path

Path is a concept used to describe a course or route that connects points, allowing movement from one location to another. The term appears in everyday speech to describe trails, routes, or lines of travel, and it is also used across various disciplines with specialized meanings.

In mathematics, a path is a continuous function from the unit interval [0,1] to a space X, with the endpoints often designated as the start and end of the path. Path-connectedness describes spaces where any two points can be joined by a path. A closed path begins and ends at the same point, and a simple path has no repeated vertices or points (besides the endpoints in the closed case).

In graph theory and computer science, a path is a sequence of vertices connected by edges. A simple path has no repeated vertices; a cycle starts and ends at the same vertex. The shortest path problem seeks the minimal-length path between two vertices and is solved with algorithms such as Dijkstra’s, Bellman-Ford, or A*.

In computing, a path refers to a route in a hierarchical file system or in a URL. A file path specifies how to locate a file, using absolute paths that start from the root and relative paths that depend on a current directory. Path notation varies by operating system (for example, slashes vs backslashes) and by context, such as environment variables like PATH.

Other uses include biological pathways that describe series of biochemical reactions or signaling events, and historical or metaphorical paths such as routes of study or development. In statistics, path analysis is a technique for modeling causal relationships using path diagrams.