the smallest total weight in a weighted graph. The term is used in routing, network design, and transportation planning to provide multiple candidate routes ranked by efficiency or cost. Variants distinguish whether paths must be simple (no repeated vertices) or may contain cycles.

In the simple-path variant, the objective is to enumerate K loopless s–t paths in nondecreasing order of their total weight. This problem is typically approached with specialized algorithms that reuse shortest-path computations. Yen’s algorithm, introduced in the 1970s, generates the K shortest loopless paths by iteratively exploring deviations from the current best path. Eppstein’s algorithm, published in the 1990s, uses a shortest-path tree and a priority structure to achieve efficiency gains for large graphs. Martins and other researchers have contributed additional techniques and refinements.

Practical implementations rely on subroutines such as Dijkstra’s algorithm for nonnegative weights or Bellman-Ford when negative weights are present, with reweighting often used to enable faster shortest-path queries. The choice of algorithm affects whether the output paths are guaranteed to be simple, the handling of equal-weight paths, and overall performance as K grows.

Applications of Kshort include providing multiple viable routes in GPS navigation, designing redundant communication paths to improve reliability, and evaluating alternative plans in logistics and traffic management. See also: shortest path problem, Yen’s algorithm, Eppstein’s algorithm, K shortest paths.