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 FloydWarshall 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)  
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) 

nlogn  Dijkstra's algorithm with Fibonacci heap (Johnson 1981; Karlsson & Poblete 1983) (1981)  
Linear  [ Gabow's algorithm (1983)]
[ Gabow Ahuja algorithm (1990)] 

logn 