Henzinger et . al . extended Frederickson's " r "-division technique for the single source shortest path algorithm in planar graphs for nonnegative edge-lengths and proposed a linear time algorithm.
12.
The main advantage of using this approach is that efficient shortest path algorithms introduced for the deterministic networks can be readily employed to identify the path with the minimum expected travel time in a stochastic network.
13.
Using the Floyd Warshall algorithm all pairs shortest path algorithm, we include intermediate nodes iteratively, and get \ Theta ( n ) time, using \ Theta ( n ^ 2 ) processors and \ Theta ( n ^ 3 ) work.
14.
For example, if vertices represent the states of a puzzle like a Rubik's Cube and each directed edge corresponds to a single move or turn, shortest path algorithms can be used to find a solution that uses the minimum possible number of moves.
15.
Marc van Kreveld suggested the algorithmic problem of computing shortest paths between vertices in a line arrangement, where the paths are restricted to follow the edges of the arrangement, more quickly than the quadratic time that it would take to apply a shortest path algorithm to the whole arrangement graph.
16.
These algorithms are based on two different principles, either performing a shortest path algorithm such as Dijkstra's algorithm on a visibility graph derived from the obstacles or ( in an approach called the " continuous Dijkstra " method ) propagating a wavefront from one of the points until it meets the other.
17.
If one represents a nondeterministic abstract machine as a graph where vertices describe states and edges describe possible transitions, shortest path algorithms can be used to find an optimal sequence of choices to reach a certain goal state, or to establish lower bounds on the time needed to reach a given state.
18.
In order to add each new edge in a way that minimizes the number of crossings it forms, one can use a shortest path algorithm in the dual graph of the current embedding, in order to find the shortest sequence of faces of the embedding and edges to be crossed that connects the endpoints of the new edge to each other.
