shortestdistance

Shortest distance refers to the minimum possible length of a path connecting two points in a given space. In Euclidean space, the shortest distance between two points is the straight line segment joining them, and its length is given by the distance formula d = sqrt((x2−x1)^2 + (y2−y1)^2) in two dimensions, with generalizations to higher dimensions.

More generally, distance is defined by a metric on a set, a function that assigns a nonnegative real number to each pair of points and satisfies nonnegativity, identity of indiscernibles, symmetry, and the triangle inequality. On curved surfaces, the shortest path between two points is a geodesic; the distance along a geodesic is the geodesic distance, the locally shortest route on the surface.

In graph theory, the shortest distance between two nodes is the minimum sum of edge weights along a path, and finding it is the shortest-path problem. Algorithms include Dijkstra’s algorithm for nonnegative weights, Bellman–Ford for graphs with possible negative weights, A* for heuristic search, and Floyd–Warshall for all-pairs distances.

Common distance measures include Euclidean distance, Manhattan distance, and great-circle distance for points on a sphere; the latter is often computed with the Haversine formula. The choice of distance measure depends on the geometry of the space and the application.

Applications span navigation, transportation, robotics, geographic information systems, and computer networks. Shortest distance may be non-unique when symmetry or multiple geodesics exist.