Weglänge

Weglänge, literally meaning "path length" in German, is a standard concept in mathematics and related disciplines describing the length of a path. In graph theory, the Weglänge of a path is the sum of the weights on the edges it traverses. If all edges are unweighted, the Weglänge equals the number of edges, also known as the hop count.

For a continuous curve gamma: [a,b] → R^n, the Weglänge is defined as L = ∫_a^b ||gamma'(t)|| dt, provided gamma is differentiable. This integral gives the geometric length of the curve in Euclidean space.

Distances and shortest paths: the Weglänge between two vertices in a weighted graph induces a distance measure, with the shortest-path Weglänge being the minimum possible sum over all connecting paths. Computing minimal Weglängen uses algorithms such as Dijkstra’s or Bellman-Ford for weighted graphs, while breadth-first search yields minimal hop counts in unweighted graphs. The A* algorithm can accelerate shortest-path computations with heuristics.

Variants and properties: in directed graphs, path length depends on edge directions, and negative weights require careful handling. In many standard problems, weights are nonnegative to ensure well-defined shortest paths. In geometry, the Weglänge of a curve satisfies metric properties under appropriate conditions, defining a distance notion on the relevant space.

Applications include network routing, transportation planning, geographic information systems, and robotics, where Weglänge informs routing decisions, resource optimization, and motion planning.