Shortest path problem in the context of A* search algorithm

Edsger Wybe Dijkstra (/ˈdaɪkstrə/ DYKE-strə; Dutch: [ˈɛtsxər ˈʋibə ˈdɛikstraː] ; 11 May 1930 – 6 August 2002) was a Dutch computer scientist, programmer, mathematician, and science essayist.

Born in Rotterdam in the Netherlands, Dijkstra studied mathematics and physics and then theoretical physics at the University of Leiden. Adriaan van Wijngaarden offered him a job as the first computer programmer in the Netherlands at the Mathematical Centre in Amsterdam, where he worked from 1952 until 1962. He formulated and solved the shortest path problem in 1956, and in 1960 developed the first compiler for the programming language ALGOL 60 in conjunction with colleague Jaap A. Zonneveld. In 1962 he moved to Eindhoven, and later to Nuenen, where he became a professor in the Mathematics Department at the Technische Hogeschool Eindhoven. In the late 1960s he built the THE multiprogramming system, which influenced the designs of subsequent systems through its use of software-based paged virtual memory. Dijkstra joined Burroughs Corporation as its sole research fellow in August 1973. The Burroughs years saw him at his most prolific in output of research articles. He wrote nearly 500 documents in the "EWD" series, most of them technical reports, for private circulation within a select group.

In the mathematical field of graph theory, the distance between two vertices in a graph is the number of edges in a shortest path (also called a graph geodesic) connecting them. This is also known as the geodesic distance or shortest-path distance. Notice that there may be more than one shortest path between two vertices. If there is no path connecting the two vertices, i.e., if they belong to different connected components, then conventionally the distance is defined as infinite.

In the case of a directed graph the distance d(u,v) between two vertices u and v is defined as the length of a shortest directed path from u to v consisting of arcs, provided at least one such path exists. Notice that, in contrast with the case of undirected graphs, d(u,v) does not necessarily coincide with d(v,u)—so it is just a quasi-metric, and it might be the case that one is defined while the other is not.

The stretch factor (i.e., bilipschitz constant) of an embedding measures the factor by which the embedding distorts distances. Suppose that one metric space S is embedded into another metric space T by a metric map, a continuous one-to-one function f that preserves or reduces the distance between every pair of points. Then the embedding gives rise to two different notions of distance between pairs of points in S. Any pair of points (x,y) in S has both an intrinsic distance, the distance from x to y in S, and a smaller extrinsic distance, the distance from f(x) to f(y) in T. The stretch factor of the pair is the ratio between these two distances, d(f(x),f(y))/d(x,y). The stretch factor of the whole mapping is the supremum of the stretch factors of all pairs of points. The stretch factor has also been called the distortion or dilation of the mapping.

The stretch factor is important in the theory of geometric spanners, weighted graphs that approximate the Euclidean distances between a set of points in the Euclidean plane. In this case, the embedded metric S is a finite metric space, whose distances are shortest path lengths in a graph, and the metric T into which S is embedded is the Euclidean plane. When the graph and its embedding are fixed, but the graph edge weights can vary, the stretch factor is minimized when the weights are exactly the Euclidean distances between the edge endpoints. Research in this area has focused on finding sparse graphs for a given point set that have low stretch factor.

Turn-by-turn navigation is a feature of some satellite navigation devices where directions for a selected route are continually presented to the user in the form of spoken or visual instructions. The system keeps the user up-to-date about the best route to the destination, and is often updated according to changing factors such as traffic and road conditions. Turn-by-turn systems typically use an electronic voice to inform the user whether to turn left or right, the street name, and the distance to the next turn.

Mathematically, turn by turn navigation is based on the shortest path problem within graph theory, which examines how to identify the path that best meets some criteria (shortest, cheapest, fastest, etc.) between two points in a large network.

In computer science, a state space is a discrete space representing the set of all possible configurations of a system. It is a useful abstraction for reasoning about the behavior of a given system and is widely used in the fields of artificial intelligence and game theory.

For instance, the toy problem Vacuum World has a discrete finite state space in which there are a limited set of configurations that the vacuum and dirt can be in. A "counter" system, where states are the natural numbers starting at 1 and are incremented over time has an infinite discrete state space. The angular position of an undamped pendulum is a continuous (and therefore infinite) state space.

Pathfinding or pathing is the search, by a computer application, for the shortest route between two points. It is a more practical variant on solving mazes. This field of research is based heavily on Dijkstra's algorithm for finding the shortest path on a weighted graph.

Pathfinding is closely related to the shortest path problem, within graph theory, which examines how to identify the path that best meets some criteria (shortest, cheapest, fastest, etc) between two points in a large network.