Shortest Path(Directed graphs)

Problem Description

The shortest path problem involves finding the shortest path between two vertices (or nodes) in a graph. Algorithms such as the Floyd-Warshall algorithm and different variations of Dijkstra's algorithm are used to find solutions to the shortest path problem.

Applications of the shortest path problem include those in road networks, logistics, communications, electronic design, power grid contingency analysis, and community detection.

Bounds Chart

Step Chart

Improvement Table

Complexity Classes	Algorithm Paper Links	Lower Bounds Paper Links
Exp/Factorial
Polynomial > 3
Cubic	Bellman–Ford algorithm (Ford 1956) (1956) Bellman–Ford algorithm (Dantzig 1960) (1960)
Quadratic	Bellman–Ford algorithm (Shimbel 1955; Bellman 1958; Moore 1959) (1959) Dijkstra's algorithm with list (Whiting & Hillier 1960) (1960) Dijkstra's algorithm with binary heap (Johnson 1977) (1977) Dijkstra's algorithm with Fibonacci heap (Fredman & Tarjan 1984; Fredman & Tarjan 1987) (1984) Thorup's algorithm (2004)
nlogn	Dijkstra's algorithm with Fibonacci heap (Johnson 1981; Karlsson & Poblete 1983) (1981)
Linear	[ Gabow's algorithm (1983)] [ Gabow Ahuja algorithm (1990)]
logn